# Energy-aware Web Browsing on Heterogeneous Mobile Platforms

Jie Ren2, Ling Gao2, Hai Wang2, and Zheng Wang31  2Northwest University, China
3MetaLab, School of Computing and Communications, Lancaster University, U.K.
Emails<EMAIL_ADDRESS>2{gl<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Web browsing is an activity that billions of mobile users perform on a daily
basis. Battery life is a primary concern to many mobile users who often find
their phone has died at most inconvenient times. The heterogeneous multi-core
architecture is a solution for energy-efficient processing. However, the
current mobile web browsers rely on the operating system to exploit the
underlying hardware, which has no knowledge of individual web contents and
often leads to poor energy efficiency. This paper describes an automatic
approach to render mobile web workloads for performance and energy efficiency.
It achieves this by developing a machine learning based approach to predict
which processor to use to run the web rendering engine and at what frequencies
the processors should operate. Our predictor learns offline from a set of
training web workloads. The built predictor is then integrated into the
browser to predict the optimal processor configuration at runtime, taking into
account the web workload characteristics and the optimisation goal: whether it
is load time, energy consumption or a trade-off between them. We evaluate our
approach on a representative ARM big.LITTLE mobile architecture using the
hottest 500 webpages. Our approach achieves 80% of the performance delivered
by an ideal predictor. We obtain, on average, 45%, 63.5% and 81% improvement
respectively for load time, energy consumption and the energy delay product,
when compared to the Linux heterogeneous multi-processing scheduler.

###### Index Terms:

Mobile Web Browsing, Energy Optimisation, big.LITTLE, Mobile Workloads

##  I Introduction

Web browsing is a major activity performed by mobile users on a daily basis
[13]. However, it remains an activity of high energy consumption [7, 26].
Heterogeneous multi-core design, such as the ARM big.LITTLE architecture [2],
is a solution to energy efficient mobile processing. Heterogeneous mobile
platforms integrate multiple processor cores on the same system, where each
processor is tuned for a certain class of workloads and optimisation goals
(either performance or energy consumption). To unlock the potential of the
heterogeneous design, software applications must adapt to the variety of
different processors and make good use of the underlying hardware, knowing
what type of processors to use and at what frequency the processor should
operate. This is because the benefits of choosing the right heterogeneous core
may be large, but mistakes can seriously hurt the user experience.

The current mobile web browser implementations rely on the operating system to
exploit the heterogeneous cores. The drawback of this is that the operating
system has no knowledge of the individual web workload to be rendered by the
browser; and as a result, this often leads to poor energy efficiency, draining
the battery faster than necessary [39]. What we would like to have is a
technique that can exploit the web workload characteristics to leverage the
heterogeneous cores to meet various user requirements: whether it is load time
(responsive time), energy consumption or a trade-off between them. Given the
diversity of mobile architectures, we would like to have an automatic approach
to construct optimisation strategies for any given platforms with little human
involvement.

This paper presents such an approach to exploit the heterogeneous mobile
platform for energy efficient web browsing. In particular, it focuses on
determining – for a given optimisation goal – the optimal processor
configuration i.e. the type of processor cores to use to render the webpage
and at what frequencies the processor cores of the system should operate.
Rather than developing a hand-crafted approach that requires expert insight
into the relative costs of particular hardware and web contents, we develop an
automatic technique that can be portable across computing environments. We
achieve this by employing machine learning to automatically build predictors
based on knowledge extracted from a set of representative, training web
contents. The trained models are then used at runtime to predict the optimal
processor configuration for a given workload and an optimisation target.

Our technique is implemented as an extension for the Google Chromium browser.
It is applied to the hottest 500 webpages ranked by www.alexa.com and is
evaluated for three distinct metrics: load time, energy consumption and energy
delay product (a trade-off between load time and energy consumption). We
evaluated our technique on a representative big.LITTLE mobile platform. Our
approach delivers significant improvement over a state-of-the-art web-aware
scheduling mechanism [40] and the Linux Heterogeneous Multi-Processing (HMP)
scheduler for all the three metrics.

The key contribution of this paper is a novel machine learning based
predictive model that can be used to optimise web workloads across multiple
optimisation goals. Our results show that significant energy efficiency for
mobile web browsing can be achieved by making effective use of the
heterogeneous mobile architecture.

##  II Background

###  II-A Web Rendering Process

Figure 1: The rendering process of Chromium browser.

Our prototype system is built upon Google Chromium, an open source version of
the popular Google Chrome web browser. To render an already downloaded
webpage, the Chromium rendering engine follows a number of steps: parsing,
style resolution, layout and paint. This process is illustrated in Figure 1.
Firstly, the input HTML page is parsed to construct a Document Object Model
(DOM) tree where each node of the tree represents an individual HTML tag such
as <body> or <p>. CSS style rules that describe how the web contents should be
presented will also be translated to the style rules. Next, the styling
information and the DOM tree are combined to build a render tree which is then
used to compute the layout of each visible element. Finally, the paint process
takes in the render tree to output the pixels to the screen. In this work, we
focus _solely_ on scheduling the rendering process on heterogeneous mobile
systems.

(a)  (b)

(c)  (d)

Figure 2: Best load time (a), energy consumption (b) and EDP (c) for rendering
wikipedia and bbc over to the HMP scheduler; and the performance of using
wikipedia best configurations w.r.t to the best available performance of bbc
(d).

###  II-B Motivation Example

Consider rendering the landing page of en.wikipedia.org and www.bbc.co.uk on
an ARM big.LITTLE mobile platform. The system has a Cortex-A15 (big) and a
Cortex-A7 (little) processors, running with the Ubuntu Linux operating system
(OS) (see also Section V-A). Here, we schedule the Chromium rendering process
to run on the big or little core under various clock frequencies. We then
record the best processor configuration found for each webpage. To isolate
network and disk overhead, we have pre-downloaded and stored the webpages in
the RAM and disabled the browser’s cache.

Figure 2 compares the best configuration against the Linux HMP scheduler for
three _lower is better_ metrics: (a) load time, (b) energy consumption and (c)
the energy delay product (EDP), calculated as energy $\times$ load time. Table
I lists the best configuration for each metric. For load time, the best
configuration gives 14% and 10% reduction for wikipedia and bbc respectively
over the HMP. For energy consumption, using the right processor configuration
gives a reduction of 58% and 17% for wikipedia and bbc respectively. For EDP,
the best configuration gives a reduction of over 55% for both websites.
Clearly, there is significant room for improvement over the OS scheduler and
the best processor configuration could change from one metric to the other.

Figure 2 (d) normalises the best available performance of bbc to the
performance achieved by using the best configuration found for wikipedia for
each metric. It shows that the best processor configuration could also vary
across webpages. The optimal configuration for wikipedia fails to deliver the
best available performance for bbc. In fact, there is a reduction of 11.9%,
18.9% and 23.5% on load time, energy and EDP available respectively for bbc
when compared to using the wikipedia-best configuration. Therefore, simply
applying one optimal configuration found for one webpage to another is likely
to miss significant optimisation opportunities.

This example demonstrates that using the right processor setting has a
significant impact on web browsing experience, and the optimal configuration
depends on the optimisation objective and the workload. What we need is a
technique that automatically determines the best configuration for any webpage
and optimisation goal. In the remainder of this paper, we describe such an
approach based on machine learning.

Table I: Optimal processor configurations for web rendering | Load time | Energy | EDP
---|---|---|---
| A15 | A7 | A15 | A7 | A15 | A7
en.wikipedia.org \- GHz | 1.8 | 1.4 | 0.9 | 0.4 | 1.3 | 0.5
www.bbc.co.uk \- GHz | 1.6 | 1.4 | 1.0 | 0.3 | 1.5 | 0.4
rendering engine | ✓ |  | ✓ |  | ✓ |

##  III Overview of our approach

Figure 3: Our three-stage approach for predicting the best processor
configuration and scheduling the rendering process.

