# Multilingual Named Entity Recognition Using Pretrained Embeddings, Attention
Mechanism and NCRF

Anton A. Emelyanov
MIPT, Sberbank
Moscow, Russia
<EMAIL_ADDRESS>
Ekaterina Artemova
National Research University
Higher School of Economics
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

In this paper we tackle multilingual named entity recognition task. We use the
BERT Language Model as embeddings with bidirectional recurrent network,
attention, and NCRF on the top. We apply multilingual BERT only as embedder
without any fine-tuning. We test out model on the dataset of the BSNLP shared
task, which consists of texts in Bulgarian, Czech, Polish and Russian
languages.

##  1 Introduction

Sequence labeling is one of the most fundamental NLP models, which is used for
many tasks such as named entity recognition (NER), chunking, word segmentation
and part-of-speech (POS) tagging. It has been traditionally investigated using
statistical approaches Lafferty et al. (2001), where conditional random fields
(CRF) Lafferty et al. (2001) has been proven to be an effective framework, by
taking discrete features as the representation of input sequence Sathiya and
Sellamanickam (2007). With the advances of deep learning, neural sequence
labeling models have achieved state-of the-art results for many tasks Peters
et al. (2017).

For the purpose of this paper, we consider neural network solution for
multilingual named entity recognition for Bulgarian, Czech, Polish and Russian
languages for the BSNLP 2019 Shared Task Piskorski et al. (2019). Our solution
is based on BERT language model Devlin et al. (2018), use bidirectional LSTM
Hochreiter and Schmidhuber (1996), Multi-Head attention Vaswani et al. (2017),
NCRFpp Yang and Zhang (2018) (being neural network version of CRF++framework
for sequence labelling) and Pooling Classifier (for language classification)
on the top as additional information.

##  2 Task Description

###  2.1 Data Format

The data consists of raw documents and the annotations, separately provided by
the organizers. Each annotation contains a set of extracted entities and their
types without duplication. We convert each raw document and corresponding
annotations to labeled sequence and predict named entity label for each token
in the input sentence. The documents are categorized into topics. There are
two topics in the dataset released first: named “brexit” and “asia_bibi”.

###  2.2 Tasks

The BSNLP Shared Task has three parts Piskorski et al. (2019):

  1. 1.

Named Entity Mention Detection and Classification;

  2. 2.

Name Lemmatization;

  3. 3.

Cross-lingual entity Matching.

For more details about the dataset and the task refer to the description on
the web page111Full BSNLP Shared Task description available at
http://bsnlp.cs.helsinki.fi/shared_task.html.. We focused on Named Entity
Mention Detection (Named Entity Recognition) in this work.

##  3 System Description

We propose modeling the task as both sequence labeling and language
classification jointly with a neural architecture to learn additional
information about text. The model consists of one encoder, which on its own is
build from the pretrained multilingual BERT model, followed by several
trainable layers and two decoders. While the first decoder generates output
tags, the second decoder identifies the language of the input sentence222Our
code is available at https://github.com/anonymize/slavic-ner. This code is
based on https://github.com/sberbank-ai/ner-bert.. The system architecture is
presented in Figure 1 and consists of seven parts:

  1. 1.

BERT Embedder as pretrained multilingual language model;

  2. 2.

Weighted aggregation of BERT output;

  3. 3.

Recurrent BiLSTM layer to be trained for the NER task;

  4. 4.

Multi-Head attention to take shorter dependencies between words into account;

  5. 5.

linear layer as the head of the encoder part;

  6. 6.

NCRF++ inference layer for decoding, i.e. final sequence labelling;

  7. 7.

Concatenation operation of Max Pooling, Average Pooling and last output of
Multi-Head attention layer, later passed to linear layer for classification as
a second decoder for language identification.

Figure 1: The system architecture

###  3.1 Neural Network Architecture

####  3.1.1 BERT Embedder

The BERT embeddings layer contains Google’s original implementation of
multilingual BERT language model. Each sentence is preprocessed as described
in BERT paper Devlin et al. (2018):

  1. 1.

Process input text sequence to WordPiece embeddings Wu and Mike Schuster
(2016) with a 30,000 token vocabulary and pad to 512 tokens.

  2. 2.

Add first special BERT token marked “[CLS]”.

  3. 3.

Mark all tokens as members of part “A” of the input sequence.

But instead of BERT’s original paper Devlin et al. (2018) we keep “B”
(“Begin”) prefix for labels and do a prediction for “X” labels on training
stage. BERT neural network is used only to embed input text and don’t fine-
tune on the training stage. We freeze all layers except dropout here, that
decreases overfitting.

We take hidden outputs from all BERT layers as the output of this part of the
neural network and pass to the next level of the neural network. So the shape
of output is $12\times 768$ for each token of $512$ length’s padded input
sequence.

####  3.1.2 BERT Weighting

Here we sum all of BERT hidden outputs from previous part:

$o_{i}=\gamma\times\sum_{i=0}^{m-1}b_{i}s_{i}$ (1)

where

  * •

$o_{i}$ is output vector of size $768$;

  * •

$m=12$ is the number hidden layers in BERT;

  * •

$b_{i}$ is output from $i$ BERT hidden layer;

  * •

$\gamma$ and $s_{i}$ is trainable task specific parameters.

As we do not fine-tune BERT, we should adapt its outputs for our specific
sequence labeling task. The suggested weighting approach is similar to ELMo
Peters et al. (2018), with a lower number of weighting vectors parameters
$s_{i}$. This approach can help to learn importance of each BERT output layer
for this task and and network doesn’t lose too much information about text,
that was stored in all BERT outputs.

####  3.1.3 Recurrent Part

This part contains two LSTM networks for forward and backward passes with
$512$ hidden units so that the output representation dim is $1024$ for each
token. We use a recurrent layer for learning dependencies between tokens in an
input sequence Hochreiter and Schmidhuber (1996).

####  3.1.4 Multi-Head Attention

After applying the recurrent layer, we use Self-attention mechanism to learn
any other dependencies in a sequence for each token. This can be denoted as
$D(d_{h}|S)$, where $D$ is some hidden dependency; $d_{h}$ is the $h$ head of
attention, and $S$ is all sequence. each head can learn its dependencies such
as morphological, syntactic or semantic relationships between words (tokens).
Presumably, dependencies may look as shown at Figure 2. Also, mechanism
attention can compensate limitations of the recurrent layer when working with
long sequences Bahdanau et al. (2015). In our architecture, we use multihead-
attention block as proposed in the paper “attention is all you need” Vaswani
et al. (2017). We took $6$ heads and value and key dim $64$.

####  3.1.5 Inference for NER Task

After the input sequence was encoded, we achieve the final representation of
each token in a sequence. This representation is passed to Linear layer with
$tanh$ activation function and gets a vector with $14$ dim, that equals to the
number of entities labels (include supporting labels “pad” and “[CLS]”). The
inference layer takes the extracted token sequence representations as features
and assigns labels to the token sequence. As the inference layer, we use
Neural CRF++ layer instead of vanilla CRF. That captures label dependencies by
adding transition scores between neighboring labels. NCRF++ supports CRF
trained with the sentence-level maximum log-likelihood loss. During the
decoding process, the Viterbi algorithm is used to search the label sequence
with the highest probability. But also, NCRF++ extends the decoding algorithm
with the support of $nbest$ output Yang and Zhang (2018). We chose the $nbest$
parameter equal to $11$, because we have $11$ meaningful labels. In this
decision we followed the original article Yang and Zhang (2018).

####  3.1.6 Inference for Language Classification

We train our system for language classification. For the classification
inference, we use Pooling Linear Classifier block as proposed in ULMFiT paper
Howard and Ruder (2018). We pass output sequence representation $H$ from
Multihead-attention part to different Poolings and concat (as shown in Figure
1):

$h_{c}=[h_{0},maxpool(H),meanpool(H)]$ (2)

where $[]$ is concatenation;

$h_{0}$ is first output significant vector of Multihead-attention part (which
does have “[CLS]” label).

