# Impacts and Integration of Remote-First Working Environments

Christopher Atti1, Cliff Cross1, Ahmet Bugra Dogan1,Christopher Hubbard 1,
Cameron Page 1, Stephen Montague 1, Elnaz Rabieinejad2
1College of Computing and Software Engineering,
Kennesaw State University, Marietta, GA USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
2Cyber Science Lab, School of Computer Science, University of Guelph, Ontario,
Canada
<EMAIL_ADDRESS>

###### Abstract

Due to the Covid-19 pandemic in 2020 or other business decisions, remote work
is becoming increasingly popular. “Remote first” working environments exist
within companies where most employees work remotely. This paper takes a deep
dive into the remote-first mentality. It investigates its effects on employees
at varying stages in their careers, day-to-day productivity, and working
relationships with team members. We found that the remote-first mentality most
impacts seasoned employees and managers, potentially due to trouble adjusting
to a new way of working compared to the rest of their careers and the “always
on” mentality associated with working from home. Regarding productivity, we
found that while software development productivity appears unimpacted, the
effectiveness of communication and employee wellbeing saw declines which are
generally associated with lowered productivity. Finally, we looked closer at
the communication side of things and how remote work impacts relationship
building. We found that the most significant impacts on relationship building
centered around “trust” and “credibility” being harder to build due to a lack
of non-verbal cues during social interactions.

###### Index Terms:

Remote-first, Work from home, Virtual relationship building, Remote
communication, Covid-19 impacts, Challenges of remote work, Remote work
productivity, Virtual collaboration.

##  1 Introduction

In today’s world, many companies face the dilemma of allowing or becoming a
remote-first work environment. Companies have allowed people to work from home
for quite some time now. Most likely, it was one, maybe two, days a week.
Therefore, “work from home” is not exactly a new phenomenon. On the other
hand, one could argue that working in an office is the new ideology [1, 2].
According to WeWorkRemotely.com, “Before the Industrial Revolution, everyone
worked out of their homes [3]. Skilled blacksmiths, carpenters, leather
workers, and potters each set up shop at their residence and sold their goods
from there.” The Industrial Revolution was when big machines and the need to
mass-produce forced people to go into warehouse-type buildings where they
could work in such fashion. Of course, some still require large machinery and
mass production, but most desk jobs do not require employees to be in a
centralized building to get their work done productively. In fact, “’43
percent of employed Americans said they spent at least some time working
remotely’ according to one Gallup study” [3]. The article posted this quote
was written in 2017, long before COVID-19 forced every non-essential worker to
work remotely [4]. Then, companies had to adapt quickly to having almost all
their employees in remote positions, which has become a revolution. However,
as we head towards a light at the end of the tunnel regarding the pandemic
that forced millions of people to work remotely, companies require their
employees to return to the office at total capacity. A new term is being
coined in response to employees who are being forced back into the office. It
is known as “The Great Resignation.” According to Forbes, “Dubbed ‘The Great
Resignation,’ workplaces are seeing a trend in employees quitting their jobs.
A record four million workers called it quits in April alone, according to the
Labor Department. Pandemic burnout, a collective reassessment about priorities
and what matters most, a labor shortage and more companies calling workers
back into the office maybe some of the reasons behind this trend” [5]. A
direct result of people being asked to return to the office is causing “a
record four million workers” to quit their jobs in search of something that
allows for more flexibility [6]. Now, people want to work from home or have
the flexibility to choose a remote-first environment, but is that the best-
case scenario for all employees or companies? In this paper, firstly, we look
to determine the effects remote-first work has on employees at different
career stages. Then, we discuss an employee’s productivity level in the office
versus at home and the process and difficulty it takes in getting a remote-
first team to gel and work together well. Finally, we examine the benefit of
Artificial intelligence (AI) in the remote working environment.

##  2 CHALLENGES OF REMOTE WORK AT VARYING CAREER STAGES

###  2.1 Seasoned Versus New Employees

While working remotely, many employees face new challenges that you may not be
subject to in an office setting [7, 8]. In recent years, seasoned employees
have adjusted to becoming fully remote while newer employees most likely have
been hired on for a start to work full remote; both have their challenges.
Seasoned employees could have a more challenging time adjusting to working
from home [9]. Real-world distractions around them that they did not have
before, such as housework, pets, tv, family, and outside noise, could all play
a part in entirely focusing on the job. This is because this is a newer
problem that these workers face. Some remote employees find it helpful to have
a dedicated home office where they go to do work. This can help those feeling
like they never get away from work getaway. Once they leave their office, they
leave work there. Seasoned employees that have been used to going to the
office sometimes can have difficulty adjusting to working remotely. The
transition from the office environment where they have gotten used to
interactions daily with people can also be challenging [10]. While working
remotely, many of the day-to-day interactions will be conducted virtually.
Some remote employees also feel distanced from their team because they do not
see them in person. If available, these employees may enjoy a hybrid style
work environment where they can work in the office one or two days a week. On
the flip side, newly hired employees were most likely already engaged to be
remote or have a hybrid role at home and in the office. However, these new
employees may not understand the company rules and standards needed to do the
job. According to [10], One of the biggest challenges would most likely fall
under communication issues. While working remotely, one has to be very
connected to those they work with, especially if they have a team-oriented
role. Being dialed into what one’s team is doing is essential for completing
tasks correctly. Sometimes not having a direct face-to-face interaction about
a project can limit one’s understanding of what it takes to complete and not
ask the right questions to get the correct answers [11]. That could mean going
back and forth to fix mistakes along the way.

###  2.2 Seasoned Versus New Employees

Managers face different types of challenges when it comes to their employees.
One thing that must be done by managers that have remote teams is to make sure
they are communicating consistently with them about projects and what is
expected of them [12]. In addition, managers must keep their experienced
employees in the loop of expectations. If their employees are experienced,
there will be less concern about getting their job done than losing
productivity or having other distractions happen [13, 14]. Managers tend to
have an “always on” approach to work, especially while being remote. After
hours, remote workers may answer emails or calls more than if one left the
office and went home. That feeling of not disconnecting can be a challenge for
many managers because they feel they can get more completed or try to answer
that last email or finish up that previous request [15]. However,
disconnecting can be very important to unwind and keep that work and life
balance separate.

##  3 EMPLOYEE PRODUCTIVITY IN OFFICE AND REMOTE

###  3.1 Measures of Remote Work Productivity

Productivity can be measured by different metrics, both quantitative and
qualitative. Typical measurements are deployment frequency, review to merge
time, TLOC, etc. The degree to which a remote employee is productive is more
scrutinized than employees in the office and more reliant on complex data.
Observing an in-office employee toiling away can lend more credence to visible
productivity versus seeing an “online” status indicator. As noted by “Darja,
Anastasiia, Nils, Efi, Eriks, and Marte, cited from Smite, Russo, and Bezerra.
“The analysis of all GitHub projects (open source and corporate) concludes
that the developer activity in terms of the number of pushes, pull requests,
code reviews, and commented issues remained similar or slightly increased
compared to the pre-pandemic year. This finding is consonant with numerous
other studies that conclude that software companies have nothing to worry
about since working from home is per se not a significant challenge for
software engineers” [16]. They express how the value of activity metrics
cannot answer questions regarding factors such as how to measure periods of
inactivity between actions and engagement and concentration periods [17, 18].

###  3.2  Reported Productivity

Self-reported productivity among engineers who work from home differs based on
personal preference and everyone’s barriers and drivers. Darja, Anastasiia,
nils, Efi, Eriks, and Marte, analyze six corporate surveys conducted in four
Scandinavian companies to uncover “emotional issues” and factors stemming from
perceived isolation and communication style/preference are barriers for those
employees who report decreased productivity. Better work-life balance,
“flexibility and planning the work hours,” and the removal of commuting and
pop-up conversations were among the most prevalent reported benefits, and
drivers of working from home [16]. The above group explained how personal
biases could further skew the reliability of remote work survey data from a
top-down organizational level to individual reflections.

###  3.3  Productivity in an Agile Environment

Issues in standard work methodologies like Scrum become compounded when
workers are remote. Mak and Krutchen study the problems with productivity via
task coordination. ”Agile techniques such as features or story cards tend not
to adapt very well [19]”. The impact of Techniques that utilize physical
presence, such as ”Dot Voting” and ”Fist of Five,” are minimized. The names of
these activities, ex. ”StandUp” becomes more labels than actions, but the
scrum guide does not explicitly state that a team must be positioned in the
same place. Along with difficulties in team dynamics, there is the perception
that trust suffers under a distributed agile team. In establishing an
empirically developed framework for building trust in remote agile teams:
Tyagi, Sibal, and Suri report, ”Trust is one of the key factors that
influences team performance and acts as a foundation for effective teamwork”
in agile teams [20]. The authors conceive a framework that assists and
supports agile teams that can benefit from trust-building techniques and
exercises. Communication challenges can become the bottleneck of an effective
agile environment in remote development environments. Bundhun and Roopesh
create a communication framework to address productivity attributes such as”
differences in geographical locations and trust and cultural issues between
team members and poor coordination and communication,” as observed by a
Mauritian-based software company [21]. In leveraging agreed-upon communication
standards, these authors could provide a framework and survey results to
accomplish adaptable guidelines.