Figure 3 depicts our three-stage approach for predicting the right processor
configuration when rendering a webpage. After the web contents (e.g. the HTML
source, CSS files and javascripts) are downloaded, they will be parsed to
construct a DOM tree together with style rules. This is performed by the
default parser of the browser. Our approach begins from extracting information
(_termed as feature extraction_), from the DOM tree and style data to
characterise the web workload. This information (or features) includes
counting different HTML tags, DOM nodes and style rules. A complete list of
the features is given in Table IV. Next, a machine learning based predictor
(that is built off-line) takes in these feature values and predicts which core
to use to run the rendering engine and at what frequencies the processors of
the platform should operate. Finally, we configure the processors and schedule
the rendering engine to run on the predicted core.

Our approach is implemented as a web browser extension which will be invoked
as soon as a DOM tree is constructed. Re-prediction and rescheduling will be
triggered if there are significant changes of the DOM tree structure, so that
we can adapt to the change of web contents. Note that we let the operating
system to schedule other web browser threads such as the input/output process.

##### Optimisation Goals

In this work we target three important optimisation metrics: (a) load time
(which aims to render the webpage as quick as possible), (b) energy
consumption (which aims to use as less energy as possible) and (c) EDP (which
aims to balance the load time and energy consumption). For each metric, we
construct a predictor using the same learning methodology described in the
next section.

##  IV Predictive Modeling

Our model for predicting the best processor configuration is a Support Vector
Machine (SVM) classifier using a radial basis function kernel [28]. We have
evaluated a number of alternative modeling techniques, including regression,
Markov chains, K-Nearest neighbour, decision trees, and artificial neural
networks. We chose SVM because it gives the best performance, can model both
linear and non-linear problems and the model produced by the learning
algorithm is deterministic. The input to our model is a set of features
extracted from the DOM tree and style rules. The output of our model is a
label that indicates the optimal core to use to run the rendering engine and
the clock frequencies of the CPU cores of the system.

Building and using such a model follows the well-known 3-step process for
supervised machine learning: (i) generate training data (ii) train a
predictive model (iii) use the predictor, described as follows.

###  IV-A Training the Predictor

Figure 4: Training the predictor. Table II: Useful processor configurations per metric. Load time | Energy | EDP
---|---|---
A15 | A7 | A15 | A7 | A15 | A7
1.6 | 1.4 | 0.8 | 0.4 | 1.3 | 0.4
1.7 | 1.4 | 0.9 | 0.4 | 1.4 | 0.4
1.8 | 1.4 | 1.0 | 0.4 | 1.5 | 0.4
1.9 | 1.4 | 0.3 | 1.1 | 0.5 | 1.2
- | - | 0.3 | 1.2 | 0.5 | 1.3
- | - | 0.3 | 1.3 | 0.5 | 1.4
Table III: Raw web features DOM Tree | #DOM nodes | depth of tree
---|---|---
#each HTML tag | #each HTML attr.
| #rules | #each property
Style Rules | #each selector pattern |
Other | webpage size (KB) |

Figure 4 depicts the process of using training webpages to build a SVM
classifier for one of the three optimisation metrics. Training involves
finding the best processor configuration and extracting feature values for
each training webpage, and learning a model from the training data.

##### Generate Training Data

We use over 400 webpages to train a SVM model. These webpages are the landing
page of the top 500 hottest websites ranked by alexa [1]. These websites cover
a wide variety of areas, including shopping, video, social network, search
engine, E-commerce, news etc. Whenever possible, we used the mobile version of
the website. Before training, we have pre-downloaded the webpages from the
Internet and stored the content in a RAM disk. For each webpage, we
exhaustively execute the rendering engine with different processor settings
and record the best performing configuration for each optimisation metric. We
then label each best-performing configuration with a unique number. Table II
lists the processor configurations found to be useful on our hardware
platform. For each webpage, we also extract the values of a selected set of
features (described in Section IV-B).

##### Building The Model

The feature values together with the labelled processor configuration are
supplied to a supervised learning algorithm. The learning algorithm tries to
find a correlation from the feature values to the optimal configuration and
outputs a SVM model. Because we target three optimisation metrics in this
paper, we have constructed three SVM models – one for each optimisation
metric. Since training is only performed once at the factory, it is a _one-
off_ cost. In our case the overall training process takes less than a week
using two identical hardware platforms.

One of the key aspects in building a successful predictor is finding the right
features to characterise the input data. This is described in the next section
which is followed by sections describing how to use the predictor at runtime.

###  IV-B Web Features

Table IV: Selected web features #HTML tag |  a, b, br, button, div, h1, h2, h3, h4, i, iframe, li, link, meta, nav, img, noscript, p, script, section, span, style, table, tbody
---|---
#HTML attr |  alt, async, border, charset, class, height, content, href, media, method, onclick, placeholder, property, rel, role, style, target, type, value, background, cellspacing, width, xmlns, src
#Style selector |  class, descendant, element, id
#Style rules |  background.attachment/clip/color/image, background.repeat.x/y, background.size, background.border.image.repeat/slice/source/width, font.family/size/weight, color, display, float
Other info. |  DOM tree depth, #DOM nodes, #style rules, size of the webpage (Kilobytes)

Our predictor is based on a number of features extracted from the HTML and CSS
attributes. We started from 948 raw features that can be collected at runtime
from Chromium. Table III lists the raw features considered in this work. These
are chosen based on our intuitions of what factors can affect scheduling. For
examples, the DOM tree structures (e.g. the number of nodes, depth of the
tree, and HTML tags) determine the complexity and layout of the webpage; the
style rules determine how elements (e.g. tables and fonts) of the webpage
should be rendered; and the larger size of the webpage the longer the
rendering time is likely to be.

##### Feature Selection

To build an accurate predictor using supervised learning, the training sample
size typically needs to be at least one order of magnitude greater than the
number of features. Given the size of our training examples (less than 500
webpages), we would like to reduce the number of features to use. We achieve
this by removing features that carry little or redundant information. For
instances, we have removed features of HTML tags or attributes that are found
to have little impact on the rendering time or processor selections. Examples
of those tags are <def>, <em> and <body>. We have also constructed a
correlation coefficient matrix to quantify the correlation among features to
remove similar features. The correlation coefficient takes a value between
$-1$ and $1$, the closer the coefficient is to $+/-1$, the stronger the
correlation between the features. We removed features that have a correlation
coefficient greater than 0.75 (ignore the sign) to any of the already chosen
features. Exemplary similar features include the CSS styles <marginTop> and
<marginRight> which often appear as pairs. Our feature selection process
results in 73 features listed in Table IV.

##### Feature Extraction

To extract features from the DOM tree, our extension first obtains a reference
for each DOM element by traversing the DOM tree and then uses the Chromium
API, document.getElementsByID, to collect node information. To gather CSS
style features, it uses the document.styleSheets API to extract CSS rules,
including selector and declaration objects.

##### Feature Normalisation

Before feeding the feature values to the learning algorithm, we scale the
value of each feature to the range of 0 and 1. We also record the _min_ and
_max_ values used for scaling, which then can be used to normalise feature
values extracted from the _new_ webpage during runtime deployment (described
in the next sub-section).

###  IV-C Runtime Deployment

Figure 5: Runtime prediction and processor configuration.

Once we have built the models as described above, we can use them to predict
the best processor configuration for any _new_ , _unseen_ web contents. The
prediction is communicated to a scheduler running as an OS service to move the
rendering process to the predicted core and set the processors to the
predicted frequencies.

Figure 5 illustrates the process of runtime prediction and task scheduling.
During the parsing stage, which takes less than 1% of the total rendering time
[17], our extension firstly extracts and normalises the feature values, and
uses a SVM classifier to predict the optimal processor configuration for a
given optimisation goal. The prediction is then passed to the runtime
scheduler to perform task scheduling and hardware configuration. The overhead
of extracting features, prediction and configuring processor frequency is
small. It is less than 20 ms which is included in our experimental results.