The result of concat Pooling ($3\times 1024$) is passed to Linear layer, and
that predicts probability for four language classes (Bulgarian, Czech, Polish
and Russian).

###  3.2 Postprocessing Prediction

After getting labels for the sequence of WordPiece tokens, we should convert
prediction to word level labels extraction named entities. Each WordPiece
token in the word is matched with neural network label prediction. We use
ensemble classifier on labels by count all predicted labels for one word
except “X” and select label for a word with the higher number of votes.

For final prediction we unite token’s sequences which have not “O” (“Other”)
label to spans and write to result of entities set.

##  4 Training the System

###  4.1 Data Conversion

On the training stage we divide the input data into two parts: the training
set (named “brexit”) and development set (named “asia_bibi”). Hence we train
the system on one topic and evaluate the system on another topic. Because the
input contains raw text and annotation, but BERT take words sequence as input,
we convert data to word level IOB markup Ramshaw and Marcus (1995). After
that, each word was tokenized by WordPiece tokenizer and word label matched
with IOBX labels.

On the prediction stage result, labels were received by voice classifier.
After this, we transform word predictions to spans markup. The results of
develop evaluation stage described in Table 1.

After evaluation stage we train our network on all input data (“brexit” and
“asia_bibi”) to make final predictions on the blind test set.

###  4.2 Training Procedure

The proposed neural network was trained with joint loss:

$\mathcal{L}=\mathcal{L}_{SL}+\mathcal{L}_{clf}$ (3)

where $\mathcal{L}_{SL}$ is maximum log-likelihood loss Yang and Zhang (2018)
for the sequence labeling task and $\mathcal{L}_{clf}$ is Cross Entropy Loss
for the language classification.

We use Adam with a learning rate of $1e-4$, $\beta_{1}=0.8$, $\beta_{2}=0.9$,
$L2$ weight decay of $0.01$, learning rate warm up, and linear decay of the
learning rate. Also, gradient clipping was applied for weights with
$clip=1.0$.

Training of proposed neural network architecture was performed on one GPU with
the batch size equal to $16$, the number of epochs equal to $150$, but stopped
at epoch number $80$ because the loss function has ceased to decrease. The
model required only around $3$ GB of memory instead of fine-tuning all BERT
model, which would have required more than $8$ GB GPU memory. All training
procedure lasted around five hours on one GPU with the evaluation of
development set on each epoch.

The final model was trained on unit of training and development datasets.

##  5 Results and Discussion

###  5.1 Evaluation Results

As baseline for BSNLP Shared Task we use a simple CRF tagger and obtain exact
word level f1-score $0.372$ on the development dataset.

Finally we use joint model for named entity recognition task and language
classification task because the model without part of the classification gave
a result by several percent less than proposed final model. This means that
the joint model pays attention to a specific language morphology and some
connections between words within one language.

label | precision | recall | f1-score
---|---|---|---
PER | 0.733 | 0.725 | 0.729
PRO | 0.384 | 0.547 | 0.451
EVT | 0.385 | 0.370 | 0.377
LOC | 0.648 | 0.872 | 0.744
ORG | 0.550 | 0.630 | 0.587
avg/total | 0.540 | 0.629 | 0.578
Table 1:  Evaluation metrics on development dataset

For proposed neural network architecture the evaluation of the training stage
was produced on development dataset. Table 1 shows span-level metrics
precision, recall, and f1-measure. For development set, we obtained the
following scores: language classification quality (f1-score): $0.998$ and
Multilingual Named Entity Recognition quality (f1-score): $0.70$ for exact
word level matching and $0.578$ for exact full entities matching. Also we
train model without language classification, which resulted in f1-score equal
to $0.66$ . This confirms the impact of language classification. Our model
significantly outperforms the CRF baseline.

The evaluation of test dataset presented in Table 2 (relaxed partial matching)
and Table 2 (relaxed exact matching) is measured by the BSNLP Shared Task
organizers.

Relaxed partial matching
---
label | precision | recall | f1-score
PER | 0.84955 | 0.87119 | 0.86023
LOC | 0.77526 | 0.93197 | 0.84642
ORG | 0.62642 | 0.87170 | 0.72898
PRO | 0.42079 | 0.81416 | 0.55483
EVT | 0.24074 | 0.15476 | 0.18841
All | 0.90142 | 0.69917 | 0.78752
Relaxed exact matching
label | precision | recall | f1-score
PER | 0.76835 | 0.74023 | 0.73317
LOC | 0.87747 | 0.73014 | 0.79705
ORG | 0.71390 | 0.52295 | 0.60369
PRO | 0.34439 | 0.18506 | 0.24075
EVT | 0.10714 | 0.16667 | 0.13043
All | 0.56225 | 0.46901 | 0.50102
Table 2:  Evaluation metrics on test dataset

###  5.2 Error Analysis

First of all, we face some errors with converting from origin data format (raw
and annotations) to word markup and back to origin format after predictions
were made. This problems stand for extra spaces, bad Unicode symbols and
symbols, absent in WordPiece vocabulary. Other errors are caused by neural
network prediction failures. The model turns to be overfitted on the negative
label “O” so that there are many false positives in the prediction. Lastly,
the infrequent labels “PRO” and “EVT” are often confused.

##  6 Related Work

The related work has several parts: firstly, our work follows the recent trend
of using pretrained neural languages models, such as Devlin et al. (2018);
Peters et al. (2018); Howard and Ruder (2018). The main difference between
original BERT’s approach for named entity recognition task Devlin et al.
(2018) we use its only as input embeddings of sequence without fine-tuning.
From ELMo paper Peters et al. (2018) we use weighting approach for different
outputs from network and getting final representation of sequence. From ULMFiT
work we took part which is related to the final decoding for classification
(Pooling Classifier) without proposed language model Howard and Ruder (2018).
Secondly we model the task of NER as a joint sequence labeling and
classification task following other joint architectures Liu and Lane (2016);
Nguyen et al. (2016).

##  7 Conclusion and Future Work

We have proposed neural network architecture that solves Multilingual Named
Entity Recognition without any additional labeled data for Bulgarian, Czech,
Polish and Russian languages. This implementation allows to train the model
even on a modern personal computer with GPU. This neural network architecture
can be used for other tasks, that can be reformulated as a sequence labeling
task for any other language.

As the next steps in the study of the underlying architecture, we can increase
or decrease the number of units on each layer or remove the recurrent layer or
multihead-attention layer. As improvements of the system, we can fine-tune
BERT embeddings and put additional layers on top of BERT or pass other modern
language models as an input.

## Acknowledgments

The article was prepared within the framework of the HSE University Basic
Research Program and funded by the Russian Academic Excellence Project
“5-100”. We are thankful to the Muppets and to the BSNLP shared task
organizers.