###  3.4  Employee Wellbeing Related to Productivity

It is generally accepted that higher well-being is related to a more
performant employee. However, general well-being can be attributed to many
variables, including timeline pressures, work-from-home conflicts, and work-
family conflicts. This is demonstrated by Darouei and Pluut, who surveyed 34
professional workers to test their five hypotheses.
”Hypothesis 1. Working from home (compared with the office) will be negatively
associated with work-family conflict.
Hypothesis 2. Within individuals, time pressure mediates the negative
relationship between working from home and work-family conflict experienced at
home.
Hypothesis 3. Within individuals, work-family conflict experienced at home in
the evening is positively related to emotional exhaustion the next morning.
Hypothesis 4. Within individuals, work-family conflict experienced at home in
the evening is negatively related to work engagement the next morning.
Hypothesis 5a. Within individuals, work-family conflict experienced at home in
the evening is positively related to negatively affecting the organization the
next morning.
Hypothesis 5b. Within individuals, work-family conflict experienced at home in
the evening is negatively related to positive affect towards the organization
the next morning”. The research survey results generally supported the
hypotheses presented, with either a negative or positive outcome affecting
performance directly [22].

##  4  DIFFICULTIES IN TEAMWORK

Semi-related to productivity is a team’s ability to “gel” and foster healthy
personal and working relationships [23]. Of course, many factors go into team
dynamics and working relationships. However, for the sake of analysis and
keeping separate from the “productivity” discussion, we can narrow this down
to the ability to get to know your team and perform collaborative software
development within a given toolset.

###  4.1  Difficulties in Getting to Know Your Team

As Wojahn, Taylor, and Blicharz identified, remote-first employees, interact
with colleagues from all over the country and the globe. They do this using a
combination of voice, text, and video communication, with the primary means of
communication being text or voice only. This leads to complications when
attempting to gauge others’ reactions to us due to the lack of visual clues
such as body language, facial expressions, and other non-verbal clues. Even
more difficult than maintaining existing relationships and communication with
colleagues when a new team comes together remotely and meets each other for
the first time [24]). The complications with non-visual communication are
exacerbated when team members have no prior history of working experience
together. In their paper, Moster, Ford, and Rodeghero identify that trust-
building is one of the most challenging parts of remote work. Communication is
vital for building trust, and “communication for virtual teams is often less
frequent than in-person teams.” To combat complications due to inconsistent
communication and lack of initial faith, ad-hoc personal conversations and
video conferencing early in team formation can lessen the impact of virtual
teamwork [25]. In a remote-first environment, the primary means of
communication will be digital, whether email, instant message, voice/video
chat, or some other form of virtual communication that is not in-person [26].
According to Yang, the previously mentioned communication mediums fall under
the “computer mediated communication” category and are supported by many tools
to help facilitate relationship building [27]. Trust is difficult to build
virtually, but “moreover, when all social interactions are mediated by
technology, people’s impressions and perception of each other can be affected
by the mediated social cues transmitted by different media. For example, the
profile images people use, the description of one’s self-introduction, and
visible online activities can influence how people perceive one another and
their willingness to form further relationships to various extents.” He
conducted multiple experiments involving virtual interactions and found that
the lack of non-verbal cues can impact relationships with team members.
However, poorly chosen ambient cues also add a factor to non-verbal cues.
Virtual backgrounds or avatars deemed “unprofessional” were found to impact
perceptions of interactions and employee credibility negatively [28].

###  4.2  Difficulties in Collaborative Software Development

The most critical factor in remote-first work is communication, but
communication alone will not solve all collaboration issues. Gupta and
Fernandez group collaboration in software engineering into four categories:
model-based collaboration to reduce ambiguity and increase error detection,
process-centric tools to help define and facilitate the software process,
awareness tools to enable team members to be easier informed of changes to
help avoid conflicts and infrastructure tools which assist in improving
interoperability by streamlining data and control integration. Notice the
heavy emphasis on tools. Due to the sheer volume of available tools, there is
no shortage of options. However, Gupta and Fernandez also found through
experience that “the choices of how to collaborate is usually an ad hoc
decision based on current practices, the available collaboration mechanisms
and sometimes, client-related constraints. Their investigations reveal few
examples in which software teams have planned a collaboration strategy based
on conscious analysis of collaboration needs, constraints in the project,
tools availability, etc.” This lack of emphasis on tool analysis before
beginning projects can heavily impact productivity and lead to frustration and
a decline in morale [29]. The open-source software development space is one
place to look for difficulties in software collaboration and how to combat
them. Braunschweig and Seaman took an in-depth look at open-source software.
They determined that developers must have a shared understanding of the goals
and plans of the project to avoid complications and misunderstandings. To
evaluate increasing shared knowledge, they attempted to quantify four
attributes of communication: Synchrony, Proximity, Proportionality, and
Maturity. They found that highly mature contact was typically associated with
“a high degree of structure and standardization. Standards, tools, explicit
rules, and mentoring of new developers help ensure that work is performed
following consistent processes, without much ongoing discussion of those
processes.” This finding also supported their other results, indicating that
having standard processes and tools helps all aspects of communication and
remote software development [30]. Reaching this maturity is the tricky part
and is impacted by the factors previously discussed, such as team trust, team
dynamic. The ability to select a proper toolset and standards for the team
[31].

##  5 The role of AI in shifting companies to remote working

As mentioned earlier, in the new era, they are moving towards working
remotely, while this method has many challenges that must be taken into
account. With the advancement of information and communication technology, new
technologies such as AI can play a complementary role in remote working and
reduce existing challenges [32]. AI leads processes in a company to be more
intelligent and more automatic [33] and, in this way, can assist remote
working by improving efficiency [34]. In the following, we will review some
applications of AI in remote first companies. In a remote-first company,
meetings are an essential part of work. These meetings can be between
employees of a team with customers, employees of a team with each other, and
also with other teams of the company. During these meetings, critical
information is exchanged. So, meetings should be high-quality in terms of
sound, resolution, etc., so employees can provide and receive important
information. The settings of meeting software may be very frustrating for IT
leaders. If something goes wrong in the setting, the meeting should be
disrupted to set them again, which is time-consuming. Recently, AI-driven
software can assist IT leaders in their job by setting light, resolution,
sound, giving access to the right member, etc. [35]. In addition, during the
meeting, AI can collect and transfer information automatically [36]. Another
critical challenge in remote working is communicating across teams at
different levels and following up with them [37]. In an AI-driven application,
employees can enter their requests into an AI bot, and the AI bot can
communicate with the right person and ask for their task status [38]. This can
be time-consuming and confusing without AI. Moreover, in remote work,
employees usually struggle with many technical problems, and because they
don’t access a rapid IT expert, they have to solve these problems by
themselves. In this field, AI can play a useful role. Companies usually use
AI-driven agents for online troubleshooting. In this way, when remote
employees have a technical problem, contact an AI bot and enter their problem.
The AI bot provides suggestions, and a step-by-step solution to the problem
[39]. These examples show that AI offers different remote working services and
can help companies to bridge the gap of remote working.

##  6  CONCLUSION

The remote-first working environment is still an open discussion. Since we
have the human factor involved in the discussion, it is almost impossible to
say whether remote-first is the best idea. As we mentioned in section III,
every individual has their preferences, which is one of the biggest reasons
this discussion is open. The remote-first work environment has positive and
negative sides, no matter how experienced or seasoned employees are. In terms
of productivity, on the other hand, based on the number of pushes, pull
requests, code reviews, and commented issues on remote repositories made by
software engineers, they kept their pace in a remote environment compared to
the office, which proves us well organized remote teams can still be as
productive as they were on-site. [8] While the article explains productivity
through GitHub pushes, pull requests, and other measurements, there are many
other factors to look at when it comes to productivity. Based on the findings
in this paper, overall productivity has declined due to the struggles
associated with working with a remote team. One of the biggest challenges in
remote-first work environments is harmony within the remote team. The
influential factors which make it challenging are team trust and dynamics. As
natural behavior of us human beings, we tend not to trust people we just met
at the beginning. Based on the nature of humans, this process becomes even
more challenging when people/teams have their first meeting via online
platforms where they cannot see each other’s facial impressions or read their
body language as clearly as they do in physical meetings. Choosing the proper
toolset to improve communication and collaboration within the team companies
can ease this challenge.