As the DOM tree is constructed incrementally by the parser, it can change
throughout the duration of rendering. To make sure our approach can adapt to
the change of available information, re-prediction and rescheduling will be
triggered if the DOM tree is significantly different from the one used for the
last prediction. The difference is calculated by counting the number of DOM
nodes between the currently used tree and the newly available one. If the
difference is greater than 30%, we will make a new prediction using feature
values extracted from the new DOM tree and style rules. We have observed that
our initial prediction often remains unchanged, so rescheduling and
reconfiguration rarely happen in our experiments.

###  IV-D Example

Table V: None-zero feature values for en.wikipedia.org. Feature | Raw value | Normalised value
---|---|---
#DOM nodes | 754 | 0.084
depth of tree | 13 | 0.285
number of rules | 645 | 0.063
web page size | 2448 | 0.091
#div | 131 | 0.026
#h4 | 28 | 0.067
#li | 52 | 0.031
#link | 10 | 0.040
#script | 3 | 0.015
#href | 148 | 0.074
#src | 36 | 0.053
#background.attachment | 147 | 0.040
#background.color | 218 | 0.058
#background.image | 148 | 0.039
#class | 995 | 0.045
#descendant | 4454 | 0.0168
#element | 609 | 0.134
#id | 4 | 0.007

As an example, consider rendering the landing page of wikipedia for energy
consumption. This scenario is most useful when the mobile phone battery is low
but the user still wants to retrieve information from wikipedia. For this
example, we have constructed a SVM model for energy using “cross-validation”
(see Section V-B) by excluding the webpage from the training example.

To determine the optimal processor configuration for energy consumption, we
first extract values of the 73 features listed in Table IV from the DOM tree
and CSS style objects. The feature values will then be normalised as described
in Section IV-B. Table V lists some of the non-zero values for this website,
before and after normalisation. These feature values will be fed into the
offline-trained SVM model which output a labeled processor configuration
($<$A15, 0.9, 0.4$>$ in this case) indicating the optimal configuration is
running the rendering process on the big core at 900 MHz and the little core
should operate at the lowest possible frequency 400 MHz. This prediction is
indeed the optimal configuration (see also Section II-B). Finally, the
processor configuration is communicated to a runtime scheduler to configure
the hardware platform. For this example, our approach is able to reduce 58% of
the energy consumption when comparing to the Linux HMP scheduler.

(a)  (b)

Figure 6: The #DOM nodes (a) and webpage size (b) for the webpages used in the
experiments.

(a) Load time (b) Energy reduction (c) EDP

Figure 7: Achieved performance for load time (a), energy consumption (b) and
EDP (c) over the Linux HMP scheduler.

##  V Experimental Setup

###  V-A Hardware and Software

##### Evaluation System

Our hardware evaluation platform is an Odroid XU3 mobile development board
with an A15 (big) processor and an A7 (little) processor. The board has 2 GB
LPDDR3 RAM and 64 GB eMMC storage. Table VI gives detailed information of the
hardware platform. We chose this platform as it is a representative big.LITTLE
architecture implementation. For example, the Samsung Galaxy S4 phone uses the
same architecture. The board runs the Ubuntu 14.04 LTS Linux OS. We
implemented our model as an extension to Chromium (version 48.0) which was
compiled using the gcc compiler (version 4.6). As the current implementation
of Google Chromium for Android does not support extensions, we did not
evaluate our approach on the Android OS.

##### Webpages

We used the landing page of the top 500 hottest websites ranked by
www.alexa.com. Whenever possible, we used the mobile version of the website
for evaluation. To isolate network and disk overhead, we have downloaded and
stored the webpages in a RAM disk. We also disabled the browser’s cache in the
experiments. Figure 6 shows the number of DOM nodes and the size for the 500
webpages used in our evaluation. As can be seen from this diagram, the
webpages range from small (4 DOM nodes and 40 Kilobytes) to large (over 8,000
DOM nodes and over 5 MB).

Table VI: Hardware platform | big CPU | LITTLE CPU | GPU
---|---|---|---
Model | Cortex-A15 | Cortex-A7 | Mali-T628
Core Clock | 2.0 GHz | 1.4 GHz | 533 MHz
Fore Count | 4 | 4 | 8

###  V-B Evaluation Methodology

##### Predictive Modelling Evaluation

We use _leave-one-out_ cross-validation to evaluate our machine learning
model. This means we remove the target webpage to be predicted from the
training example set and then build a model based on the remaining webpages.
We repeat this procedure for each webpage in turn. It is a standard evaluation
methodology, providing an estimate of the generalisation ability of a machine-
learning model in predicting _unseen_ data.

##### Comparisons

We compare our approach to two alternative approaches, a state-of-the-art web-
aware scheduling mechanism [40] (_referred as WS hereafter_), and the Linux
Heterogeneous Multi-Processing (HMP) scheduler which is designed for the
big.LITTLE architecture to enable the use of all different CPU cores at the
same time. WS uses a regression model built from the training examples to
estimate webpage load time and energy consumption under different processor
configurations. The model is used to find the best configuration by
enumerating all possible configurations.

##### Performance Report

We profiled each webpage under a processor configuration multiple times and
report the _geometric mean_ of each evaluation metric. To determine how many
runs are needed, we calculated the confidence range using a 95% confidence
interval and make sure that the difference between the upper and lower
confidence bounds is smaller than 5%. We instrumented the Chromium rendering
engine to measure the load time. We excluded the time spent on browser
bootstrap and shut down. To measure the energy consumption, we have developed
a lightweight runtime to take readings from the on-board energy sensors at a
frequency of 10 samples per second. We then matched the readings against the
rendering process’ timestamps to calculate the energy consumption.

Figure 8: Our performance w.r.t. performance of an oracle predictor. We
achieve over 80% of the oracle performance.

##  VI Experimental Results

In this section, we first compare our approach against WS and the HMP
scheduler. We then evaluate our approach against an ideal predictor, showing
that our approach delivers over 80% of the oracle performance. Finally, we
analyse the working mechanism of our approach.

###  VI-A Overall Results

Figure 7 shows the performance results of our approach and WS for the three
evaluation metrics across all websites. For each metric, the performance
improvement varies for different webpages. Hence, the _min-max_ bars in this
graph show the range of improvement achieved across the webpages we used. The
baseline in the diagram is the HMP scheduler.

##### Load Time

Figure 7 (a) shows the achieved performance when fast response time is the
first priority. For this metric, WS achieves an averaged speedup of 1.34x but
it causes significant slowdown (up to 1.26x) for some websites. By contrast,
our approach never leads to deteriorative performance with up to 1.92x
speedup. Overall, our approach outperforms WS with an average speedup of 1.45x
vs 1.34x over the HMP scheduler, and has constantly better performance across
websites.

##### Energy Consumption

Figure 7 (b) compares the achieved performance when having a long battery life
is the first priority. In this scenario, adaptive schemes (WS and our
approach) can significantly reduce the energy consumption through dynamically
adjusting the processor frequency. Here, WS is able to reduce the energy
consumption for most websites. It achieves on average an energy reduction of
57.6% (up to 85%). Once again, our approach outperforms WS with a better
averaged reduction of 63.5% (up to 93%). More importantly, our approach uses
less energy for all testing websites compared to HMP, while WS sometime uses
more energy than HMP. This is largely due to the fact that our approach can
better utilise the webpage characteristics to determine the optimal
frequencies for CPU cores. In addition, for several webpages, WS estimates the
big core gives better energy consumption, which are actual a poor choice.

##### EDP

Figure 7 (c) shows the achieved performance for minimizing the EDP value, i.e.
to reduce the energy consumption without significantly increasing load time.
Both adaptive schemes achieve improvement on EDP when compared to HMP. WS
delivers on average a reduction of 69% (up to 84%), but it fails to deliver
improved EDP for some websites. Unlike WS, our approach gives constantly
better EDP performance with a reduction of at least 20%. Overall, we achieve
on average 81% reduction (up to 95%) of EDP, which translates to 38%
improvement over WS on average.