## References

  * Bahdanau et al. (2015) Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. 2015.  Neural Machine Translation by Jointly Learning to Align and Translate.  _CoRR_ , abs/1409.0473.
  * Devlin et al. (2018) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018.  Bert: Pre-training of Deep Bidirectional Transformers for Language Understanding.  In _NAACL-HLT_.
  * Hochreiter and Schmidhuber (1996) Sepp Hochreiter and Jürgen Schmidhuber. 1996.  Lstm can Solve Hard Long Time Lag Problems.  In _NIPS_.
  * Howard and Ruder (2018) Jeremy Howard and Sebastian Ruder. 2018.  Universal Language Model Fine-tuning for Text Classification.  In _ACL_.
  * Lafferty et al. (2001) John D. Lafferty, Andrew McCallum, and Fernando Pereira. 2001.  Conditional Random Fields: Probabilistic Models for Segmenting and Labeling Sequence Data.  In _ICML_.
  * Liu and Lane (2016) Bing Liu and Ian Lane. 2016.  Attention-Based Recurrent Neural Network Models for Joint Intent Detection and Slot Filling.  In _INTERSPEECH_.
  * Nguyen et al. (2016) Thien Huu Nguyen, Kyunghyun Cho, and Ralph Grishman. 2016.  Joint event extraction via recurrent neural networks.  In _Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , page 00–309, San Diego, California. Association for Computational Linguistics.
  * Peters et al. (2017) Matthew E. Peters, Waleed Ammar, Chandra Bhagavatula, and Russell Power. 2017.  Semi-supervised sequence tagging with bidirectional language models.  In _ACL_.
  * Peters et al. (2018) Matthew E. Peters, Mark Neumann, Mohit Iyyer, Matt Gardner, Christopher Clark, Kenton Lee, and Luke S. Zettlemoyer. 2018.  Deep contextualized word representations.  In _NAACL-HLT_.
  * Piskorski et al. (2019) Jakub Piskorski, Laska Laskova, Michał Marcińczuk, Lidia Pivovarova, Pavel Přibáň, Josef Steinberger, and Roman Yangarber. 2019.  The second cross-lingual challenge on recognition, classification, lemmatization, and linking of named entities across Slavic languages.  In _Proceedings of the 7th Workshop on Balto-Slavic Natural Language Processing_ , Florence, Italy. Association for Computational Linguistics.
  * Ramshaw and Marcus (1995) Lance A. Ramshaw and Mitchell P. Marcus. 1995.  Text Chunking using Transformation-Based Learning.  _CoRR_ , cmp-lg/9505040.
  * Sathiya and Sellamanickam (2007) Keerthi Sathiya and Sundararajan Sellamanickam. 2007.  _Crf versus svm-struct for sequence labeling_ , volume 1.  Yahoo Research Technical Report.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017.  Attention Is All You Need.  In _NIPS_.
  * Wu and Mike Schuster (2016) Yonghui Wu and Quoc VLe Mohammad Norouzi Wolfgang Macherey Maxim Krikun Yuan Cao Qin Gao Klaus Macherey Mike Schuster, Zhifeng Chen. 2016.  _Google’s neural machine translation system: Bridging the gap between human and machine translation_ , volume arXiv:1609.08144.
  * Yang and Zhang (2018) Jie Yang and Yue Zhang. 2018.  Ncrf++: An Open-source Neural Sequence Labeling Toolkit.  In _ACL_.


# Melting of DNA in confined geometry

Arghya Maity<EMAIL_ADDRESS>Navin Singh
<EMAIL_ADDRESS>Department of Physics, BITS-Pilani, Pilani
campus, India-333031

###### Abstract

The stability of DNA molecule during the encapsulation process is a topic of
intense research. We study the thermal stability of the double-stranded DNA
molecule of different lengths in a confined space. Using a statistical model
we evaluate the melting profile of DNA of different length in two geometries:
conical and cylindrical. Our results show that not only the confinement but
also the geometry of the confined space plays a prominent role in the
stability and opening manner of the molecule.

###### pacs:

87.14.gk, 87.15.Zg, 87.15.A-

##  I Introduction

Deoxyribonucleic acid (DNA) is one of the interesting and complex
biomolecules. The most abundant conformation of this molecule is a double-
helical structure (also known as the Watson-Crick double helix). The genetic
information of the entire organism is coded in the sequence of four
nucleotides Adenine (A), Guanine (G), Cytosine (C), Thymine (T) Watson and
Crick (1953). It has been observed that on increasing the temperature of the
solution containing a DNA molecule, the double-strand takes a conformational
change into the single-stranded configuration. This is known as DNA melting or
denaturation Zhang  _et al._ (1997); Williams (2016); Vologodskii and Frank-
Kamenetskii (2018); Frank-Kamenetskii and Prakash (2014a); Poland and Scheraga
(1966); Fisher (1984); Richard and Guttmann (2004). The same can be achieved
by changing the pH of the solution as well as by pulling either of the strands
by a mechanical device Sadhukhan and Bhattacharjee (2014); Hatch  _et al._
(2008); Kumar and Li (2010). The importance of this molecule lies in its wide
applications in the molecular motor, DNA computer, origami formation, DNA
chip, and biomedicine, etc. To perform gene therapy Jour Putnam (2006),
nanorobotics Yurke  _et al._ (2000), diagnostics Dharmadi and Gonzalez (2004)
performance, DNA degradation is a major obstacle for the effective and
efficient use of DNA. DNA degradation eventuates in chemical breakdown Jour
Putnam (2006) or sometimes through mechanical forces Murphy  _et al._ (2006).
In gene therapy technique DNA is protected by a physical barrier. Many
different techniques have to get better results like complexation with
polycations Jour Putnam (2006), charged copolymers of different architecture,
cationic lipids or liposomes de Lima  _et al._ (2001). As other options, DNA
can be confined within gel Goh  _et al._ (2004); des Rieux  _et al._ (2009),
polymeric nanocapsules(micelles) Haladjova  _et al._ (2012); Csaba  _et al._
(2005), microparticles. There are many elegant and versatile approach for DNA
encapsulation Dimitrov  _et al._ (2011).

In DNA encapsulation, the ability to preserve the DNA and efficiently release
it, are the processes which are inversely related. The objective of good DNA
encapsulation is to find an optimal balance between these two issues.
Researchers are trying to balance it in numerous ways; one of them is short
DNA encapsulated by a spherical inorganic nanoshell with an overall thickness
of $\sim$10 nm Park  _et al._ (2010). Carbon nanotubes also have been proved
itself as a potential candidate for DNA encapsulation Cruz  _et al._ (2014).
The thermodynamical spontaneity of DNA encapsulation of carbon nanotube under
different conditions is still a significant area of research. The threshold
diameter of this tube is also a vital issue to investigate since below the
threshold encapsulation is inhibited Cruz  _et al._ (2015). Many sensitive
parameters are involved in DNA encapsulation techniques like the medium and
topology of the carrier, thermodynamic parameters, etc.

In vivo, the DNA molecule is confined in a limited space such as the cell
chamber or a channel and is in highly dense solvent conditions Li  _et al._
(2014); Kumar  _et al._ (2017); Turner  _et al._ (2002); Singh and Singh
(2017). This confinement restricts the conformation and movement of DNA
molecules in the cell. The thermodynamic properties of DNA molecules highly
depend on the confined space Akabayov  _et al._ (2013). It is known that
conformational properties of biopolymers under confinement are the crucial
relevance in living systems like DNA packing in eukaryotic chromosomes, viral
capsids, etc. Kumar  _et al._ (2011).

In most of the studies, either rectangular or spherical geometry Li  _et al._
(2014) is considered as confinement. Despite recent progress, still little is
known about the confinement nature of DNA Wanunu (2012). In the present
manuscript, motivated by all these different kinds of experiments in different
geometries, we attempt to understand the thermal stability of a DNA molecule
that is confined in different geometries through a statistical model. The
present manuscript is an following work of our recent published research work
Maity  _et al._ (2019). It helps to understand confinement effect more
immensely. The present work is divided into the following sections: with a
brief introduction to the statistical model in Sec. II, we discuss the melting
of DNA in confined geometry in Sec. IV. The paper ends with a brief discussion
and future scope of the work in Sec. V.