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# Testing the Hubble law with Pantheon+

Deng Wang<EMAIL_ADDRESS>National Astronomical Observatories, Chinese
Academy of Sciences, Beijing, 100012, China

###### Abstract

The Hubble law (HL) governs the low-redshift (low-z) evolution of the distance
of an object. However, there is a lack of an investigation of its validity and
effective radius for a long time, since the low-z background data with a high
precision is scarce. The latest Type Ia supernovae sample Pantheon+ having a
significant increase of low-z data provides an excellent opportunity to test
the HL. We propose a generalized HL and implement the first modern test of the
HL with Pantheon+. We obtain the constraint on the deviation parameter
$\alpha=1.00118\pm 0.00044$, confirm the validity of linear HL with a $0.04\%$
precision and give the transition redshift $z_{t}=0.03$ and luminosity
distance $D_{L,t}=123.13\pm 1.75$ Mpc, which means that HL holds when $z<0.03$
and breaks down at a distance of $D_{L}>123.13$ Mpc. Comparing the ability of
Type Ia supernovae and HII galaxies in testing the HL, we stress the
uniqueness and strong power of Type Ia supernovae in probing the low-z
physics.

Introduction. In 1922, Friedman first derived the so-called Friedmann
equations characterizing the expanding universe in the framework of general
relativity Friedman:1922kd . After five years, Lemaître Lemaitre:1927
independently derived the solution of cosmic expansion from gravitational
field equations, observed the proportionality between the distance to a
celestial body and the recessional velocity of this body, and estimated the
proportionality coefficient to be $625$ km s-1 Mpc-1. Two years later, Hubble
Hubble:1929ig  confirmed the existence of cosmic expansion using observations,
gave a more accurate value of $500$ km s-1 Mpc-1 to this coefficient, and
derived the following well-known HL

$v=H_{0}D_{L},$ (1)

where $v$ is the recessional velocity of an object, $H_{0}$ is the Hubble
constant and $D_{L}$ is the luminosity distance of the same object. It is
worth noting that Hubble obtained this relationship by deriving the recession
velocities of objects from their redshifts, many of which were earlier
measured and related to velocities by Slipher in 1917 Slipher:1917 . After
making a simple Taylor series expansion, the HL can be expressed as

Figure 1: The schematic plot for the effective range of the HL. Below the transition redshift $z_{t}$, the HL $cz=H_{0}D_{L}$ holds but it breaks down beyond $z_{t}$. Figure 2: Upper panel. The redshift ($z$)-apparent magnitude ($m_{b}$) diagram of Pantheon+ (blue) and Pantheon (red). It is clear to see that Pantheon+ has more low-z SNe Ia and Pantheon has no data point in the range $z<0.01$. Lower panel. The evolution of the extent of the deviation from the standard HL prediction $\alpha=1$ over $z$. The dashed lines denote the redshift $z=0.01$ (magenta) and $z=0.03$ (cyan), respectively.  Table 1: The $1\,\sigma$ (68%) constraining results of the generalized HL in different low-z redshift ranges. The number of SNe Ia and the extent of the deviation from $\alpha=1$ in each range is shown in the second and fourth columns, respectively. Data range |  Number |  $\alpha$ |  Tension
---|---|---|---
$z<0.01$ |  111 |  $1.00330\pm 0.00110$ |  $3.00\,\sigma$
$z<0.02$ |  265 |  $1.00251\pm 0.00068$ |  $3.69\,\sigma$
$z<0.03$ |  468 |  $1.00118\pm 0.00044$ |  $2.68\,\sigma$
$z<0.04$ |  590 |  $1.00190\pm 0.00036$ |  $5.28\,\sigma$
$z<0.05$ |  645 |  $1.00186\pm 0.00034$ |  $5.47\,\sigma$
$z<0.06$ |  678 |  $1.00202\pm 0.00032$ |  $6.31\,\sigma$
$z<0.07$ |  697 |  $1.00207\pm 0.00032$ |  $6.47\,\sigma$
$z<0.08$ |  727 |  $1.00223\pm 0.00031$ |  $7.19\,\sigma$
$z<0.09$ |  736 |  $1.00230\pm 0.00030$ |  $7.67\,\sigma$
$z<0.1$ |  741 |  $1.00235\pm 0.00030$ |  $7.83\,\sigma$
$z<0.2$ |  948 |  $1.00468\pm 0.00024$ |  $19.50\,\sigma$
$z<0.3$ |  1207 |  $1.00688\pm 0.00020$ |  $34.40\,\sigma$

$cz=H_{0}D_{L},$ (2)

where $c$ and $z$ are the speed of light and the redshift of an observed
object, respectively. This equation is a low-z approximate version of Eq.(1).
At low redshifts, one can easily use it to derive the accurate redshift of an
object when knowing its luminosity distance, and vice versa. However, in
practice, with more and more high precision low-z observations, there are two
problems occurs: (i) whether is actually HL valid for low-z observations at
all? (ii) if HL is valid for low-z data, what is its effective radius? Koranyi
and Strauss Koranyi:1997  investigated correctly the former problem by using
the IRAS 1.2 Jy Redshift Survey in 1997, and confirmed the validity of HL.
However, their data quality is very limited and they assume a large cosmic
expansion rate $H_{0}=100$ km s-1 Mpc-1, which affects the correctness of
their conclusions, to implement their analysis. Although HL as one of the
milestones of modern cosmology tells us the fact that the universe is during
an expanding phase, the validity of HL must be precisely tested.
Unfortunately, so far, there is still a lack of an accurate test of HL with
modern astronomical observations. Since we do not have enough data points
covering the whole low-z range in the past several decades, the latter problem
can not be solved well. Due to the rapid process of modern Type Ia supernovae
(SNe Ia) observations, we can not only implement an accurate test of HL, but
also determine well the luminosity distance $D_{L}(z_{t})$ at the transition
redshift $z_{t}$, beyond which $cz=H_{0}D_{L}$ will break down (see Fig.1 for
a better illustration). Particularly, the recent release of the largest SNe Ia
sample called Pantheon+ Scolnic:2021amr ; Brout:2022vxf , which shows a
significant increase of low-z data relative to the original Pantheon sample
Pan-STARRS1:2017jku , lets us have enough high-precision data points in the
redshift range $z<0.1$ to test the HL. As a consequence, we implement the fist
modern test of HL using the Pantheon+ data in this study. We confirm the
validity of linear HL with a $0.04\%$ precision find that the HL holds when
$z<0.03$ and breaks down at a distance of $D_{L}>123.13$ Mpc.

Figure 3: The 1-dimensional posterior distributions of the deviation parameter
$\alpha$ in different low-z ranges of SNe Ia. The vertical bands are the
$1\,\sigma$ (dark) and $2\,\sigma$ (light) confidence ranges of $\alpha$. The
dashed lines denote $\alpha=1$ (red) and the best fits (blue), respectively.
Figure 4: The redshift ($z$)-distance modulus ($\mu$) diagram of Pantheon+
(blue) and HII galaxy measurements (pink). The dashed line denotes the
redshift $z=0.01$. Figure 5: The 1-dimensional posterior distribution of the
deviation parameter $\alpha$ for HII galaxies. The vertical bands are the
$1\,\sigma$ (dark) and $2\,\sigma$ (light) confidence ranges of $\alpha$. The
dashed blue line denote the best fit.

Data and method. Recently, Refs.Scolnic:2021amr ; Brout:2022vxf  release the
latest and largest SNe Ia sample called Pantheon+, which consists of 1701
light curves of 1550 spectroscopically confirmed SNe Ia coming from 18
different sky surveys. Pantheon+ has a significant increase relative to the
original Pantheon Pan-STARRS1:2017jku  at low redshifts and covers the
redshift range $z\in[0.00122,\,2.26137]$. To describe the difference of two
datasets better, we show the redshift-apparent magnitude diagram in Fig.2.
Especially, Pantheon+ has 111 SNe Ia when $z<0.01$ but Pantheon does not. If
considering a larger redshift range $z<0.03$, Pantheon+ has 468 points which
are more than 95 SNe Ia from Pantheon by a factor of $\sim$ 5\. This means
that Pantheon+ can give an unprecedented constraint on the HL, and that one
can test the HL at extremely low redshifts via Pantheon+ at present.

To test the HL with low-z Pantheon+ data, we propose a generalized form of HL
as

$(cz)^{\alpha}=H_{0}D_{L},$ (3)

where $\alpha$ characterizes the deviation from HL and it reduces to the
standard HL when $\alpha=1$. $D_{L}$ is expressed as

$D_{L}=10^{\frac{\mu-25}{5}},$ (4)

where $\mu$ is the distance modulus of a SNe Ia where the absolute magnitude
has been determined by the SH0ES Cepheid host distances Scolnic:2021amr ;
Brout:2022vxf . To implement the Bayesian analysis, we express the $\chi^{2}$
at low redshifts as

$\chi^{2}=\sum\limits_{i}\left[\frac{(cz_{i})^{\alpha}-(H_{0}D_{L})_{obs}}{\sigma_{H_{0}D_{L}}}\right]^{2},$
(5)

where $z_{i}$ is the redshift of $i$-th SNe Ia, and $(H_{0}D_{L})_{obs}$ is
our observable and it can be derived by combining the newest local measurement
Riess:2021jrx  $H_{0}=73.04\pm 1.04$ km s-1 Mpc-1 with the distance modulus
observations from Pantheon+ Scolnic:2021amr ; Brout:2022vxf , and its error
$\sigma_{H_{0}D_{L}}=\sqrt{H_{0}^{2}\sigma_{D_{L}}^{2}+D_{L}^{2}\sigma_{H_{0}}^{2}}$,
where $\sigma_{H_{0}}$ and $\sigma_{D_{L}}$ denote the $H_{0}$ error and the
uncertainty of luminosity distance of a SNe Ia at $z$, respectively.