###  VI-B Compare to Oracle

In Figure 8, we compare our scheme to an ideal predictor (_oracle_) that
always gives the optimal processor configuration. This comparison indicates
how close our approach is to the theoretically perfect solution. As can be
seen from the diagram, our approach achieves 85%, 90% and 88% of the oracle
performance for load time, energy consumption and EDP respectively. The
performance of our approach can be further improved by using more training
webpages together with more useful features to better characterise some of the
web workloads to improve the prediction accuracy.

###  VI-C Analysis

####  VI-C1 Optimal Configurations

(a) Load time (b) Energy consumption (c) EDP

Figure 9: The distribution of the optimal processor configurations for load
time (a), energy consumption (b) and EDP (c).

Figure 9 shows how the distribution of optimal processor configurations
changes from one metric to the other. To optimise for load time, we should
always run the rendering process on the fast, big core (A15) with a frequency
of at least 1.6 GHz. For this optimisation goal, nearly half of the websites
benefit from using the A15 core at 1.9 GHz while others benefit from a lower
frequency (1.6 to 1.8 GHz). We believe this is because for some webpages using
a lower frequency can reduce CPU throttling activities [3] (i.e. the OS will
greatly clock down the processor frequency to prevent the CPU from over-
heating). We also found that running the rendering process at 2.0 GHz (a
default setting used by many performance-oriented schedulers) does not give
better load time. When optimising for energy consumption, around 30% of the
simple websites benefit from the low-power A7 core. Furthermore, for the
websites where the A15 core is a good choice, they are in favour of a lower
clock frequency over the optimal one for load time. For EDP, using the A7 core
benefits some websites but the optimal clock frequency leans towards a median
value of the available frequency range. This is expected as EDP is a metric
for quantifying the trade-off between load time and energy consumption. This
diagram shows the need to adapt the processor settings to different web
workloads and optimisation goals.

####  VI-C2 Performance for each configuration

Figure 10: The achieved performance for all configurations listed in Table II.

Figure 10 shows the performance for using each of the processor configurations
listed in Table II across optimisation metrics. It shows that a “one-size-
fits-all” scheme fails to deliver the optimal performance. For example, while
the A15(0.8GHz)-A7(0.4GHz) configuration is able to reduce the energy
consumption by 40% on average, it is outperformed by our approach that gives a
reduction of 63.5%. This is confirmed by Figure 9 (b), showing that running
the A15 core at 0.8GHz only benefits 20% of the websites. Similar results can
be found for the other two optimisation metrics. This experiment shows that an
adaptive scheme significantly outperforms a hardwired strategy.

####  VI-C3 Prediction Accuracy

Our approach gives correct predictions for 82.9%, 88% and 85% of the webpages
for load time, energy consumption and EDP respectively. For those webpages
that our approach makes a misprediction, the resulting performance is not far
from the optimal, where we still achieve a reduction of 24%, 21% and 56% for
load time, energy consumption and EDP when compared to HMP.

Figure 11: Breakdown of runtime overhead to rendering time.

####  VI-C4 Breakdown of Overhead

Figure 11 shows the breakdown of runtime overhead. The runtime overhead of our
approach is low – less than 4% with respect to the rendering time. Since the
benefit of our approach is significant, we believe such a small overhead is
acceptable. Most of the time (15 ms) is spent on moving the rendering process
from one processor to the other. This is expected as task migration involves
initialising the hardware context (e.g. cache warm up), which can take a few
micro-seconds. The overheads of other operations, i.e. feature extraction,
predicting and frequency setting, is very low, which are less than 5 ms in
total.

####  VI-C5 Feature Importance

Figure 12: A Hinton diagram showing the importance of selected web features to
the prediction accuracy. The larger the box, the more important a feature is.

Figure 12 shows a Hinton diagram illustrates some of the most important
features that have an impact on the load time, energy and EDP specific models.
Here the larger the box, the more significantly a particular feature
contributes to the prediction accuracy. The x-axis denotes the features and
the y-axis denotes the model for each metric. The importance is calculated
through the information gain ratio. It can be observed that HTML tags and
attributes (e.g. <li>, <img>, <bgcolor>) and style rules are important when
determining the processor configurations for all optimisation metrics. Other
features are extremely important for some optimisation metrics (such as #DOM
nodes is important for energy, and #HTML.tag.table is important for load time
and energy) but less important for others. This diagram shows the need for
distinct models for different optimisation goals.

####  VI-C6 Adapt to Different Network Environments

Figure 13: Webpage rendering time to the time spent on downloading the
contents in different network environments.

In all the previous experiments, we have isolated the network impact by
storing the webpage into a RAM disk. In practice, the device can be used in
different network environments. A natural question to ask is: which of the
three models developed in this work best suits for a particular environment?
Figure 13 shows the webpage rendering time with respect to the download time
under different network settings: 2G, 3G, 4G and WiFi (802.11). We further
breakdown each environment into two groups: poor and good. A network
environment is considered to be poor if the packet loss is greater than 30%,
otherwise it is considered to be good. As can be seen from the diagram, the
download time dominates the total processing time in poor and good 2G network
environments. In such environments, our energy-tuned model can be used to
trade rendering performance for energy consumption without compromising the
user experience, by moving the rendering process to run on an low power
processor at a low clock frequency. Our EDP-tuned model is mostly suitable for
a good 3G network environment with a limited download bandwidth. Finally, our
load-time-tuned model can be used in good 4G and Wifi environments to satisfy
the performance requirement if load time is the first priority. This diagram
demonstrates the need of an adaptive scheme in different network environments.

##  VII Related Work

Our work lies at the intersection of numerous areas: web browsing
optimisation, task scheduling, energy optimisation and predictive modeling.

##### Web Browsing Optimisation

A number of techniques have been proposed to optimise web browsing, through
e.g. prefetching [31] and caching [22] web contents, scheduling network
requests [21], or re-constructing the web browser workflow [38]. Most of the
prior work are built for a homogeneous mobile systems where the processors are
identical. Furthermore, prior work often targets one single optimisation goal
(either performance or energy consumption). Unlike previous research, our work
targets a heterogeneous mobile system with different processing units and
multiple optimisation goals. The work presented by Zhu _et al._ [40] is the
nearest work, which uses linear regression models to estimate the load time
and energy consumption for each web event to determine where to run the
rendering process. While promising, there are two significant shortcomings
with this approach. Firstly, it schedules the webpage to the big core with the
highest frequency if no configuration meets the cut-off latency. This leads to
poor performance as can be seen in Section VI-C6 in some networking
environments. Secondly, their linear regression models only capture the linear
correlation between the web workload characteristics and the processor
configuration, leading to a low prediction accuracy for some webpages. Our
work addresses both of these issues by dynamically configuring all CPU cores
of the system and modelling both linear and non-linear behaviour.

##### Task Scheduling

There is an extensive body of work on scheduling application tasks on
homogeneous and heterogeneous multi-core systems e.g. [36, 4, 24]. Most of the
prior work in the area use heuristics or analytical models to determine which
processor to use to run an application task, by exploiting the code or runtime
information of the program. Our approach targets a different domain by using
the web workload characteristics to optimise mobile web browsing for a number
of optimisation metrics.

##### Energy Optimisation

Many techniques have been proposed to optimise web browsing at the application
level. These include aggregating data traffic [12], bundling HTTP requests
[15], and exploiting the radio state mechanism [37]. Our approach targets a
lower level, by exploiting the heterogeneous hardware architecture to perform
energy optimisation. Work on application-level optimisation is thus
complementary to our approach. Studies on energy behaviour of web workloads
[7, 26, 18] are also orthogonal to our work.

##### Predictive Modelling

Machine learning based predictive modelling is rapidly emerging as a viable
way for systems optimisation. Recent studies have shown that this technique is
effective in predicting power consumption [23], program optimisation [33, 34,
10, 29, 19, 20], auto-parallelisaton [27, 32, 30], task scheduling [11, 9, 8,
35, 16], benchmark generation [6], estimating the quality of service [5],
configuring processors using DVFS [14], and grouping communication traffics to
reduce power consumption [25]. No work so far has used machine learning to
predict the optimal processor configuration for mobile web browsing across
optimisation goals. This work is the first to do so.