##  II Model and Methods

To investigate the effect of confinement on the thermal denaturation of DNA
molecule, we adopt the well known Peyrard-Bishop-Dauxois model (PBD). The
model is quasi-one-dimensional and describes the motion of the molecules
through the stretching of the hydrogen bonds between the bases in a pair
Peyrard and Bishop (1989); Dauxois  _et al._ (1993). The model underestimates
the entropy associated with the different conformations of the molecule Frank-
Kamenetskii and Prakash (2014b). In the linear form of the model, it ignores
the effect of helicoidal nature of the molecule as well as the solvent effect
of the solution. Despite of these shortcomings, the model still has enough
details to describe the denaturation/unzipping process of DNA molecule. The
interactions in the DNA, containing $N$ base pairs, are represented as,

$H=\sum_{i=1}^{N}\left[\frac{p_{i}^{2}}{2m}+V_{M}(y_{i})\right]+\sum_{i=1}^{N-1}\left[W_{S}(y_{i},y_{i+1})\right],$
(1)

here $y_{i}$ represents the stretching from the equilibrium position of the
hydrogen bonds. First term of the model is the momentum term which is
$p_{i}=m\dot{y}_{i}$. The reduced mass, $m$, is considered the same for both
$AT$ and $GC$ base pairs. The interaction between the nearest base pairs along
the chain, the stacking interaction, is represented by,

$W_{S}(y_{i},y_{i+1})=\frac{k}{2}(y_{i}-y_{i+1})^{2}[1+\rho
e^{-b(y_{i}+y_{i+1})}].$ (2)

The single strand elasticity is represented by $k$, while the anharmonicity in
the strand elasticity is represented by $\rho$. The parameter, $b$, describes
the range of anharmonicity. For our studies, we choose model parameters $k$ =
0.015 eVÅ-2 , $\rho=50.0$, $b$ = 0.35 Å-1 for the current investigation
Theodorakopoulos (2010). In past it has been shown that value of $k$ and
$\rho$ defines the sharpness in the transition from doble strand to single
strand Cocco and Monasson (1999); Singh and Singh (2005). The hydrogen bonding
between the two bases in the $i^{\rm th}$ pair is represented by the Morse
potential as,

$V_{M}(y_{i})=D_{i}(e^{-a_{i}y_{i}}-1)^{2}.$ (3)

where $D_{i}$ represents the potential depth, and $a_{i}$ represents the
inverse of the width of the potential well. These two parameters have a
crucial role in DNA denaturation. The dissociation energy, $D_{i}$, is a
representation of the hydrogen bond energy that binds the $A−T$ and $G−C$
pairs while $a_{i}$ represents bond stiffness. From previous results, we know
that the bond strengths of these two pairs are in an approximate ratio of
$1.25$ \- $1.5$ as the GC pairs have three while AT pairs have two hydrogen
bondsAres  _et al._ (2005); Weber  _et al._ (2009); Zoli (2011, 2012, 2014);
Tapia-Rojo  _et al._ (2010); Bergues-Pupo  _et al._ (2014a, b); Frank-
Kamenetskii and Prakash (2014b); Macedo  _et al._ (2014). The potential
parameters are taken as $a_{AT}=4.2\;{\rm\AA^{-1}},a_{GC}=1.5\times a_{AT}$
and $D_{AT}$ = 0.075 eV while, $D_{GC}=1.5\times D_{AT}$.

Table 1: Set of model parameters: The model parameters are tuned in such a way that the melting temperature of 120 base pairs chain in the free environment is $\sim$300 K. We find the $T_{m}^{free}=315.38$ K with these parameters for the free chain. $D_{\rm AT}$ | $D_{\rm GC}$ | $a_{\rm AT}$ | $a_{\rm GC}$ | $\rho$ | $\kappa$ | $b$
---|---|---|---|---|---|---
(eV) | (eV) | (${\rm\AA^{-1}})$ | (${\rm\AA^{-1}}$) | (-) | (${\rm eV/\AA^{2}}$) | (${\rm\AA^{-1}}$)
0.075 | $1.5D_{\rm AT}$ | 4.2 | $1.5a_{\rm AT}$ | 50.0 | 0.015 | 0.35

We can study the thermodynamics of the transition by evaluating the partition
function. For a sequence of $N$ base pairs, the canonical partition function
can be written as Singh and Singh (2005):

$Z=\int\prod_{i=1}^{N}\left\\{dy_{i}dp_{i}\exp(-\beta H)\right\\}=Z_{p}Z_{c},$
(4)

where $Z_{p}$ corresponds to the momentum part of the partition function while
the $Z_{c}$ contributes as the configurational part of the partition function.
The momentum part is nothing but $(2\pi mk_{B}T)^{N/2}$. The configurational
part of the partition function, $Z_{c}$, is defined as Singh and Singh (2005),

$Z_{c}=\int\left[\prod_{i=1}^{N-1}dy_{i}K(y_{i},y_{i+1})\right]dy_{N}K(y_{N})$
(5)

where $K(y_{i},y_{i+1})=\exp\left[-\beta H_{c}(y_{i},y_{i+1})\right]$. For the
homogeneous chain, one can evaluate the partition function by transfer
integral (TI) method by applying the periodic boundary condition Zhang  _et
al._ (1997). For the homogeneous chain, one can evaluate the partition
function using the transfer integral (TI) method by applying the periodic
boundary condition. For a chain with random sequence of $AT$ and $GC$ pairs
and open boundaries, the calculation of partition function is little bit
tricky. In past, various researchers have addressed the issue of solving the
partition function of the chain with heterogeneous sequence and open
boundaries Cule and Hwa (1997); Zhang  _et al._ (1997); Srivastava and Singh
(2011); Campa and Giansanti (1998); Singh and Singh (2013). In addition to
these issues, the partition function in the PBD model is divergent in nature.
To overcome this problem, an upper cut-off for the integration needs to set up
Zhang  _et al._ (1997); van Erp  _et al._ (2006); Dauxois and Peyrard (1995);
Zhang  _et al._ (1997); Singh and Singh (2015). In our previous studies, we
found that an integration limit of -5.0 ${\rm\AA}$ to 200 ${\rm\AA}$ is
sufficient to overcome the divergence issue of the partition function. Once
the proper cut-offs are introduced, the task is to discretise the integral in
Eq.5. In order to get a precise value of melting temperature ($T_{m}$) we have
observed that Gaussian quadrature is the most effective quadrature. We have
found that discretization of the space with 900 points is sufficient to get an
accurate value of $T_{m}$ Singh and Singh (2015); Zhang  _et al._ (1997). Once
we are able to evaluate the partition function, we can determine the
thermodynamic quantities of interest by evaluating the Helmholtz free energy
of the system. The Helmholtz free energy per base pair is defined as,

$f(T)=-\frac{1}{2}k_{B}T\ln\left(2\pi mk_{B}T\right)-\frac{k_{B}T}{N}\ln
Z_{c}$ (6)

The specific heat, $C_{v}$, of the system, in the thermal ensemble, is
evaluated by taking the second derivative of the free energy, as
$C_{v}=-T(\partial^{2}f/\partial T^{2})$. The melting temperature ($T_{m}$) of
the chain is evaluated from the peak in the specific heat. Another important
quantity of interest is the average separation, $\langle y_{j}\rangle$, of the
$j^{th}$ pair of the chain which is given by,

$\langle y_{j}\rangle=\frac{1}{Z}\int\prod_{i=1}^{N}y_{j}\exp(-\beta H)dy_{i}$
(7)

##  III Introducing the Confinement in The Model

In this work, we study the effect of confined geometries on the melting
profile of $phage-\lambda$(GenBank: J02459.1) DNA chain of first 120 base
pairs sequence. In past, researchers have studied the effect of rectangular
confinement on the melting profile of the DNA molecules Mogurampelly and Maiti
(2013); Reiter-Schad  _et al._ (2015). In this manuscript our focus is on
effect of conical as well as cylindrical geometry (as shown in Fig.1) on the
thermal stability of the DNA molecule.

Figure 1:  (a) The schematic representation of the DNA molecule in a confined
cell of conical shape. The distance $x$ represents the distance between a pair
and the confining wall while $\theta$ is the angle at which the cellular wall
is suppose to confine the DNA. Here $z=3.4\times i$, the distance along
$x$-axis. (b) The schematic representation of the DNA molecule in a confined
cell of cylindrical geometry. The $r$ is the distance of the confined wall
from the DNA strand. The radius of the cylinder becomes $R_{c}$ = $r$ \+ DNA
radius (10 ${\rm\AA}$).