Constraints from SNe Ia. To explore the impacts of different low-z ranges on
constraints on the generalized HL in Pantheon+, we start with $z<0.01$ and
increase gradually the upper limit of the range by 0.01 each time. The
constraining results of $\alpha$ are shown in Fig.3 and Tab.1. Using the
unique data lying in $z<0.01$ in Pantheon+, we obtain $\alpha=1.00330\pm
0.00110$, which surprisingly gives a $0.1\%$ determination on $\alpha$. Its
best fitting value 1.00330 just has a $0.33\%$ difference relative to the
standard HL prediction $\alpha=1$. Basically, this result means that the HL
holds in $z<0.01$. Interestingly, we observe a $3\,\sigma$ inconsistency with
$\alpha=1$ at such a high precision, which may indicate that the original HL
needs a very small modification. When $z<0.02$, we have $\alpha=1.00251\pm
0.00068$, which is now a $0.07\%$ determination on $\alpha$ and the best fit
of which has a $0.25\%$ difference relative to $\alpha=1$. It shows a
$3.69\,\sigma$ inconsistency to $\alpha=1$ that is a little higher than the
case of $z<0.01$ does. It is interesting that the constraint
$\alpha=1.00118\pm 0.00044$ for $z<0.03$ including 468 SNe Ia are just in a
$2.68\,\sigma$ tension with the standard HL, while it determines $\alpha$ with
a $0.04\%$ precision and its best fit has only $0.12\%$ deviation from
$\alpha=1$. Interestingly, when considering $z<0.04$, the constraint
$\alpha=1.00190\pm 0.00036$ shows a beyond $5\,\sigma$ inconsistency to the
original HL and its best fit has a larger deviation ($0.2\%$) from $\alpha=1$
than the case of $z<0.03$. Note that this is a transition point in our
analysis. Staring from $z<0.04$, for the cases of
$z<0.05,\,0.06,\,0.07,\,0.08,\,0.09,\,0.1$, this discrepancy is more and more
significant, the best fits prefer a positive deviation from $\alpha=1$, and
the errors of $\alpha$ decrease continuously (see Tab.1).

Until $z<0.1$, we find that the HL seems to describe data well in the Hubble
diagram. However, when $z>0.1$, the HL exhibits a significant deviation from
Pantheon+ data. To show this feature, we also constrain $\alpha$ for $z<0.2$
and $z<0.3$, and find that these two cases give $19.5\,\sigma$ and
$34.4\,\sigma$ tensions. With the increase of SNe Ia number in a considered
high-redshift range, the best fit of $\alpha$ will be larger, its error
becomes compressed by increasing sample size, and consequently, the tension
will be larger. Here we define the tension as
$T\equiv\left|\alpha-1\right|/\sigma_{\alpha}$, where $\sigma_{\alpha}$ denote
the error of $\alpha$ derived from observations. This statistical quantity
describes the extent of the deviation from the standard HL for a given
dataset. From the lower panel of Fig.3 and Tab.1, it is easy to observe that
$z=0.03$ can serve as a safe breaking point of HL, which is mainly ascribed to
the small $T$ value (2.68) and small difference ($0.12\%$) when $z<0.03$.
Hence, we choose conservatively the transition redshift $z_{t}=0.03$ for
Pantheon+ and the accompanied transition luminosity distance
$D_{L,t}=123.13\pm 1.75$ Mpc. This gives the following relation

$\begin{cases}cz=H_{0}D_{L},&\quad\quad\quad z\in(0,\,\,0.03]\\\ cz\neq
H_{0}D_{L},&\quad\quad\quad z\in[0.03,\,\,+\infty]\end{cases}$ (6)

which indicates that HL breaks down when $z>0.03$ and we should consider the
cosmology-dependent recession velocity.

Constraints from HII galaxies. In light of the above success of Pantheon+ in
testing the HL, an important task is to compare the ability of alternative
modern background probes with that of SNe Ia in constraining $\alpha$. This
alternative probe should at least include two features: (i) it can give
competitive constraints on $\Lambda$CDM relative to SNe Ia; (ii) its sample
should cover $z<0.03$ with enough data points. To the best of our knowledge,
HII galaxies (hereafter HII) measuring the distance via the
$L(H\beta)$-$\sigma$ relation (see Gonzalez-Moran:2021drc  for details) just
satisfy both requirements. Therefore, we take HII to implement a cross check
and its Hubble diagram is shown in Fig.4 for $z<0.03$. We observe that HII
data not only has large errors, but also distributes in an incompact way
unlike SNe Ia. This implies that HII can not give a good enough test of HL.
Confronting Eq.(3) with HII, we obtain $\alpha=0.9283\pm 0.0051$ (see Fig.5),
which gives a $0.55\%$ determination on $\alpha$, and shows, respectively,
$14.06\,\sigma$ and $14.24\,\sigma$ tensions to $\alpha=1$ and Pantheon+
constraint for $z<0.03$.

Discussions and Conclusions. Using the latest and largest Pantheon+ SNe Ia
sample, we implement a modern test of the HL and confirm the linear HL with a
precision of less than $0.1\%$. We find that the transition redshift
$z_{t}=0.03$ and distance $D_{L,t}=123.13\pm 1.75$ Mpc, which implies that HL
holds when $z<0.03$ and breaks down at a distance of $D_{L}>123.13$ Mpc.

To provide another perspective, the HL can be expressed as $D_{L}=cz/H_{0}$,
which is the 1st-order Taylor expansion of $D_{L}$ on $z$. If calculating
$D_{L}$ at high redshifts, one must consider the cosmology dependence. The
cosmological implication of $z_{t}=0.03$ is that any background probe can not
distinguish different cosmological models when $z<0.03$. Although the
constraint from HII has a precision of less than $1\%$, it can not give an
persuasive test since its incompact distribution in the late-time Hubble
diagram. Consequently, we should stress the strong power of SNe Ia in
exploring the low-z physics. However, except unknown systematic uncertainties,
we do not rule out the possibility that the $\alpha$ tension between SNe Ia
and HII predicts the existence of new physics at low redshifts. Our results
are closely related to the validity of relativistic Doppler shift formula,
which indicates that our constraint is degenerate with the validity of special
relativity and classical Doppler shift formula at a cosmological distance.

Acknowledgements. DW thanks Adam Riess for helpful discussions. DW is
supported by the National Science Foundation of China under Grants No.11988101
and No.11851301.

## References

  * (1) A. Friedman, “On the Curvature of space,” Z. Phys. 10, 377-386 (1922).
  * (2) G. Lemaître, “Expansion of the universe, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae,” Mon. Not. Roy. Astron. Soc. 91, 483 (2011).
  * (3) E. Hubble, “A relation between distance and radial velocity among extra-galactic nebulae,” Proc. Nat. Acad. Sci. 15, 168-173 (1929).
  * (4) V. M. Slipher, “Radial velocity observations of spiral nebulae,” The Observatory 40, 304 (1917).
  * (5) D. M. Koranyi and M. A. Strauss, “Testing the Hubble Law with the IRAS 1.2 Jy Redshift Survey,” Astrophys. J. 933, 34 (1997).
  * (6) D. M. Scolnic et al., “The Pantheon+ Analysis: The Full Dataset and Light-Curve Release,” [arXiv:2112.03863 [astro-ph.CO]].
  * (7) D. Brout et al., “The Pantheon+ Analysis: Cosmological Constraints,” [arXiv:2202.04077 [astro-ph.CO]].
  * (8) D. M. Scolnic et al. [Pan-STARRS1 Collaboration], “The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample,” Astrophys. J. 859, 101 (2018).
  * (9) A. G. Riess et al., “A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s-1 Mpc-1 Uncertainty from the Hubble Space Telescope and the SH0ES Team,” Astrophys. J. Lett. 934, L7 (2022).
  * (10) A. L. González-Morán et al., “Independent cosmological constraints from high-z H ii galaxies: new results from VLT-KMOS data,” Mon. Not. Roy. Astron. Soc. 505, 1441-1457 (2021).