##  VIII Conclusions

This paper has presented an automatic approach to optimise mobile web browsing
on heterogeneous mobile platforms, providing a significant performance
improvement over state-of-the-art. At the heart of our approach is a machine
learning based model that provides an accurate prediction of the optimal
processor configuration to use to run the web browser rendering process,
taking into account the web workload characteristics and the optimisation
goal. Our approach is implemented as an extension to the Google Chromium web
browser and evaluated on an ARM big.LITTLE mobile platform for three distinct
metrics. Experiments performed on the 500 hottest websites show that our
approach achieves over 80% of the oracle performance. It achieves over 40%
improvement over the Linux HMP scheduler across three evaluation metrics: load
time, energy consumption and the energy delay product. It consistently
outperforms a state-of-the-art webpage-aware scheduling mechanism. Our future
work will explore further refinement to prediction accuracy and to dynamically
adapt to different networking environments.

## Acknowledgments

This work was performed while Jie Ren was a visiting PhD student with the
School of Computing and Communications at Lancaster University. The research
was partly supported by the UK Engineering and Physical Sciences Research
Council under grants EP/M01567X/1 (SANDeRs) and EP/M015793/1 (DIVIDEND), and
the National Natural Science Foundation of China under grant agreements
61373176 and 61572401. The corresponding author of this article is Zheng Wang
(email: z.wang@lancaster.ac.uk).

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# DNA thermal denaturation by polymer field theory approach: effects of the
environment

Yu. Holovatch  $|$ ${}^{\textsf{\footnotesize{\mbox{?},\penalty
1000\mbox{?},\penalty 1000\mbox{?}}}}$, C. von Ferber  $|$
${}^{\textsf{\footnotesize{\mbox{?},\penalty 1000\mbox{?}}}}$, Yu. Honchar
$|$ ${}^{\textsf{\footnotesize{\mbox{?},\penalty 1000\mbox{?},\penalty
1000\mbox{?}}}}$ Corresponding author<EMAIL_ADDRESS>
(Received June 9, 2021, in final form July 19, 2021)

###### Abstract

We analyse the effects of the environment (solvent quality, presence of
extended structures — crowded environment) that may have impact on the order
of the transition between denaturated and bounded DNA states and lead to
changes in the scaling laws that govern conformational properties of DNA
strands. We find that the effects studied significantly influence the strength
of the first order transition. To this end, we re-consider the Poland-Scheraga
model and apply a polymer field theory to calculate entropic exponents
associated with the denaturated loop distribution. For the $d=3$ case, the
corresponding diverging $\varepsilon=4-d$ expansions are evaluated by
restoring their convergence via the resummation technique. For the space
dimension $d=2$, the exponents are deduced from mapping the polymer model onto
a two-dimensional random lattice, i.e., in the presence of quantum gravity. We
also show that the first order transition is further strengthened by the
presence of extended impenetrable regions in a solvent that restrict the
number of the macromolecule configurations.

Key words: DNA denaturation, Poland-Scheraga model, polymer networks, crowded
environment

###### Abstract

Ми розглянули вплив середовища (якість розчинника, присутність витягнутих
структур (перешкод) — «зайняте» середовище), який може змінити рід переходу
між денатурованим та зв’язаним станами ДНК і привести до змін законів
скейлінґу для конформаційних властивостей ланцюжків ДНК. Показано, що
досліджені ефекти значним чином впливають на інтенсивність переходу першого
роду. З цією метою, ми розглянули модель Поланда-Шераги і застосували підхід
теорії поля для полімерів, щоби обчислити ентропійні показники, пов’язані з
розподілом денатурованих петель на ланцюгу. Для випадку $d=3$ проаналізовано
відповідні розбіжні $\varepsilon=4-d$ розклади, оцінюючи їх за допомогою
відновлення збіжності методами пересумовування степеневих рядів. Для
вимірності $d=2$ їх обчислено завдяки проектуванню полімерної моделі на
двовимірну випадкову ґратку, тобто розглянуто систему за присутності квантової
ґравітації. Ми також показуємо, що інтенсивність переходу першого роду
посилюється за наявності у розчиннику протяжних непроникних областей, що
обмежують кількість конфігурацій макромолекули.

Ключов слова: денатурація ДНК, модель Поланда-Шераги, полімерні мережі,
невпорядковане середовище

Nucleic acids together with proteins and carbohydrates belong to
macromolecules essential to all known forms of life. Enormous experimental,
theoretical, and simulational efforts have been involved to understand and
qualitatively describe their physical, chemical and biological properties. In
this paper we show how an insight from polymer field theory helps to shed
light on properties of a DNA helix-to-coil (also called denaturation,
unwinding or unzipping) transition: a phenomenon, that lies at the origin of
biological processes involving DNA, as duplication or transcription. The
latter phenomena occur in a cell and are complex biological protein-mediated
processes. An analogue of DNA unwinding is also observed in vitro: when
purified DNA solution is heated above the room temperature, the cooperative
transition from the hydrogen bound double-stranded helix structure to a single
stranded one occurs, see [1] and references therein for review. This
phenomenon is known as DNA thermal denaturation and is the subject of our
study.

In statistical physics, the DNA thermal denaturation is described in terms of
the Poland-Scheraga model [2, 3, 4] that allows its treatment in terms of
phase transition theory. In a recent paper [5] we have shown that changes in
the solvent quality may cause an essential impact on the order of the phase
transition between denaturated and bounded DNA states. To quantify this
impact, we have calculated $\varepsilon=4-d$ expansions for the entropic
exponents that govern the denaturated loop distribution in a good solvent and
in the $\theta$-solvent regimes and evaluated these (divergent) expansions in
$d=3$. In this paper, we complement such analysis by offering exact results
for the exponents at $d=2$. Moreover, we further analyse possible reasons that
may have impact on the order of the transition. In particular, we are
interested in the effects caused by the presence of extended structures that
restrict the swelling of polymer chains. By such analysis we make an attempt
to consider the situation in a more realistic condition of macromolecules in a
crowded environment of a cell [6].

a. b.

Figure 1: Model of DNA thermal denaturation (unzipping) transition discussed
in this paper. At low temperatures $T$, two DNA strands remain connected by
hydrogen bonds and form a single long flexible polymer chain (figure a). With
an increase of $T$, the chain unzips and a loop emerges: now the whole
structure consists of two different ‘species’: double stranded side chains,
shown by solid lines, and a single stranded loop shown by a thin line in the
middle (figure b).

The rest of the paper is organized as follows. In the next section we repeat
some of our results for the scaling relations and entropic scaling exponents
[5] and give their numerical estimates at $d=3$. Based on the exact conformal
scaling dimensions for two-dimensional copolymers derived from an algebraic
structure existing on a random lattice (quantum gravity) [7, 8, 9] we derive
exact values of the exponents at $d=2$ and discuss the whole sector
$2\leqslant d\leqslant 4$ in section 2. Crowded environment effects are
analysed in section 3, conclusions are summarized in section 4. It is our
pleasure and a big honour to contribute by this paper to the Festschrift
devoted to Prof. Yu. Kalyuzhnyi on the occasion of his 70th birthday. Doing so
we deeply acknowledge his seminal contributions to the soft matter physics in
general and to the subject discussed in this paper in particular, see e. g.,
[10, 11, 12, 13, 14, 15]. CvF and YuH also are indebted to the jubilee for a
long-lasting friendship, numerous discussions about physics and not only.