How the confinement can be realized in the PBD model? We adopt the following
scheme. We restrict the configuration space of the system as shown in Fig. 1.
There is no change in the lower limit of integral for both the geometry,
however, the upper limit of integral for each base pair is modified according
to the distinct geometry. For the conical confined geometry, the upper limit,
$x$ is evalauted as, $x=3.4\times i\times\tan(\theta)$ where $i$ is the site
index ($i=1,2,3,...,N$), and $N$ is the total number of base pairs in the
chain. The term $\theta$ is the angular separation between the confining wall
and the DNA strand. Since two base pairs are approximately 3.4 ${\rm\AA}$
distance apart we take this as the multiplication factor in calculation. Thus,
the configurational space for our calculation extends from -5 ${\rm\AA}$ to
$x\;{\rm\AA}$. There is another important factor that affects the melting
transition is the cross section of the pore. The cross section of the
confining wall (or pore width) is represented by the term, $\delta$. The value
of $\delta=0\;{\rm\AA}$ means the distance between the vortex and the DNA
strand is 0 ${\rm\AA}$ (as shown in Fig. 1a). For the cylindrical geometry the
upper limit is defined as $r$. The quantity $r$ denotes the distance between
the confined wall and the DNA strand. Using the modified scheme we calculate
the partition function and hence evaluate all the thermodynamical properties
of the system for both geometries.

##  IV Results

We consider the first 120 bp of the phage-$\lambda$ DNA chain sequence and
restrict it according to the confinement geometry. By evaluating the partition
function, we calculate free energy per base pair hence specific heat as a
function of temperature. The melting temperature $T_{m}$ of chain at various
angular separations($\theta$) for conical confined geometry and in different
diameter of the cylinder for cylindrical confined geometry are calculated
through the peak in specific heat as a function of temperature Zhang  _et al._
(1997); Maity  _et al._ (2017). The results are plotted in Fig. 2. The
obtained results clearly show that the confinement has a strong effect on the
stability of the molecule. We find that the melting temperature ($T_{m}$) of
dsDNA increases as the confinement becomes stronger. We vary the anglular
separation and found that $T_{m}$ decreases as the angle changes from $5^{0}$
to $6^{0}$. The melting temperature for an angular separation of $5^{0}$ is
found to be 349.7 K while it reduces to 347 K for an angle of $6^{0}$, i.e. a
reduction of approximately $2.7$ K in the melting temperature. The melting
temperature further reduces for the higher angular separations and get
saturated at an angle of $\approx 15^{0}$ (342K). The stability of DNA
confinement in a cylindrical symmetric wall has been investigated by various
researchers using Poland Scheraga model Reiter-Schad  _et al._ (2015); Werner
_et al._ (2015).

Figure 2: (on left)The change in the melting temperature with the change in
the angular separation between the confining wall and DNA molecule. The pore
width $\delta=0\;{\rm\AA}$. Figure shows that beyond a certain angle,
$\theta_{c}$ there is no effect of confining wall on the melting temperature,
$T_{m}$ of the molecule. (on right) The DNA is confined in a cylinder and the
plot shows the change in melting temperature with radius of the cylinder.

To investigate the effect of geometries on the stability of the DNA molecule,
we also consider a DNA chain confined in a cylindrical confinement. We change
the radius of the cylinder ($R_{c}$ = $r$ \+ DNA radius) and calculate the
change in the melting temperature of the system. The results are plotted in
Fig.2. When the DNA is confined in a cylinder at $r=10\;{\rm\AA}$, we find the
melting temperature of the system as $\sim 359.6$ K. The melting temperature
reduces drastically to a value of $\sim 350$ K for the radial separation of
$11\;{\rm\AA}$(reduce to 9.6K). On further increasing the separation the
$T_{m}$ reduces and almost get saturated at value of $\sim 318$ K. Since the
two geometries supress the entropy of the system in a different way, one can
believe that DNA which is confined in cylindrical geometry is more stable than
the DNA that is confined in conical geometry upto a certain separation. The
point to note that while for shorter separation stability is higher for
cylindrical confinement but it is lower at higher separation (please see Fig.
2 for a detail comparision).

Figure 3: Relative $T_{m}$ shift follows as $\theta_{c}^{-v}$ and
$R_{c}^{-v}$, where $v$ is 0.296 for the conical confinement and 4.3 for the
cylindrical confinement. For the cylindrical confinement, the simulation
result of reference Li  _et al._ (2014) is also plotted where $v$ is 1.323.
The exponential index $v$ is achieved by fitting the straight lines.

We also compare our results with the results obtained by Huaping et al Li  _et
al._ (2014). They have calculated the relative change in melting temperature,
$\frac{T_{m}-T_{m}^{free}}{T_{m}}$, with the radius of cyliner, $R_{c}$. With
the help of numerical simulation, they found the relative change in $T_{m}$
scales with the radius of cylinder as $R_{c}^{-v}$ with $v=1.323$. To compare
our results, we also calculate the relative change in $T_{m}$ with the change
in the radius of cylinder ($R_{c}$) and the angular separation of the conical
confinement ($\theta_{c}$). The obtained results are plotted in the figure 3.
Our results show that the relative change in $T_{m}$ scales with the radius of
cylinder as $R_{c}^{-v}$ with $v=4.3$ for cylindrical confinement while
$v=0.296$ for conical confinement. Although the our system is little different
in terms of sequence and chain length, our results are also showing similar
power law behaviour with different exponent.

To understand more about the microscopic details of opening of the DNA chain
in a confined space, we calculate the average separation of the chain,
$\langle y_{i}\rangle$, with the change in the angular separation ($\theta$).
From the opening profile, the effect of confinement angle is more apparent.
When the angular separation is $5^{0}$, the effect lasts for $\sim$25 base
pairs while for the $10^{0}$ angular separation, the effect lasts for $\sim$
base pairs.

Figure 4:  The opening of the chain in different angular
separation($\theta$=$5^{0}$& $10^{0}$). When the angular separation is
$5^{0}$, the effect lasts for $\sim$25 base pairs while for the $10^{0}$
angular separation, the effect lasts for $\sim$20 base pairs at 380K.

How the different sections of the chain open below the melting temperature is
also the matter of interest here. From the plot (fig. 4), the sequence and
effect of conical confinement are visible. The weaker section of the chain
consists of AT pairs is more entropic than the section containing GC pairs.
While the opening of the homogeneous chain is very smooth, the opening of the
heterogeneous chain has bubbles. One can see that how these bubbles disappear
above the melting temperature of the chain when the DNA is in the denatured
state.

Depending on the geometry the opening profile differs. The opening profile of
the chain in these two different geometry are shown through 3D plots in Fig.5
which shows the variation of $\langle y_{i}\rangle$ with temperature and base
pair.

Figure 5:  The opening profile of the chain is shown through the 3D plots
which shows a variation in $\langle y_{j}\rangle$ for the 120bp chain. Here we
show the profile for two different angular separation, $\theta=5^{0}$ and
$\theta=10^{0}$ (for pore width $\delta=0\;{\rm\AA}$) and for the cylindrical
geometry $r=20\;{\rm\AA}$. The $x$-axis is site index while $y$-axis is
temperature (in K) and $z$-axis denotes the average separation, $\langle
y_{j}\rangle$ in ${\rm\AA}$. Figure 6:  How the effect of confinement changes
with the linear size of the molecule is shown here. We consider the phage-
lambda DNA molecule sequence of different lengths as 120, 180, 240 and 300.
The pore diameter $\delta=0\;{\rm\AA}$.