# An RNA Sequencing Analysis of Glaucoma Genesis in Mice

Jai Sharma  Student Intern Researcher, Robinson Lab, Stanford University;
269 Campus Drive, Palo Alto, CA 94305; Email<EMAIL_ADDRESS>Vidhyacharan Bhaskar  Professor, School of Engineering, San Francisco State
University;
1600 Holloway Avenue, San Francisco, CA 94132  Corresponding Author’s Email:
<EMAIL_ADDRESS>
###### Abstract

Glaucoma is the leading cause of irreversible blindness in people over the age
of 60, accounting for 6.6 to 8% of all blindness in 2010, but there is still
much to be learned about the genetic origins of the eye disease. With the
modern development of Next-Generation Sequencing (NGS) technologies,
scientists are starting to learn more about the genetic origins of Glaucoma.
This research uses differential expression (DE) and gene ontology (GO)
analyses to study the genetic differences between mice with severe Glaucoma
and multiple control groups. Optical nerve head (ONH) and retina data samples
of genome-wide RNA expression from NCBI (NIH) are used for pairwise comparison
experimentation. In addition, principal component analysis (PCA) and
dispersion visualization methods are employed to perform quality control tests
of the sequenced data. Genes with skewed gene counts are also identified, as
they may be marker genes for a particular severity of Glaucoma. The gene
ontologies found in this experiment support existing knowledge of Glaucoma
genesis, providing confidence that the results were valid. Future researchers
can thoroughly study the gene lists generated by the DE and GO analyses to
find potential activator or protector genes for Glaucoma in mice to develop
drug treatments or gene therapies to slow or stop the progression of the
disease. The overall goal is that in the future, such treatments can be made
for humans as well to improve the quality of life for human patients with
Glaucoma and reduce Glaucoma blindness rates.

###### Index Terms:

mRNA Sequence Data; Statistical Analysis; Differential Expression Analysis;
Gene Ontology Analysis; Glaucoma; Ophthalmology

##  I Introduction

###  I-A Background

The neurodegenerative eye disease Glaucoma is a part of a group of eye
diseases which can cause blindness. Glaucoma in particlar is one of the
leading causes of blindness in people over the age of 60 [1]. The most common
type of Glaucoma, Primary Open Angle Glaucoma (POAG)111This paper will be
discussing only Primary Open Angle Glaucoma, so the terms “Glaucoma” and
“Primary Open Angle Glaucoma” will be used interchangeably unless otherwise
specified., is caused when resistance in the Trabecular Meshwork leads to
increased Intraocular Pressure (IOP) in the eye. As a result, aqueous humor,
which supplies nutrients to the cornea, is unable to flow from the Posterior
Chamber to the Anterior Chamber, through the Trabecular Meshwork, and back
into the body. In POAG, a gradual increase in IOP leads to damage to the optic
nerve and the chronic and irreversible onset of Glaucoma. As POAG progresses,
the IOP increase leads to cupping of the optic disc; while the ratio between
the diameter of the optic cup to the diameter of the optic disc is 0.5 in a
normal eye, this ratio increases during cupping in POAG [1]. Left untreated,
Glaucoma takes on average 25 years to progress from normal vision to complete
blindness. Even with treatment, 9% to 19% of patients reach complete
blindness, implying a need for treatment improvements [2].

Glaucoma in general is characterized by the cupping and atrophy of the optic
nerve head, visual field loss of the patient, and often the increased IOP [3].
Thapa, et al. studied a subset of the Nepalese population and found that the
occurrence rate of Glaucoma was 1.9%. Approximately 68% of these cases were
Primary Open Angle Glaucoma, 22.67% of these cases were Primary Angle Closure
Glaucoma, and 9.33% of these cases were secondary Glaucoma [4]. Casson, et al.
highlighted that while Glaucoma used to be characterized by high IOP, many
cases of Glaucoma without high IOP and high IOP without Glaucoma disputed this
assumption. Now, high IOP is simply considered a risk factor rather than a
defining feature, and reducing IOP is currently the only treatment strategy
for Glaucoma [5].

###  I-B Risk Factors and Symptoms

It is known that old age, family history, Black ethnic origin, and Myopia
(nearsightedness) are all risk factors for Glaucoma. Once a patient is
diagnosed with Glaucoma, routine screening by an optometrist is used to ensure
the disease’s slow progression. As POAG progresses, the patient experiences a
loss of peripheral vision (tunnel vision), fluctuating pain, headaches, and
blurred vision. Symptoms are usually worse at nighttime or in dark spaces [1].

###  I-C Current Diagnosis and Treatment Methods

While 111.8 million people are expected to have Glaucoma by 2040, there is
much to be learned about the genetic origins and development of this disease.
Glaucoma (all types collectively) have a 10% diagnosis rate worldwide and less
than 50% diagnosis rate in developed nations [2].

Currently, there are several ways to diagnose Glaucoma. First, Non-Contact
Tonometry can be used to shoot puffs of air and measure puff response from the
cornea. A small reaction to the puff of air would be associated with high IOP,
and an according Glaucoma diagnosis can be made. This method is relatively
less accurate, but it is a good estimation for eye pressure. The industrial
standard for Glaucoma diagnosis is the Goldmann Applanation Tonometry method,
which utilizes advanced equipment to make contact with the cornea and apply
different amounts of pressure to find the IOP, from which a Glaucoma diagnosis
can be made. Other forms of diagnosis include Fundoscopy, which can help
examine the eye for cupping, and the Visual Field Assessment Test, which tests
for peripheral vision loss [1]. Glaucoma is asymptomatic until it reaches an
advanced stage, so it is recommended that patients undergo regular screening
examinations from age 40 onward. The rate of false positives for Glaucoma
diagnosis is high, so it is also recommended that any positive finding is
followed up with further testing [6].

The IOP for a normal eye ranges from 10 mmHg to 21 mmHg, and treatment is
often started when IOP reaches 24 mmHg. Current treatments include
Prostaglandin Analogues (e.g. Latanoprost) which increase uveoscleral outflow,
Topical Beta-Blockers (e.g. Timolol) and Carbonic Anhydrase Inhibitors (e.g.
Dorzolamide) which reduce the production of aqueous humor, and
Sympathomimetics (e.g. Brimonidine) which increase uveoscleral outflow and
reduce the production of aqueous humor. Surgical methods of treatment include
Trabeculectomy which creates a new channel for aqueous humor to drain from the
eye; however, this procedure is risky and its effects are only temporary [1].
Conlon, et al. analyzed trends in Glaucoma treatments, concluding that there
has been a recent increase in Glaucoma medications and the use of laser
trabeculoplasty over the past decade. Additionally, there has been a noticable
decrease in the frequency of invasive incisional surgery. New methods such as
microinvasive Glaucoma surgery have also been introduced [7]. Despite advances
in laser and incisional surgery, Schwartz, et al. claim that medical therapy
is still the primary method of treatment and that there is good evidence that
laser trabeculoplasty is just as good as initial medical therapy [8].

###  I-D Research Goal

Crucial to the development of new cures for POAG is the more sophisticated
understanding of the disease at the genetic level. The goal of this research
is to find the genes responsible for the genesis of Glaucoma in mice so that
this information may be used to develop new treatments in the future which
target these genes or corresponding gene pathways in mice. The overall goal is
that future research can develop new and more effective POAG treatments for
humans.

This research is focused on generating candidate gene lists which may contain
potential activator or protector genes for Glaucoma in mice. However, it
should be noted that doing intensive biological studies to study these gene
lists manually is outside the scope of this research.

Note that mice were used here because mice have eyes which are similar to
human eyes in that they develop hereditary Glaucoma and high IOP, as will be
explained in the next section. This research and similar research on Glaucoma
have taken advantage of this unique property of mice.

##  II Literature Review

Research has been conducted to explore the function of ncRNA, transcriptional
mRNA, and similar biological molecules in Glaucoma, Cataracts, and different
types of cancers. This research has shown to successfully identify targets for
further therapeutic research to develop treatments for the corresponding
disease.

Chen, et al. explored time-series circRNA, lncRNA, miRNA, and mRNA expression
profiles of developing mice retina samples. The aim of this research was to
find the key functional ncRNA and ceRNA which regulated retinal neurogenesis.
The research found that several ncRNAs in the circRNA/lncRNA-miRNA-mRNA
network, including circCDYL, circATXN1, circDYM, circPRGRIP, lncRNA Meg3, and
lncRNA Vax2os, involved neurotransmitter transport and multicellular organism
growth during retinal development [13]. Zhou, et al. aimed to identify
differing biological mechanisms involved in each Glaucoma subtype. Results
from the research indicated that human participants with high tension Glaucoma
had an enrichment of genes associated with unfolded protein response. Further,
the research identified the differential expression of genes in certain IOP
regulating tissues [2]. Howell, et al. used house mice mRNA data set to
investigate the genetic origins of Glaucoma using temporally ordered sequences
of Glaucoma states. The overall goal was to identify molecular events which
can be targeted to provide effective new treatments for human glaucoma. A
major finding from this study was that mice with a mutation in the C1QA gene
were protected from Glaucoma. Additionally, it was found that inhibition of
endothelin system with bosentan, an endothelin receptor antagonist, was
strongly protective against damage from Glaucoma [14].

Current research methods perform genetic comparisons between mice with
Glaucoma and mice without Glaucoma, but one of the drawbacks of this research
is that it fails to stratify these comparison based on specific features such
as age or strain. Thus, we lack insight into how these features specifically
contribute to or protect from Glaucoma. The research presented in this paper
intends to solve this issue.