##  1 Poland-Scheraga model: scaling relations and $\varepsilon$-expansion

The model suggested by Poland and Scheraga in middle-sixties [2, 3] describes
the DNA thermal denaturation by a proper account of energy-entropy interplay:
at low temperatures $T$, the bound state, figure 1 a, is favoured by energy
whereas at high $T$ the unbound state, figure 1 b, is favoured by entropy as
the one having more configurations. Poland and Scheraga’s theoretical works
lead to a whole family of DNA denaturation models [4, 6, 16, 17, 18, 19, 20,
21, 22, 23, 24]. It was shown that the unzipping transition mechanism is
governed by the universal loop exponent $c$ which describes scaling of the
partition function of a single-stranded DNA loop in double stranded side
chains, see figure 1 b:

${\cal Z}_{\rm loop}\sim\mu^{\ell}\ell^{-c},$ (1.1)

here, $\ell$ is loop length (number of unbound segments) and $\mu$ is non-
universal fugacity. For $c>1$, the model predicts the denaturation transition
whereas for $0\leqslant c\leqslant 1$ the order parameter (average number of
ordered bound pairs in a chain) is a continuous function of $T$ smoothly
changing between 1 and 0 when $T$ increases from 0 to $\infty$. In turn, for
larger values of $c$, the order parameter either continuously vanishes at
$T=T_{c}$ for $1<c\leqslant 2$ or disappears abruptly at $T=T_{c}$ for $c>2$.
The last two types of behaviour correspond to the second and first order phase
transitions, respectively. However, the value of $c$ is not obvious. First
papers on the model suggested $c=d/2$, which lead to the second order
transition and $d=3$ [2, 3]. Later Fisher has considered taking into account
the self-avoiding nature of chains that lead to $c=d\nu$ [16], where $\nu$ is
polymer mean square end-to-end distance scaling exponent. Still, with
$\nu(d=3)\simeq 0.588$ [25], the phase transition remains the second order.
This result contradicts experimental observations of the first order nature of
the transition [1]. A more general approach to analyze scaling properties of
the macromolecule configurations shown in figure 1 was based on polymer
network theory, as interaction between the loop and the chain was taken into
account [17, 18, 19, 20, 21, 26, 27]. Considering both the side chains and the
loop as self-avoiding walks (SAWs), it was shown that the phase transition is
of the first order for $d=2$ and above. This result was further supported by
numerical simulations [26] and it was also suggested that possible
heterogeneity in chain structure may strengthen the transition.

Depending on temperature, the asymptotic scaling behaviour of a long flexible
polymer macromolecule in a good solvent belongs either to random walk (RW),
$T=T_{\theta}$, or to self-avoiding walk (SAW), $T>T_{\theta}$ universality
classes ($T_{\theta}$ denoting the $\theta$-point) [28, 29]. Therefore, the
only difference that may be observed in asymptotic scaling of chains of
different species (in our case these are the double- and single-stranded
chains) is due to the difference in asymptotic scaling properties of mutually
interacting SAWs and RWs. Based on this fact, recently [5] we have applied
polymer field theory [9, 25, 32, 33, 34] to derive scaling relations that
express the loop exponent $c$ (1.1) in terms of the familiar copolymer star
exponents $\eta_{f_{1}f_{2}}$. The latter govern the scaling of star-like
polymer structures that are created by linking together the end points of
polymer chains of two different species at a common core, as shown in figure
2. When such a copolymer star is immersed in a good solvent, its asymptotic
properties are universal in the limit of long chains. In particular, the
partition function (the number of configurations) of a copolymer star made of
two sets of $f_{1}$ and $f_{2}$ mutually avoiding RWs scales with its size $R$
as [9, 32, 33, 34]:

$Z^{G}_{f_{1}f_{2}}\sim R^{\eta^{G}_{f_{1}f_{2}}}\,.$ (1.2)

In turn, the partition function of a copolymer star made of mutually avoiding
sets of $f_{1}$ SAWs and $f_{2}$ RWs scales as:

$Z^{U}_{f_{1}f_{2}}\sim R^{\eta^{U}_{f_{1}f_{2}}-f_{1}\eta^{U}_{20}}\,.$ (1.3)

The third case which is of interest here is the star of two sets of $f_{1}$
and $f_{2}$ SAWs. For its partition function, one gets:

$Z^{S}_{f_{1}f_{2}}\sim
R^{\eta^{S}_{f_{1},f_{2}}-(f_{1}+f_{2})\eta^{S}_{20}}\,.$ (1.4)

Indices $G,U,S$ in the above formulae refer to the fixed points (FPs) of the
renormalization group transformation that govern the scaling of corresponding
mutually avoiding structures: Gaussian FP for RWs, unsymmetric FP for RW and
SAW, and symmetric FP for SAWs, see [32, 33, 34] for more details. Exponents
$\eta^{S}_{f_{1},f_{2}}$ are related to $\eta^{U}_{f_{1},f_{2}}$ and to the
homogeneous star exponents $\eta_{f}$ [30, 31] via:
$\eta^{S}_{f_{1},f_{2}}=\eta^{U}_{f_{1}+f_{2},0}=\eta_{f_{1}+f_{2}}$.

a. b.

Figure 2: a: block copolymer of two polymer chains of different species, shown
by solid and thin lines, linked together. b: copolymer star consisting of
$f_{1}$ chains species 1 and $f_{2}$ chains of species 2 tied together at
their end points. Its scaling properties are governed by universal copolymer
star exponents $\eta_{f_{1}f_{2}}$. Note that the block copolymer gives a
trivial example of a two-arm copolymer star with an exponent $\eta_{11}$.

With the above considerations in mind, one is led to four different cases that
account for possible inhomogeneities and, therefore, for different scaling
exponents of the DNA denaturation model shown in figure 1 b:

  1. 1.

both bound chains and the unbound loop are SAWs (SAW-SAW-SAW);

  2. 2.

bound chains are SAWs, the loop is RW (SAW-RW-SAW);

  3. 3.

the chains are RW-like, while the loop is SAW (RW-SAW-RW);

  4. 4.

both the chains and the loop are RW-like, though they do not intersect each
other (RW-RW-RW).

The scaling relations that express the loop exponent $c$ (1.1) in terms of
copolymer star exponents $\eta_{f_{1}f_{2}}$ for these four cases read [5]:

1\. SAW-SAW-SAW: $\displaystyle c_{1}=\nu_{\rm
SAW}(3\eta^{S}_{20}+d-2\eta^{S}_{12})\,,$ (1.5) 2\. SAW-RW-SAW: $\displaystyle
c_{2}=\nu_{\rm RW}(\eta^{S}_{20}+d-2\eta^{U}_{12})\,,$ 3\. RW-SAW-RW:
$\displaystyle c_{3}=\nu_{\rm SAW}(2\eta^{S}_{20}+d-2\eta^{U}_{21})\,,$ 4\.
RW-RW-RW: $\displaystyle c_{4}=\nu_{\rm RW}(d-2\eta^{G}_{21})\,.$

Here, $\nu_{\rm RW}=1/2$ and $\nu_{\rm SAW}$ are the mean square end-to-end
distance exponents for the random and self-avoiding walks, correspondingly,
and $d$ is space dimension. The exponents $\eta_{f_{1}f_{2}}$ have been
calculated within field-theoretical renormalization group approach [9, 32, 33,
34] and are currently know in the fourth order of the $\varepsilon=4-d$
expansion [35]. Below, we list them together with the $\varepsilon$-expansion
for the exponent $\nu_{\rm SAW}$ [36] in the corresponding order:

$\displaystyle\eta^{S}_{20}(\varepsilon)=$
$\displaystyle-\varepsilon/4-9\varepsilon^{2}/128+\varepsilon^{3}[264\zeta(3)-49]/2048$
(1.6)
$\displaystyle+\varepsilon^{4}[704\piup^{4}-297600\zeta(5)+38160\zeta(3)+235]/655360\,,$
$\displaystyle\eta^{S}_{12}(\varepsilon)=$
$\displaystyle-3\varepsilon/4-3\varepsilon^{2}/128+3\varepsilon^{3}[40\zeta(3)+23]/2048$
(1.7)
$\displaystyle+\varepsilon^{4}[64\piup^{4}-32640\zeta(5)-6480\zeta(3)+3333]/131072\,,$
$\displaystyle\eta^{U}_{12}(\varepsilon)=$
$\displaystyle-3\varepsilon/4+\varepsilon^{2}[42\zeta(3)-13]/128+\varepsilon^{3}[384\zeta(3)-5]/2048$
(1.8)
$\displaystyle+\varepsilon^{4}[1024\piup^{4}-528000\zeta(5)+14880\zeta(3)+7655]/655360\,,$
$\displaystyle\eta^{U}_{21}(\varepsilon)=$
$\displaystyle-\varepsilon+\varepsilon^{2}[42\zeta(3)+1]/64+17\varepsilon^{3}/1024$
(1.9) $\displaystyle-\varepsilon^{4}[1056\zeta(3)-721]/65536\,,$
$\eta^{G}_{21}(\varepsilon)=-\varepsilon,$ (1.10) $\displaystyle\nu_{\rm
SAW}(\varepsilon)=$ $\displaystyle
1/2+\varepsilon/16+15\varepsilon^{2}/512+\varepsilon^{3}[135/8192-(33/1024)\zeta(3)]+\varepsilon^{4}[3799/524288$
(1.11)
$\displaystyle-(873/32768)\zeta(3)-(11/40960)\piup^{4}+(465/4096)\zeta(5)]\,,$

where $\zeta(x)$ is Riemann zeta-function. Note that the formula for the
exponent $\eta^{G}_{21}$ contains only linear in $\varepsilon$ term and is
exact.

Substituting expressions (1.6)-(1.11) into the scaling relations (1.5) one can
evaluate loop exponents $c_{i}$ at any value of $d$. It is well known,
however, that $\varepsilon$-expansions of the field theory are asymptotic at
best and proper resummation technique is required to get a reliable numerical
information on their basis [38, 37]. Applying resummation technique based on
the Borel-Leroy transformation enhanced by conformal mapping of a cut-plane on
a disc [39, 40, 41], we arrived at the following values of the loop exponents
$c_{i}$ for $d=3$ [5]:

$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle 2.147\pm 0.009,\hskip
10.00002ptc_{2}=2.169\pm 0.004,$ (1.12) $\displaystyle c_{3}$ $\displaystyle=$
$\displaystyle 2.76\pm 0.03,\hskip 20.00003ptc_{4}=2.5.$

Clearly, $c>2$ in all configurations, which confirms the first order
transition.

Figure 3: (Colour online) Loop closure exponents $c_{i}$ at different space
dimension $d$. Triangles show exact results at $d=2$ (2.3) and squares show
most accurate results obtained by resummation at $d=3$ [5]. The lines show
analytic continuation to non-integer $d$ via $\varepsilon$-expansion in the
first and second orders, thick solid and thin dashed lines, correspondingly.
Note that in the case of mutually avoiding RWs (blue line), loop closure
exponents $c_{4}(\varepsilon)$ are exact and linear in $\varepsilon$.

##  2 Exact results at $d=2$ and quantum gravity

As has been discussed above, the $\varepsilon$-expansions for the $c_{i}$ may
serve as a basis for reliable numerical estimates at $d=3$ provided
appropriate resummation technique is applied. With the perturbative expansions
and their numerical estimates at hand, it is instructive to corroborate the
results by comparing them with the data for other space dimensions, if
available. One obvious result is obtained for $d=4$: there, as it is easy to
check via equations (1.5), all exponents are equal: $c_{i}(d=4)=2$. Besides,
there is a tempting opportunity to get exact values for the exponents at
$d=2$. Indeed, to this end one can make use of the exact results for the
scaling exponents of $d=2$ copolymer stars of mutually avoiding bunches of SAW
and RW [7, 8]. There, the relations between exponents in fluctuating geometry
(quantum gravity) and flat $d=2$ geometry have been used to extract the exact
values of the exponents. In notations of the previous section, the exponents
read:111Cf. equations (100), (101) of [9].

$\displaystyle\eta^{G}_{f_{1}f_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{48}\Big{\\{}4-\Big{[}\sqrt{24f_{1}+1}+\sqrt{24f_{2}+1}-2\Big{]}^{2}\Big{\\}}\,,$
(2.1) $\displaystyle\eta^{U}_{f_{1}f_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{48}\Big{\\{}4+5f_{1}-\Big{[}3f_{1}+\sqrt{24f_{2}+1}-1\Big{]}^{2}\Big{\\}}\,.$
(2.2)

Substituting these formulae into equations (1.5) and taking into account that
$\nu_{\rm SAW}(d=2)=3/4$ [25], one gets the following exact values of the
exponents $c_{i}$ at $d=2$:

$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle\frac{77}{32}\simeq
2.406,\hskip 10.00002ptc_{2}=\frac{109}{48}\simeq 2.271,$ (2.3) $\displaystyle
c_{3}$ $\displaystyle=$ $\displaystyle\frac{7}{2},\hskip 10.00002ptc_{4}=3\,.$

These values are shown by triangles in figure 3. The obtained result for the
exponent $c_{3}$ recovers the value predicted at $d=2$ by the exact formula
that follows from equation (1.5) and is also valid for other values of $d$:
$c_{3}=2+\varepsilon/2$.

Comparing the values of the loop closure exponents $c_{i}$ at $d=2$ and at
$d=3$ one can arrive at certain conclusions about an impact of chain
heterogeneity on the strength of the DNA thermal denaturation transition. The
first observation is that passing from the homogeneous SAW composition within
the Poland-Scheraga model (as described by the exponent $c_{1}$) usually leads
to strengthening of the first order transition. When the SAW side chains are
substituted by RWs, the strength of the transition increases: $c_{3}>c_{1}$,
$c_{4}>c_{2}$. In turn, when the side chains remain unchanged, the change of
the SAW loop to the RW loop decreases the strength of the first order
transition: $c_{3}>c_{4}$ and $c_{1}>c_{2}$. The last effect is more
pronounced for the RW side chains and at space dimension $d=2$. In general,
the following rule holds: $c_{2}<c_{1}<c_{4}<c_{3}$ (with $c_{2}\sim c_{1}$ at
$d=3$).

Another striking feature that follows from the comparison of the exact and
perturbative results shown in figure 3 is a rather unusual behaviour of the
$\varepsilon$-expansion curves. Indeed, the first order
$\varepsilon$-expansion for the exponents $c_{i}$ (solid lines in the plot)
nicely corresponds to the resummed $\varepsilon^{4}$-data at $d=3$ and to the
exact values at $d=2$. Such a behaviour is obvious for the exponent $c_{3}$,
where the first order $\varepsilon$-expansion provides an exact number.
However, for the other exponents, an account of the higher orders of the
perturbative expansion needs careful application of the resummation technique.
Being evaluated naïvely by simple adding higher order contribution, the
$\varepsilon$-expansion holds only very close to the upper critical dimension
$d=4$, as shown in the figure by the thin dashed lines for the case of
$\varepsilon^{2}$-data. Therefore, the first order $\varepsilon$-expansion
provides the so-called optimal truncation [38] for the $c_{i}(\varepsilon)$
series.

##  3 Crowded environment

In two former sections, 1 and 2, we discussed an impact of the solvent
quantity on the order of the DNA thermal denaturation transition. Another
factor that may modify the scaling exponents of long flexible polymer
macromolecules is the presence of impurities — impenetrable regions in a
solvent that restrict the number of polymer configurations, see e.g., [42] and
references therein. Statistics of polymers in disordered medium is of interest
for a number of reasons. In the context of our study it is important to
mention its relevance for treating macromolecules in a cell, composed of many
different kinds of biochemical species [43, 44, 45].