We also investigate the effect of confinement on different sized DNA molecule
by varying the chain size. We consider the chain sequence of 120, 180, 240,
and 300 base pairs of the phage-$\lambda$ DNA sequence and investigate the
variation in melting temperature with varying angular separation. The
resulting plots are shown in Fig. 6.The qualitative feature of all the plots
are same, however, the decrease in melting temperature for all the chains is
not same. For the chain of 120 bp, the $T_{m}$ is 349.7 K for an angular
separation of $5^{0}$ which reduces to 347 K for and angle of $6^{0}$. The
asympototic value of $T_{m}$ is 342 K which occur at $\approx 15^{0}$ of
angular separation. This means for a span of $10^{0}$ ($15^{0}-5^{0}=10^{0}$)
the variation in melting temperature vanishes and the decrease in $T_{m}$ is
$\approx$5 K($349.7-342=7.7$ K). For the 180 bp chain,this shift is about 3.3
K ($337.3-334=3.3K$) for a span of $7^{0}$ ($12^{0}-5^{0}=7^{0}$) as it gets
the asympototic value of $T_{m}$ at $12^{0}$, while, for 300 bp chain the
melting temperature reduces from 329 K (at $5^{0}$) to 328 K (at $9^{0}$) so
the shift of $T_{m}$ is $1$ K ($329-328=1$ K). One can clearly see that as the
length of the chain increases the change in melting temperature with angle is
reducing. For chain of length more than 350 base pairs, we observe that there
is no change in the melting temperature with the change in the angular
separation. The probable arugement for the small variation is the entropy of
the system that varies with size. The smaller the chain, the stronger the
suppression of entropy. For a bigger chain, a large section of the chain might
not be seeing the confining wall because the distance between the wall and the
DNA chain increases (please refer to Fig.01 a).

Next, we study the thermal stability of the system with the change in the pore
width,$\delta$ for conical geometry. Here we calculate the change in the
melting temperature with the change in the pore width, $\delta$ for a fix
value of $\theta$ (the angular separation). The results are shown in Fig. 7.

Figure 7:  (On left) The change in the melting temperature ($T_{m}$) with the
pore width ($\delta$) for a fix value of angular separation ($\theta=5^{0}$).
(On right) The change in melting temperature ($T_{m}$) with the confinement
angle ($\theta$) for two values of the pore width ($\delta$).

From the Fig. 7 it is clear that with the increase in pore width ($\delta$)
for a same value of angular separation ($\theta$), the melting temperature
decreases. This is due to fact that when the pore width increases the end at
the vortex of the cone gets more space which increases the end entropies.
While for the lower pore diameter, only one of the ends of the DNA has
sufficient entropy to contribute to the opening of DNA molecule, for higher
values of pore diameter, the other end also contribute to the opening of the
molecule. Thus, the DNA molecule now melts at lower temperature.

##  V Conclusions

In this manuscript, we have studied, theoretically, the stability of
phage-$\lambda$ DNA molecule of different lengths that are confined in conical
as well as cylindrical geometry. Probably this is the first model-based study
that studies the impact of angular confinement on the stability of the DNA
molecule in a thermal ensemble. Our studies show that not only the confinement
but also the geometry of the confinement plays a crucial role in the stability
as well as the overall activity of the DNA molecule.

When the confining wall becomes closer, the configuration space for the DNA
molecule, in cylindrical geometry, is restricted, which suppresses the entropy
of the chain. Hence the melting temperature of the system in cylindrical
geometry is more than the melting temperature of the DNA that is confined in
conical geometry up to a certain extends. If we compare two cases of the
confinement (conical and cylindrical) and find the rate of decrease in the
$T_{m}$, then we see that the DNA confined in cylindrical geometry has a
higher decrease rate. Through simulation of the short DNA molecule that is
confined in a cylindrical as well as in spherical geometry, Huaping et al. Li
_et al._ (2014) has found that the melting temperature of the system decreases
with the increasing diameter of the cylinder/sphere. We have found through the
PBD model that the relative change in $T_{m}$ scales with the radius of
cylinder as $R_{c}^{-v}$ with $v=4.3$ for cylindrical confinement while
$v=0.296$ for conical confinement. Since our system is little different in
terms of sequence and chain length than the Huaping et al, our results are
also showing similar power law behaviour with different exponent.

Our findings may attract experimentalists to design the shells of different
geometries and investigate the effect of the geometry of the shell on the
stability of encapsulated DNA. In DNA encapsulation, the ability to preserve
the DNA and efficiently release it, are the processes which are inversely
related. The objective of good DNA encapsulation is to find an optimal balance
between these two issues. Researchers are trying to balance these two issues
to find a potential material for DNA encapsulation.

## Acknowledgement

We would like to acknowledge M. Peyrard (Lyon), Amar Singh (CCB, University of
Kansas, USA), and Y. Singh (BHU) for useful discussions and drawing our
attention to some useful papers. We appreciate the financial support from
Department of Science and Technology, New Delhi.