This research focuses on finding potential activator or protector genes for
Primary Open Angle Glaucoma in house mice. This research is significant
because one control group of mice, matched by age but not strain,
interestingly does not develop Glaucoma. It is worthwhile to examine the
genetic differential expression between this control group and the severe
Glaucoma group to better understand on the genetic level how one strain is
apparently “immune” to Glaucoma. Other control groups used in this research
consists of mice which are matched by age and strain but do not have Glaucoma
and mice which are matched by strain and not age. Through these experiments,
we can more effectively perform genetic comparisons to understand the genes
which play a role in the genesis of Glaucoma in each mentioned case.

##  III Data Availability Statement

This research used a publicly available data set from NCBI (NIH) of RNA
Sequence samples from the optical nerve head (ONH) and retina of mice [15]
(citation is clickable link to data source). The following section is a
description of the data set.

Genome-wide assessment of gene expression was performed in two species of
mice: DBA/2J mice and D2-Gpnmb+ mice. DBA/2J is a breed of mice which develops
high IOP and hereditary Glaucoma similar to that in human eyes [16].
D2-Gpnmb+, on the other hand, is a breed of mice which does not develop high
IOP or Glaucoma [17]. As such, D2-Gpnmb+ mice can be used as a control group
when studying the DBA/2J mice.

The NCBI data set consisted of 40 RNA Seq samples from DBA/2J mice, each of
10.5 months of age, placed into one of the following groups based on Glaucoma
severity: 1) No or early 1, 2) No or early 2, 3) Moderate, and 4) Severe.

Additionally, the data set contained two control groups. One control group
(Control Group 1) consisted of 10 D2-Gpnmb+ mice, each of 10.5 months of age
(age and strain matched, no glaucoma control). The second control group
(Control Group 2) consisted of 10 DBA/2J mice, each of 4.5 months of age
(young, pre-glaucoma). Only ONH, not retina, RNA Seq data was provided for the
second control group.

##  IV Proposed Experimental Design

###  IV-A Methodology Description

An initial quality control analysis was used to gauge the usability of the
data. Principal Component Analysis and Dispersion Analysis were used as
initial visualizations. After this, genes with skewed expression were
identified in hopes of finding marker genes from Glaucoma.

After quality control, differential expression and gene ontology analyses were
used to find potential activator or protector genes for Glaucoma.

###  IV-B Experimental Design

Figure 1: Experimental Design clarifying the comparisons which were made in
this experiment.

The workflow of the comparisons outlined in Figure 1 will be clarified in the
next section. Here, severe ONH and retina data are compared to multiple ONH
and retina data control groups. The results of these comparisons are compared
at the end of the study. In this study, the results of the different pairwise
comparisons yielded similar differentially expressed genes and gene
ontologies, giving more confidence in these results.

To find the genes responsible for the genesis of Glaucoma, the Severe group
was compared to three other groups: the two control groups and the No or early
1 group. The following is the intuition behind these comparisons:

The comparison between the Severe and the Control 1 group could reveal which
differentially expressed (DE) genes enable the D2-Gpnmb+ breed to not develop
Glaucoma or high IOP. The comparison between the Severe and the Control 2
group could reveal which DE genes lead to the manifestation of Glaucoma as the
mice grow older. The comparison between the Severe and the No or early 1 group
could reveal which DE genes lead some mice to develop Glaucoma and not others
(even when they are the same species and age). The No or early 1 group acted
as a sort of third control group because the mice in this group do not develop
Glaucoma.

Additionally, the experiment was repeated for both the ONH and retina samples
(where data was available).

###  IV-C Experimental Workflow of Comparisons

This section clarifies the exact workflow of each experiment highlighted in
the last section.

Each workflow consisted of four main parts:

  1. 1.

Find differentially expressed (DE) genes between the two RNA Seq sample
groups. Split this list of DE genes into an upregulated and downregulated gene
list according to the value of the Log Fold Change (LogFC).

  2. 2.

Make heatmaps to compare the expression of the top 20 DE genes across all
considered samples in the comparison. The top 20 DE genes consisted of the top
10 upregulated DE genes and the top 10 downregulated DE genes (ranked by
LogFC).

  3. 3.

Make boxplots which model the expression of the top 20 upregulated DE genes
and the the top 20 downregulated DE genes. The $x$-axis had each of the eleven
RNA Seq sample groups in the data:

No or early 1 ONH, No or early 2 ONH, Moderate ONH, Severe ONH, Control Group
1 ONH, Control Group 2 ONH, No or early 1 retina, No or early 2 retina,
Moderate retina, Severe retina, and Control Group 1 retina.

The $y$-axis represented the expression (in Log Gene Counts) of the gene.
Since each boxplot visualization represented the expression of one DE gene, 40
boxplot visualizations were made per comparison. Recall that there were
repeated boxplot visualizations made since a gene that was differentially
expressed in one comparison could have been differentially expressed in
another.

  4. 4.

Gene ontology (GO) was performed on the top 1000 upregulated DE genes (ranked
by LogFC) and the Biological Process (BP), Cellular Component (CC), and
Molecular Function (MF) gene ontologies were found. The gene ontology was
similarly performed on the top 1000 downregulated DE genes. The DAVID gene
ontology tool was used to conduct the gene ontology analysis [18][19].

The results of Step 4 provide the more general result as to which group or
class of genes should be targeted to create a new Glaucoma treatment for the
DBA/2J mice.

###  IV-D Other Analysis and Visualizations

In addition to the comparisons already mentioned, several other analyses were
conducted.

First, the Principal Component Analysis (PCA) algorithm was used to visualize
the relationships between the different RNA Seq sample groups.

Second, a Dispersion Visualization was made to assess the quality of the data
and find the common biological coefficient of variation (BCV).

Third, the skewness in gene counts for each gene was calculated. Genes which
were considered to have a skewness in their expression were visualized through
histograms. Genes with skewed expression are important because they can serve
as marker genes for a particular RNA Seq sample group.

##  V Experimental Results and Analysis

###  V-A PCA Visualizations

The Principal Component Analysis (PCA) algorithm was used to visualize the
optical nerve head (ONH222In the data set, optical nerve head is abbreviated
as OHN, so the abbreviation OHN is used in the visualizations. However, these
two abbreviations are essentially the same otherwise.) and retina data. Since
these two types of data come from different parts of the eye, they were
visualized separately.

Figure 2: The PCA visualization of the ONH data using principal components 1
and 2.

As shown in Figure 2, groups Control Group 1 ONH, Control Group 2 ONH, No or
early 1 ONH, and No or early 2 ONH all coincide with each other. This made
sense due to the similarity of their nature; these were either control groups
or groups which represent mice which do not develop Glaucoma as they age. On
the other hand, the Moderate OHN and Severe OHN groups also coincided because
they represent a different class of mice which do develop Glaucoma as they
age.

Figure 3: The PCA visualization of the retina data using principal components
1 and 2.

Similar to the previous visualization, Figure 3 shows that groups Control
Group 1 retina, Control Group 2 retina, No or early 1 retina, and No or early
2 retina all coincide with each other, while the Moderate retina and Severe
retina groups are separate from the other groups.

Note that the variance maintained in these visualizations (42.55% and 42.03%)
is not especially high, so there are nuances in the data that cannot be seen
here.

First, the fact that the Severe group, Moderate group, and other groups were
separated in the visualization provided a good quality control check for the
data set.

Furthermore, the stark contrast between the Severe group and the other groups
implies that there may be significant genetic differences in these groups.
Learning and targeting these genetic differences can aid the development of
future treatments for Glaucoma. This reasoning served as a motivation for
differential expression analysis and gene ontology.

###  V-B Dispersion Visualization

The dispersion plot visualization was used to visualize the biological
coefficient of variation (BCV) for the data.

Figure 4: The dispersion visualization of the RNA Seq data to visualize the
BCV.

In Figure 4, the Dispersion Visualization is shown. Each point corresponds to
one gene, and the $x$ axis represents the average log (base 2) counts per
million (CPM) of that gene while the $y$ axis represents the biological
coefficient of variation (BCV) for that gene. The red line shows the common,
or average, BCV, and the blue line shows the trend of the BCV as the average
log CPM increases.

The common dispersion is $7.71\times 10^{-2}$, so the common BCV is
$2.78\times 10^{-1}$. The industrial standard of BCV is less than about
$3.00\times 10^{-1}$, so the common BCV of $2.78\times 10^{-1}$ was an
acceptable level of variation. Additionally, the most dense area of points is
almost entirely below the common BCV, the mid dense area of points is almost
cut in half by the common BCV, and the least dense area of points is entirely
above the common BCV. This means that the majority of genes have a BCV less
than the common BCV, but there are certain genes which skew this plot and
increase the common BCV. Nevertheless, the common BCV falls under the
industrial threshold as mentioned previously, so this dispersion plot provided
another good quality check of the data. These quality control procedures
enabled the conduction of other analyses, as will be discussed in the
following sections.