There exist different analytic frameworks to model an impact of disordered
medium on the scaling properties of (interacting) SAWs and RWs. To give a few
examples, the latter are studied on a percolation cluster [46, 47] or at
presence of quenched defects [48, 49, 50, 51, 52, 53, 54, 55]. Taking into
consideration that the uncorrelated defects do not influence polymer scaling
[48, 49], the so-called ‘extended’ or long-range correlated structural
disorder has been shown to be relevant. A model of long-range correlated
disorder has been suggested in [50] and further exploited in studies of
polymers [52, 53, 54, 55]. Within this model, one considers the defects,
characterized by the density-density pair correlation function $g(r)$ decaying
at a large distance $r$ according to the power law

$g(r)\sim r^{-a}\,.$ (3.1)

For integer values of $a$, such defects have a direct interpretation: the case
$a=d$ corresponds to point-like defects, while $a=d-1$ ($a=d-2$) correspond to
straight lines (planes) of defects of random orientation. Sometimes non-
integer values of $a$ are interpreted in terms of fractal structures.222See
also [56, 57, 58], where the relation of fractal dimension to the analytically
continued non-integer dimension is discussed in more details. Detailed
analysis of an impact of the long-range correlated disorder (3.1) on possible
changes in the exponents (1.5) and hence on the DNA thermal denaturation is
beyond the scope of this study. However, we will use some of the previously
obtained results in order to understand and qualitatively describe this
possible impact.

It is easy to see that the presence of long-range correlated impurities may or
may not be relevant and change the polymer scaling exponents depending on the
value of $a$. Indeed, large-distance asymptotics of the pair correlation
function (3.1) corresponds to the power-law behaviour of its Fourier-image at
small wave vector $k$ in the form $k^{a-d}$. Therefore, by simple power
counting, one arrives at the conclusion that such a term becomes relevant at
small $k$ for $a<d$. Applying field-theoretic renormalization group technique,
the corresponding polymer model has been analysed and the scaling exponents
were calculated in the two-loop approximation at fixed $d=3$ and different
values of the correlation parameter $a$ as well as in a one-loop order by the
double expansion in $\varepsilon=4-d$ and $\delta=4-a$ [52, 53, 54]. The
derivation given below is based on these double $\varepsilon,\delta$
expansions. In particular, it has been shown that for certain region of
parameters $\varepsilon/2<\delta<\varepsilon$, the scaling properties of a
single flexible polymer chain in porous environment with a long-range
correlated structure are governed by a new, ‘long-range’ fixed point $L$. The
mean square end-to-end distance exponent $\nu_{\rm SAW}$ in the first order of
$\varepsilon,\delta$ expansion reads [52]:

$\nu^{L}_{\rm SAW}=1/2+\delta/8+\dots\,.$ (3.2)

In turn, the $\eta_{f_{1}f_{2}}$ exponents for co-polymer stars in porous
environment with long-range correlated structure are given by:333Cf. equation
(39) from [55].

$\displaystyle\eta^{S_{L}}_{f_{1}f_{2}}$ $\displaystyle=$
$\displaystyle\frac{-(f_{1}+f_{2})(f_{1}+f_{2}-1)\delta}{4}\,,$ (3.3)
$\displaystyle\eta^{U_{L}}_{f_{1}f_{2}}$ $\displaystyle=$
$\displaystyle\frac{-f_{1}(f_{1}+3f_{2}-1)\delta}{4}\,,$ (3.4)
$\displaystyle\eta^{G_{L}}_{f_{1}f_{2}}$ $\displaystyle=$ $\displaystyle-
f_{1}f_{2}\delta\,.$ (3.5)

In equations (3.3)–(3.5), the first exponent $\eta^{S_{L}}_{f_{1}f_{2}}$
corresponds to the star of $f_{1}+f_{2}$ SAWs, the second exponent
$\eta^{U_{L}}_{f_{1}f_{2}}$ describes the star of mutually avoiding sets of
$f_{1}$ SAWs and $f_{2}$ RWs, and the third exponent
$\eta^{G_{L}}_{f_{1}f_{2}}$ describes the star of two mutually avoiding sets
of $f_{1}$ and $f_{2}$ RWs.

Two cautions are at place here. First, the ‘long-range’ fixed point $S_{L}$ is
accessible in the region where the above mentioned power counting shows that
the disorder is irrelevant. Second, the fixed points $U_{L}$ and $G_{L}$ can
be reached only for specific initial conditions. Similar situation is also
encountered when the $\varepsilon,\delta$ expansion is applied to study models
of $m$-vector magnets with long-range correlated quenched disorder [50].
However, an account of higher order contributions restores the physical region
of stability of the ‘long-range’ fixed point confirming a qualitatively
correct result of the first-order analysis, see e.g., [59] and references
therein. Therefore, with an aim of getting a qualitative description of an
impact of extended long-range correlated impurities on the DNA thermal
denaturation transition, we proceed with formulae (3.2)–(3.5) substituting
them into the scaling relations (1.5) and arrive at the following first-order
values for the $c_{i}$ exponents:

$\displaystyle c^{L}_{1}$ $\displaystyle=$ $\displaystyle
c^{L}_{2}=2-\varepsilon/2+5\delta/4\,,$ (3.6) $\displaystyle c^{L}_{3}$
$\displaystyle=$ $\displaystyle c^{L}_{4}=2-\varepsilon/2+2\delta.$ (3.7)

As it follows from equation (3.1), the smaller is $a$, the stronger are the
correlations in porous structure that restricts the volume available for the
macromolecule. Indeed, the density-density correlation function $g(r)$ decays
slower with a decrease of $a$, attaining the fat-tail features. The positive
sign at the linear in $\delta$ terms in equations (3.6), (3.7) brings about an
increase in the exponents $c_{i}$ with an increase of $\delta=4-a$. This
allows to conclude, that an increase in density correlations of the porous
structure leads to strengthening of the DNA thermal denaturation transition.
Moreover, comparing equations (3.6) and (3.7), one concludes that
$c^{L}_{3},c^{4}_{4}>c^{L}_{1},c^{L}_{2}$, similar to what was observed for
the DNA denaturation in a pure solvent without porous medium. The difference
between the exponents increases with an increase of $\delta$:
$c^{L}_{3,4}-c^{L}_{1,2}=3\delta/4$. Of course, with all cautions mentioned
above, these results should be considered as qualitative predictions, rather
than a quantitative description of DNA denaturation in a crowded environment.
The above obtained relations $c^{L}_{1}=c^{L}_{2}$ and $c^{L}_{3}=c^{L}_{4}$
may be (and perhaps indeed are) violated in the second order of the
perturbation theory. However, it is worth mentioning that the scaling
arguments supported by the renormalization group calculations predict the
effect of strengthening the order of the denaturation transition when it
occurs in presence of extended structures that restrict the swelling of the
polymer coil.

##  4 Conclusions

The value of the loop closure exponent $c$ (1.1) discriminates between
different ways the thermal denaturation of the DNA occurs: for $c>2$, the
denaturated loop emerges abruptly, in the first order phase transition manner,
for $1<c<2$, the transition is continuous, and for $c<1$, no transition
happens. Numerous attempts of theoretical description and numerical simulation
of this phenomenon finally led to the coherent picture, observed also in the
in vitro experiments and simulations [1, 60, 17, 27]: the transition is of the
first order and $c>2$. Besides, the factors that may have an impact on the
strength, and, eventually, even on the order, of this transition are discussed
in the literature [6]. In a recent paper [5], we have derived scaling
relations that express the loop closure exponent $c$ of the Poland-Scheraga
model in terms of the copolymer star exponents $\eta_{f_{1}f_{2}}$ [32, 33,
34]. This enabled us to analyse an impact of inhomogeneities in DNA chain
composition and solvent quality on the order of the transition. As it has been
shown in [5] and as it is briefly discussed in the above section 1,
consideration of the macromolecule as sets of mutually avoiding SAWs and RWs
(see figure 1) leads to an increase in value of $c$ and $d=3$ and, hence,
strengthens the first order transition. In the present paper, we support this
observation providing exact results at $d=2$. Moreover, we show that the
effect of strengthening is further enhanced by the so-called crowded
environment with the long-range correlated inhomogeneities.

##  5 Acknowledgement

We acknowledge useful discussions with Maxym Dudka, Ralph Kenna, Mariana
Krasnytska, and Dmytro Shapoval. This work was supported in part by the
National Academy of Sciences of Ukraine, project KPKBK 6541230.

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