## References

  * Watson and Crick (1953) J. D. Watson and F. Crick, Nature 171, 737 (1953).
  * Zhang  _et al._ (1997) Y.-l. Zhang, W.-M. Zheng, J.-X. Liu, and Y. Z. Chen, Phys. Rev. E 56, 7100 (1997).
  * Williams (2016) M. C. Williams, Acta Crystallographica Section D 72, 599 (2016).
  * Vologodskii and Frank-Kamenetskii (2018) A. Vologodskii and M. D. Frank-Kamenetskii, Physics of Life Reviews 25, 1 (2018).
  * Frank-Kamenetskii and Prakash (2014a) M. D. Frank-Kamenetskii and S. Prakash, Physics of Life Reviews 11, 153 (2014a).
  * Poland and Scheraga (1966) D. Poland and H. A. Scheraga, J.Chem.Phys 45, 1456–1463 (1966).
  * Fisher (1984) M. E. Fisher, Journal of Statistical Physics 34, 667 (1984).
  * Richard and Guttmann (2004) C. Richard and A. J. Guttmann, Journal of Statistical Physics 115, 925 (2004).
  * Sadhukhan and Bhattacharjee (2014) P. Sadhukhan and S. M. Bhattacharjee, Indian Journal of Physics 88, 895 (2014).
  * Hatch  _et al._ (2008) K. Hatch, C. Danilowicz, V. Coljee, and M. Prentiss, Nucleic Acids Research 36, 294 (2008).
  * Kumar and Li (2010) S. Kumar and M. S. Li, Physics Reports 486, 1 (2010).
  * Jour Putnam (2006) D. Jour Putnam, Nature Materials 5, 1476 (2006).
  * Yurke  _et al._ (2000) B. Yurke, A. J. Turberfield, A. P. M. Jr, F. C. Simmel, and J. L. Neumann, Nature 406, 605 (2000).
  * Dharmadi and Gonzalez (2004) Y. Dharmadi and R. Gonzalez, Biotechnology Progress 20, 1309 (2004).
  * Murphy  _et al._ (2006) J. C. Murphy, T. Cano, G. E. Fox, and R. C. Willson, Biotechnology Progress 22, 519 (2006).
  * de Lima  _et al._ (2001) M. C. P. de Lima, S. Simões, P. Pires, H. Faneca, and N. Düzgüneş, Advanced Drug Delivery Reviews 47, 277 (2001).
  * Goh  _et al._ (2004) S. L. Goh, N. Murthy, M. Xu, and J. M. J. Fréchet, Bioconjugate Chemistry 15, 467 (2004).
  * des Rieux  _et al._ (2009) A. des Rieux, A. Shikanov, and L. D. Shea, Journal of Controlled Release 136, 148 (2009).
  * Haladjova  _et al._ (2012) E. Haladjova, S. Rangelov, C. B. Tsvetanov, and S. Pispas, Soft Matter 8, 2884 (2012).
  * Csaba  _et al._ (2005) N. Csaba, P. Caamaño, A. Sánchez, F. Domínguez, and M. J. Alonso, Biomacromolecules 6, 271 (2005).
  * Dimitrov  _et al._ (2011) I. V. Dimitrov, E. B. Petrova, R. G. Kozarova, M. D. Apostolova, and C. B. Tsvetanov, Soft Matter 7, 8002 (2011).
  * Park  _et al._ (2010) D.-H. Park, J.-E. Kim, J.-M. Oh, Y.-G. Shul, and J.-H. Choy, Journal of the American Chemical Society 132, 16735 (2010).
  * Cruz  _et al._ (2014) F. J. A. L. Cruz, J. J. de Pablo, and J. P. B. Mota, The Journal of Chemical Physics 140, 225103 (2014).
  * Cruz  _et al._ (2015) F. J. Cruz, J. J. de Pablo, and J. P. Mota, Biochemical Engineering Journal 104, 41 (2015).
  * Li  _et al._ (2014) H. Li, Z. Wang, N. Li, X. He, and H. Liang, The Journal of Chemical Physics 141, 044911 (2014).
  * Kumar  _et al._ (2017) S. Kumar, S. Kumar, D. Giri, and S. Nath, Europhysics Letters 118, 28001 (2017).
  * Turner  _et al._ (2002) S. W. P. Turner, M. Cabodi, and H. G. Craighead, Phys. Rev. Lett. 88, 128103 (2002).
  * Singh and Singh (2017) A. Singh and N. Singh, Phys. Chem. Chem. Phys. 19, 19452 (2017).
  * Akabayov  _et al._ (2013) B. Akabayov, S. Akabayov, S. Lee, G. Wagner, and C. Richardson, Nature Communications 4 (2013).
  * Kumar  _et al._ (2011) H. Kumar, Y. Lansac, M. A. Glaser, and P. K. Maiti, Soft Matter 7, 5898 (2011).
  * Wanunu (2012) M. Wanunu, Physics of Life Reviews 9, 125 (2012).
  * Maity  _et al._ (2019) A. Maity, A. Singh, and N. Singh, EPL (Europhysics Letters) 127, 28001 (2019).
  * Peyrard and Bishop (1989) M. Peyrard and A. R. Bishop, Phys. Rev. Lett. 62, 2755 (1989).
  * Dauxois  _et al._ (1993) T. Dauxois, M. Peyrard, and A. R. Bishop, Phys. Rev. E 47, R44 (1993).
  * Frank-Kamenetskii and Prakash (2014b) M. D. Frank-Kamenetskii and S. Prakash, Physics of Life Reviews 11, 153 (2014b).
  * Theodorakopoulos (2010) N. Theodorakopoulos, Phys. Rev. E 82, 021905 (2010).
  * Cocco and Monasson (1999) S. Cocco and R. Monasson, Phys. Rev. Lett. 83, 5178 (1999).
  * Singh and Singh (2005) N. Singh and Y. Singh, The European Physical Journal E 17, 7 (2005).
  * Ares  _et al._ (2005) S. Ares, N. K. Voulgarakis, K. Ø. Rasmussen, and A. R. Bishop, Phys. Rev. Lett. 94, 035504 (2005).
  * Weber  _et al._ (2009) G. Weber, N. Haslam, J. W. Essex, and C. Neylon, Journal of Physics: Condensed Matter 21, 034106 (2009).
  * Zoli (2011) M. Zoli, The Journal of Chemical Physics 135, 115101 (2011).
  * Zoli (2012) M. Zoli, Journal of Physics: Condensed Matter 24, 195103 (2012).
  * Zoli (2014) M. Zoli, Journal of Theoretical Biology 354, 95 (2014).
  * Tapia-Rojo  _et al._ (2010) R. Tapia-Rojo, J. J. Mazo, and F. Falo, Phys. Rev. E 82, 031916 (2010).
  * Bergues-Pupo  _et al._ (2014a) A. Bergues-Pupo, J. Bergues, and F. Falo, Physica A 396, 99 (2014a).
  * Bergues-Pupo  _et al._ (2014b) A. E. Bergues-Pupo, F. Falo, and A. Fiasconaro, EPL (Europhysics Letters) 105, 68005 (2014b).
  * Macedo  _et al._ (2014) D. Macedo, I. Guedes, and E. Albuquerque, Physica A 404, 234 (2014).
  * Cule and Hwa (1997) D. Cule and T. Hwa, Phys. Rev. Lett. 79, 2375 (1997).
  * Srivastava and Singh (2011) S. Srivastava and N. Singh, The Journal of Chemical Physics 134, 115102 (2011).
  * Campa and Giansanti (1998) A. Campa and A. Giansanti, Phys. Rev. E 58, 3585 (1998).
  * Singh and Singh (2013) A. Singh and N. Singh, Physica A 392, 2052 (2013).
  * van Erp  _et al._ (2006) T. van Erp, S. Cuesta-López, and M. Peyrard, The European Physical Journal E 20, 421 (2006).
  * Dauxois and Peyrard (1995) T. Dauxois and M. Peyrard, Phys. Rev. E 51, 4027 (1995).
  * Singh and Singh (2015) A. Singh and N. Singh, Phys. Rev. E 92, 032703 (2015).
  * Mogurampelly and Maiti (2013) S. Mogurampelly and P. K. Maiti, The Journal of Chemical Physics 138, 034901 (2013).
  * Reiter-Schad  _et al._ (2015) M. Reiter-Schad, E. Werner, J. O. Tegenfeldt, B. Mehlig, and T. Ambjörnsson, The Journal of Chemical Physics 143, 115101 (2015).
  * Maity  _et al._ (2017) A. Maity, A. Singh, and N. Singh, European Biophysics Journal 46, 33 (2017).
  * Werner  _et al._ (2015) E. Werner, M. Reiter-Schad, T. Ambjörnsson, and B. Mehlig, Phys. Rev. E 91, 060702 (2015).


17). We have validated that the $w_{m+}$ measurements for the same samples
show a signal that is significant with respect to the errors. However, the
amplitude of this signal is relatively low compared to our measurements for
red galaxies, which can be expected from the $\sim 50\%$ fraction of blue
galaxies that the simulation predicts for the DES-like samples (Table 2). A
possible reason for the non-detection of the II and GI terms in MICE could be
that the angular shear statistics $\xi_{\pm}$ is less sensitive to IA
contaminations than the projected statistics $w_{m+}$, as it probes the
integrated IA signal over a wide range of redshifts, including galaxy pairs
with large physical separations that are only weakly intrinsically aligned.
The resulting decrease in the signal-to-noise ratio could lead to a null
detection for the IA terms in the angular statistics.

A weak IA signal in angular statistics lines up with findings of Wei et al.
(2018), who study IA contaminations in mock observations of the weak lensing
surveys KiDS and DLS using a semi-analytic model to implement IA in the Elucid
simulation. In contrast to our results these authors find a weak but still
significant IA contribution to the GG signal. One potential reason for that
difference may be the use of constraints on galaxy-halo misalignment angles
that were derived from LRGs for the entire KiDS sample in that work. This
could result in an overly high signal, as LRGs show the strongest alignment
signal compared to other galaxy populations (see e.g. Fig. 15).

Another reason for the weak alignment in the DES-like samples could be the
fact that predictions for the alignment of dim galaxies (which constitute a
significant part of the sample) are presumably affected by the relatively low
mass resolution, causing noise in the measured host halo orientations and
hence decreasing the predicted galaxy alignment (see Appendix A). However, a
weak contribution of IA in DES-like samples lines up with recent findings from
Secco & Samuroff et al., (2022a), who derive constraints on IA model
parameters from the cosmological analysis of the cosmological weak lensing
signal and find those to be consistent with zero.

We stress here that the IA predictions from the MICE simulation for DES-like
samples need to be taken with caution since the simulation has only been
tested directly against IA observations from red galaxies at redshift that are
well below those probed by DES. Furthermore, the DES samples contain a high
fraction of blue galaxies, in particular at high redshifts (Table 2) for which
the simulation predicts no IA signal by construction. This lack of alignment
has been observed for blue galaxies at low redshifts. However, the alignment
of blue galaxies at high redshifts remains unconstrained by observations.