###  V-C Skewed Gene Counts

The Pearson Mode Skewness was used to determine the skewness in the gene
counts of each gene across all data samples. The threshold 2 was used to
determine which genes should be considered skewed in their gene counts.
Specifically, if the Pearson Mode Skewness was less than -2 or greater than 2,
a gene was considered skewed in gene counts.

By this method, 112 genes were found to have skewed gene counts. For each
selected gene, a histogram was constructed of the log gene counts of that gene
across all samples. Specifically, the $x$-axis displayed the log gene counts,
and the $y$-axis displayed the number of samples, or frequency.

Figure 5: The histogram for the gene COL1A1 (Collagen Type I Alpha 1 Chain), a
gene which was identified to be skewed in gene counts.

As seen in Figure 5, the bimodal distribution implied that this gene could be
used as a marker gene for some of the RNA Seq groups. Similarly, other genes
found to have skewed expression were highlighted because they could
potentially be marker genes for Glaucoma. Therefore, instead of having to
perform whole genome sequencing, future researchers can focus on these marker
genes for more efficient diagnosis of Glaucoma.

Not all 112 histograms could be shown here, so the rest of the histograms were
uploaded to the GitHub repository for this paper [20].

###  V-D Pairwise Comparisons

The complete results for the five pairwise comparisons were not included, but
an example of the heatmap generated from the first pairwise comparison is
shown below. The complete generated data is available on the GitHub repository
for this paper. [20].

#### Heatmap

Figure 6: Heatmap of top 20 DE genes; Experiment 1: Severe ONH vs Control
Group 1 ONH.

##  VI Discussions

This Discussion section discusses the conclusions, key takeaways, and
implications of the pairwise comparison experiments.

###  VI-A Conclusions and Key Takeaways

First, it should be noted that apart from several genes, the majority of the
heatmaps contained genes which had clear differential expression between the
two groups compared in each comparison. The stark contrast in the expression
for the DE genes implied that the quality of the data was good, as we would
expect such contrasts when comparing mice with severe Glaucoma and control
groups.

The key takeaways from this study lie in the gene ontologies. Specifically,
common gene ontologies were found using the upregulated genes from different
pairwise comparison experiments, implying that the results found were
significant. These gene ontologies also support existing knowledge about the
development of Glaucoma, implying that the results were valid.

In biological process (BP) gene ontologies, nervous system development and
synaptic signaling were common themes. Correspondingly, cellular component
(CC) gene ontologies in locations such as the axon, synaptic membrane, and
neuronal cell body were observed. Finally, these gene ontologies corroborated
the molecular function (MF) gene ontologies, which included ion channel
activity and gated channel activity.

For each gene ontology, specific upregulated and downregulated DE gene lists
were generated which correspond to that gene ontology. These gene lists, which
can be found on the GitHub repository for this paper [20], can be studied and
then selectively targeted in mice with Glaucoma to develop more effective
treatments for the disease.

It should be noted that this Discussion section does not go into a deep
biological study of the individual genes in the generated gene lists as this
is beyond the scope of this research; this goal is addressed in detail to
future researchers in the next section.

###  VI-B Need For Further Research

The differential expression and gene ontology analyses from the pairwise
comparison experiments provide candidate gene lists which must be targeted to
develop new Glaucoma treatments. Future researchers can perform thorough
biology-oriented studies of these candidate gene lists to find potential
activator or protector genes for Glaucoma in mice. These findings can be
confirmed by inhibiting or activating such genes in mice to observe the effect
on the development of Glaucoma or high IOP.

In accordance with these results, future researchers may develop new gene
therapies or drugs which can be tested on mice to slow or stop the progression
of Glaucoma.

Once such treatments are shown to be effective on mice, the possibility of
creating treatments for human Glaucoma can be explored. Future researchers
must also keep in mind the ethical concerns with human trials and how human
based treatment testing may be different than mouse based treatment testing.

## References

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# Euclidean Path Integral, Entanglement Entropy, and Quantum Boundary
Conditions

Dong-han Yeom<EMAIL_ADDRESS>Department of Physics Education, Pusan
National University, Busan 46241, Republic of Korea
Research Center for Dielectric and Advanced Matter Physics, Pusan National
University, Busan 46241, Republic of Korea

###### Abstract

To understand the information loss paradox in a consistent way, we provide a
brief big picture that describes both outside and inside a black hole. We
summary several ideas including the Euclidean path integral, the entanglement
entropy, and the quantum gravitational treatment for the singularity. This
integrated discussion can provide an alternative point of view toward the
ultimate resolution of the information loss paradox.111Proceedings of the 17th
Italian-Korean Symposium on Relativistic Astrophysics. Talk on August 4, 2021,
Gunsan, Korea.

##  I Introduction: key ideas to obtain the Page curve

The information loss paradox of black holes is an unresolved problem in modern
theoretical physics [1]. There have been fifty years history of the debating
[2], but recently there was a big improvement in string theory [3]. In the
recent discussion, the basic idea was originated from the computation of the
entanglement entropy. The entanglement entropy can be computed from the
extreme of the generalized entropy. As a result, we obtain several saddles
that are known to be extremal surfaces. If there exist more than one extremal
surface, the contribution from the new saddle can explain the Page curve that
is deeply related to the unitary evolution of the black hole. The contribution
from the new saddle can be justified by the replica trick; the new
contribution comes from the replica wormhole [4].

However, it is still premature to conclude that we have a sufficient
justification of new saddles. A new contribution from the replica trick comes
from the path integral of the replicas of the density matrix rather than
quantum states. However, this is not a path integral of the quantum state
which is more orthodox in quantum mechanics. Therefore, the physical
interpretation is vague; how can information be emitted from the black hole
after the Page time?

In this regards, to understand the physics in the common ground, we assume the
following two contents [5].

  * –

1\. Multi-history condition: there exist at least two solutions (saddles,
steepest-descents, or whatever) that dominantly contribute to the entanglement
entropy computation, say $h_{1}$ and $h_{2}$. We further assume that $h_{1}$
is an information-losing solution while $h_{2}$ is an information-preserving
solution.

  * –

2\. Late-time dominance condition: initially, the probability of $h_{1}$, say
$p_{1}$, was dominated. However, as time goes on, the probability of $h_{2}$,
say $p_{2}$, is dominated. (The time can be measured in terms of the entropy,
or any methods.)

Based on these assumptions, the expectation value of the entanglement entropy
is approximately [5]

$\displaystyle\langle S\rangle\simeq p_{1}S_{1}+p_{2}S_{2},$ (1)

where $S_{1,2}$ denote the entanglement entropy for $h_{1,2}$, respectively.
Initially $p_{2}\ll p_{1}\simeq 1$ and eventually $p_{1}\ll p_{2}\simeq 1$.
$S_{1}$ monotonically increases or never approaches to zero, while $S_{2}$
eventually decreases to zero. Therefore, one can explain the Page curve for a
unitary evolution.

Perhaps, we may interpret as follows. If the recent development is the correct
answer to the information loss paradox, then there must exist the state-level
description. In other words, in the path-integral of quantum states, there
must be a corresponding steepest-descent of the replica wormhole
contributions; if it is not possible, then at least, there must be an analog
solution that mimics the role of the replica wormhole. Therefore, it might be
interesting to understand the entire wave function by using the Euclidean path
integral approach [6], i.e., in terms of the state-level path integral.

##  II Instantons for Hawking radiation and
tunneling to trivial geometry

In the Euclidean path integral approach [6], from the past infinity
$(h^{\mathrm{in}}_{ab},\phi^{\mathrm{in}})$ to the future infinity
$(h^{\mathrm{out}}_{ab},\phi^{\mathrm{out}})$, one can provide the propagator
by using the following path-integral

$\displaystyle\Psi_{0}\left[h^{\mathrm{out}}_{ab},\phi^{\mathrm{out}};h^{\mathrm{in}}_{ab},\phi^{\mathrm{in}}\right]=\int\mathcal{D}g_{\mu\nu}\mathcal{D}\phi\;\;e^{-S_{\mathrm{E}}[g_{\mu\nu},\phi]},$
(2)

where we sum-over all $g_{\mu\nu}$ and $\phi$ that connects from
$(h^{\mathrm{in}}_{ab},\phi^{\mathrm{in}})$ to
$(h^{\mathrm{out}}_{ab},\phi^{\mathrm{out}})$. This Euclidean path-integral
can be approximated by the steepest-descent approximation:

$\displaystyle\Psi_{0}\left[h^{\mathrm{out}}_{ab},\phi^{\mathrm{out}};h^{\mathrm{in}}_{ab},\phi^{\mathrm{in}}\right]\simeq\sum_{\mathrm{on-
shell}}e^{-S_{\mathrm{E}}^{\mathrm{on-shell}}[g_{\mu\nu},\phi]},$ (3)

where we sum-over all on-shell solutions, or so-called instantons.

Let us consider the following action

$\displaystyle S=\int
dx^{4}\sqrt{-g}\left[\frac{1}{16\pi}\mathcal{R}-\frac{1}{2}\left(\nabla\phi\right)^{2}\right]+\int_{\partial\mathcal{M}}\frac{\mathcal{K}-\mathcal{K}_{o}}{8\pi}\sqrt{-h}dx^{3},$
(4)

where $\mathcal{R}$ is the Ricci scalar, $\mathcal{K}$ is the Gibbons-Hawking
boundary term, and $\mathcal{K}_{o}$ is the Gibbons-Hawking boundary term for
the periodically identified Minkowski [7]. Note that $\phi$ is a free scalar
field and hence there is no contribution of the scalar field to the on-shell
action. For any field combinations, the on-shell Euclidean action becomes

$\displaystyle
S_{\mathrm{E}}=-\int_{\partial\mathcal{M}}\frac{\mathcal{K}-\mathcal{K}_{o}}{8\pi}\sqrt{+h}dx^{3}+\left(\mathrm{contribution\;at\;horizon}\right).$
(5)

The probability of an instanton process is $P\sim e^{-2B}$, where

$\displaystyle
B=S_{\mathrm{E}}(\mathrm{solution})-S_{\mathrm{E}}(\mathrm{background}).$ (6)

We impose the classicality condition, i.e., the reality condition of the
scalar field at the future infinity. Then the solution must be complex-valued
in the bulk region, but we do not physical observe them. The complex-valued
scalar field does not contribute to the on-shell action, but due to the energy
conservation, there appears a cusp at the horizon of the Euclidean geometry
[8]. After the regularization of the cusp, we obtain the correct on-shell
action [9]

$\displaystyle
2B=\frac{\mathcal{A}}{4}-\frac{\mathcal{A}^{\prime}}{4}=4\pi\left(M^{2}-M^{\prime
2}\right),$ (7)

where $\mathcal{A}$ and $M$ are the areal radius and mass of the initial black
hole, respectively, while the primed (′) quantities denote for the final black
hole.

In the end, if $M^{\prime}=M-\omega$ with a very small $\omega\ll M$, then

$\displaystyle 2B=8\pi M\omega$ (8)

which is the consistent result of Hawking temperature. On the other hand,
there exist a wide spectrum of instantons, e.g., $M^{\prime}=0$ [8]. In this
case, the probability is exponentially suppressed and no more dominated, but
the resulting geometry is trivial, e.g., there exists no horizon nor
singularity [10]. Therefore, we can reasonably conclude that the existence of
a tunneling channel toward a trivial geometry is very evident, although the
price is the exponentially suppressed low probability.

##  III Revisit the Page curve

As we have checked the existence of the tunneling channel toward a trivial
geometry, i.e., the multi-history condition, we need to demonstrate the late-
time dominance condition. In order to do this, we first observe the possible
variety of the Euclidean time [5].

Basically, if the shell is static, the contribution from the solution is two
parts, where from the bulk integration, we obtain $4\pi M_{+}^{2}$ and from
the boundary term at infinity, we obtain one more additional $4\pi M_{+}^{2}$.
The last term must be canceled from the boundary term at infinity of the
background solution. Eventually, we obtain the result that such a transition
is exponentially suppressed and the exponential factor is the same as the
entropy of the original black hole. Conceptually, one can write as follows:

$\displaystyle
2B=(\mathrm{bulk\;term\;of\;solution})+(\mathrm{boundary\;term\;of\;solution})-(\mathrm{boundary\;term\;of\;background}).$
(9)

The Euclidean time of the background cannot be chosen freely. If we started
from a Schwarzschild black hole, the Euclidean time period of the background
geometry is fixed. However, the Euclidean period of the solution part is
different. In principle, if the final boundary is the same, we can consider
periodically identified Euclidean time period arbitrarily. For this case, we
obtain the following factor:

$\displaystyle
2B=n((\mathrm{bulk\;term\;of\;solution})+(\mathrm{boundary\;term\;of\;solution}))-(\mathrm{boundary\;term\;of\;background}),$
(10)

where $n\geq 1$.

If we assume $n$ is an integer, then we obtain the result of the tunneling
probability:

$\displaystyle\sum_{n=1}^{\infty}e^{-S(2n-1)}=\frac{1}{e^{S}-e^{-S}},$ (11)

where $S=4\pi M_{+}^{2}$.

Therefore, if we only consider two histories, where one is a semi-classical
black hole and the other is a trivial geometry without a horizon, the
probabilities of each history, $p_{1}$ and $p_{2}$, respectively, becomes as
follows:

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\frac{e^{S}-e^{-S}}{1+e^{S}-e^{-S}},$ (12) $\displaystyle p_{2}$
$\displaystyle=$ $\displaystyle\frac{1}{1+e^{S}-e^{-S}}.$ (13)

We can finally obtain the entanglement entropy as a function of $S$:

$\displaystyle S_{\mathrm{ent}}$ $\displaystyle=$
$\displaystyle\left(S_{0}-S\right)\times p_{1}+0\times p_{2}$ (14)
$\displaystyle=$
$\displaystyle\left(S_{0}-S\right)\left(\frac{e^{S}-e^{-S}}{1+e^{S}-e^{-S}}\right),$
(15)

where $S_{0}$ is the initial entropy of the black hole. Here, for simplicity
we assume that the entanglement entropy of $h_{1}$ monotonically increases,
while the entanglement entropy of $h_{2}$ goes to zero (because there is no
black hole). This explains the unitary evolution of a black hole (see Fig. 1,
[5]).

Figure 1: Left: An example of the Page curve. The horizontal axis is the
entropy of radiation $S_{\mathrm{rad}}\equiv S_{0}-S$ and the vertical axis is
the entanglement entropy $S_{\mathrm{ent}}$. Right: At the Page time, we
estimate the ratio between the entanglement entropy and the initial entropy.
As the initial black hole size increases, the portion of the compressed
information increases. (See more details in [5].)

##  IV More on the late-time dominance condition

In the previous section, we only compared with two histories. However, in
general, there are infinitely many tunneling channels; also, the number of
histories for non-trivial geometries will be much more than those of the
trivial geometries. In this regards, can we still sure that the non-singular
geometries will eventually be dominated than the geometries with
singularities?

Perhaps, the quantum resolution of the singularity, i.e., the quantum
gravitational treatment of the singularity will help. The probability of an
arbitrary 3-hypersurface will be approximately a multiplication between the
probability of the inside and the outside the black hole [11]. Of course, in
general, the probability will be very complicated, but if we must provide the
DeWitt boundary condition for the singularity [12], the probability for inside
the horizon must approach zero; hence, the entire 3-hypersurface with a
singularity will approach zero, too.

Interestingly, there are some evidences that the DeWitt boundary condition for
the black hole singularity is very generic.

  * –1.

The quantum gravitational investigation for inside the horizon shows that
there exists a quantum bouncing surface [11]. Therefore, at this surface, we
can assign the DeWitt boundary condition; and one can interpret that the
annihilation-to-nothing process happens around the surface.

  * –2.

Due to the time-symmetry of the spacetime, in various loop quantum gravity
inspired black hole models, there exists bouncing points near the spacelike
singularity or spacelike hypersurface [13], not only static black holes, but
also gravitational collapsing cases [14]. One can interpret the quantum
bouncing point as the hypersurface for the DeWitt boundary condition.

  * –3.

In addition to them, due to the BKL conjecture of the singularity, there
exists an argument that the vanishing boundary condition of the Wheeler-DeWitt
equation is necessary [15].

Therefore, by integrating these arguments, we conclude that the late-time
dominance condition is quite reasonable and general, which is justified by
conservative quantum gravitational approaches.

##  V Future perspectives

In this paper, we summarized several ideas. First, the unitary Page curve can
be explained, if two conditions (the multi-history condition and the late-time
dominance condition) are satisfied. Second, in the Euclidean path integral
approach, the multi-history condition is quite evident. Third, the late-time
dominance condition can be supported by considering the multiple Euclidean
time period; also, the late-time dominance can be further supported by the
DeWitt boundary condition inside the black hole.

If all ideas are working consistently, we can understand the information loss
paradox in a deeper way. It is a very interesting question that what is the
relation to the island conjecture and the replica wormholes. We believe that
the island conjecture is deeply related to the previous two conditions.
However, it is less clear what is the corresponding geometry in terms of the
state-level (Euclidean) path integral. At this point, we need to investigate
further between these two approaches.

Also, it is worthwhile to find a way to (theoretically or experimentally)
confirm ideas, whether the replica wormholes exist or the other Euclidean
tunneling channels do the same role instead of the replica wormholes, etc. We
leave these interesting topics for the future research projects.

###### Acknowledgements.

The author would like to thank for stimulated discussions with Suddhasattwa
Brahma, Che-Yu Chen, Pisin Chen, Misao Sasaki, and Junggi Yoon. This work is
supported by the National Research Foundation of Korea (Grant no.:
2021R1C1C1008622, 2021R1A4A5031460).

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