Future improvements of the simulation could therefore consist in taking into
account IA observations at higher redshifts from upcoming spectrosopic
surveys. An interesting extension of our work in that regard based on current
observations would be to reduce the misalignment of blue galaxies, such that
the simulation reproduces the observed null detection for this type of
galaxies (e.g. from direct measurements in a SDSS sample (Johnston et al.,
2019) or indirect measurements in DES Y1 data (Samuroff et al., 2019)) within
the corresponding errors. Such a decrease of misalignment may lead to more
significant IA contributions to the predicted DES Y3 lensing signal.

An additional improvement of our modeling could consist in a more realistic
selection of discs and ellipticals, for instance using two different color
indices as discussed in Section III.3. In our current implementation all
galaxies defined as red by a single color index cut are treated as ellipticals
and are therefore aligned with their host halos principle axes. A more
sophisticated color cut could identify a fraction of red objects as discs and
align them with their host halo’s angular momentum vector, which might change
the predictions of our simulation. One shortcoming which is harder to address
is that a significant fraction of galaxies have an irregular morphology, which
is currently not taken into account in the modeling. Hydrodynamic simulations
may help to find an effective model that describes the intrinsic alignment of
such galaxies. More realism could further be added to the simulation by
introducing a dependence of satellite alignment on the distance to the host
halo center, which has been found in observations (Huang et al., 2018,
Georgiou et al., 2019) as well as in hydrodynamic simulations (Knebe et al.,
2020).

There are several potential applications of our IA simulation not explored
here. For instance, studies of IA in third-order lensing correlations (Schmitz
et al., 2018, Pyne & Joachimi, 2021), in particular because such statistics
have been detected at high signal-to-noise in recent DES Y3 data (Secco et
al., 2022b). Another application would be to study priors on the IA modeling,
in particular on TATT parameters that are poorly constrained otherwise.

We expect that the unique size and depth of the MICE IA simulation presented
here, and the improvements in the IA assignment model, will play a central
role in constraining our understanding of the IA contribution in ongoing and
future weak lensing observations.

## Availability of data and software

The MICE IA simulation presented in this work as well as the halo catalog used
in the modeling are publicly available at
CosmoHub121212https://cosmohub.pic.es (Carretero et al., 2017, Tallada et al.,
2020). A public version of our IA simulation code is available on GitHub
131313https://github.com/kaidhoffmann/genIAL.

###### Acknowledgements.

We thank Christopher Bonnett for the initial idea to implement IA in MICE. We
thank Sukhdeep Singh and Rachel Mandelbaum for providing $w_{g+}$ measurements
from the BOSS LOWZ sample. We thank David Navarro for useful comments on the
draft. KH acknowledges support by the Swiss National Science Foundation (Grant
No. 173716, 198674), and from the Forschungskredit Grant of the University of
Zurich (Projekt K-76106-01-01). JB and SS are partially supported by NSF grant
AST-2206563. This work was also partly supported by the program "Unidad de
Excelencia María de Maeztu CEX2020-001058-M". CosmoHub is hosted by the Port
d’Informació Científica (PIC), maintained through a collaboration of Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT) and
Institut de Física d’Altes Energies (IFAE), with additional support from
Universitat Autónoma de Barcelona (UAB). Funding for the DES Projects has been
provided by the U.S. Department of Energy, the U.S. National Science
Foundation, the Ministry of Science and Education of Spain, the Science and
Technology Facilities Council of the United Kingdom, the Higher Education
Funding Council for England, the National Center for Supercomputing.
Applications at the University of Illinois at Urbana-Champaign, the Kavli
Institute of Cosmological Physics at the University of Chicago, the Center for
Cosmology and Astro-Particle Physics at the Ohio State University, the
Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M
University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho
de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de
Desenvolvimento Científico e Tecnológico and the Ministério da Ciência,
Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the
Collaborating Institutions in the Dark Energy Survey. The Collaborating
Institutions are Argonne National Laboratory, the University of California at
Santa Cruz, the University of Cambridge, Centro de Investigaciones
Energéticas, Medioambientales y Tecnológicas-Madrid, the University of
Chicago, University College London, the DES-Brazil Consortium, the University
of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi
National Accelerator Laboratory, the University of Illinois at Urbana-
Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the Institut de
Física d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-
Maximilians Universität München and the associated Excellence Cluster
Universe, the University of Michigan, NSF’s NOIRLab, the University of
Nottingham, The Ohio State University, the University of Pennsylvania, the
University of Portsmouth, SLAC National Accelerator Laboratory, Stanford
University, the University of Sussex, Texas A&M University, and the OzDES
Membership Consortium. Based in part on observations at Cerro Tololo Inter-
American Observatory at NSF’s NOIRLab (NOIRLab Prop. ID 2012B-0001; PI: J.
Frieman), which is managed by the Association of Universities for Research in
Astronomy (AURA) under a cooperative agreement with the National Science
Foundation. The DES data management system is supported by the National
Science Foundation under Grant Numbers AST-1138766 and AST-1536171. The DES
participants from Spanish institutions are partially supported by MICINN under
grants ESP2017-89838, PGC2018-094773, PGC2018-102021, SEV-2016-0588,
SEV-2016-0597, and MDM-2015-0509, some of which include ERDF funds from the
European Union. IFAE is partially funded by the CERCA program of the
Generalitat de Catalunya. Research leading to these results has received
funding from the European Research Council under the European Union’s Seventh
Framework Program (FP7/2007-2013) including ERC grant agreements 240672,
291329, and 306478. We acknowledge support from the Brazilian Instituto
Nacional de Ciência e Tecnologia (INCT) do e-Universo (CNPq grant
465376/2014-2). This manuscript has been authored by Fermi Research Alliance,
LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy,
Office of Science, Office of High Energy Physics.

##  Appendix A Dark matter halo alignment

###  A.1 Effects of noise in halo orientations

Figure 18: 3D halo alignment statistics quantifying the average alignment of
the major axis and angular momentum of dark matter halos in MICE ($\bf A$ and
$\bf J$, solid and dashed lines respectively) and the vector $\bf r$ pointing
to neighboring halos, as defined in Equation 31. Results are shown for halos
at $z=0.25$ with more than $N_{p}=320$ particles ($M_{h}\gtrsim
10^{12.97}h^{-1}\textrm{M}_{\odot}$). $N_{rand}$ indicates the number of
random particles per halo from which $\bf A$ and $\bf J$ were measured.
$\eta=0$ corresponds to random alignment.

We investigate here the impact of noise resulting from low halo particle
numbers on halo alignment statistics in MICE. For that purpose we select halos
in the MICE light-cone with $N_{p}\gtrsim 320$ particles ($M_{h}\gtrsim
10^{12.97}h^{-1}\textrm{M}_{\odot}$), then select randomly a subset of
$N_{rand}$ particles from these halos without replacement and measure the halo
major axis and angular momentum vectors ($\bf A$ and $\bf J$ respectively) as
detailed in Section III.2.1. Finally, we measure the 3D alignment statistics
of these halos with the large-scale structure as

$\eta_{X}(r)=\langle|{\bf{\hat{X}}}_{1}\cdot{\bf\hat{r}}|\rangle(r)-1/2,$ (31)

which is the inner product of the the unit vectors ${\bf{\hat{X}}}$ and
${\bf\hat{r}}$, where $\bf{X}$ refers to either $\bf A$ or $\bf J$, $\bf{r}$
is a vector pointing to a neighboring halo and $\langle\ldots\rangle$ denotes
the average over all halo pairs separated by the distance $r$. Relative
orientations of $\bf X$ and $\bf r$ that are random, parallel and
perpendicular to each other lead to $\eta_{X}=0$, $>0$ and $<0$ respectively.
In Fig. 18 we compare our measurements of $\eta_{A}$ and $\eta_{J}$ for
various values of $N_{rand}$. We find a positive amplitude of $\eta_{A}$,
which indicates that the major axes tend to point towards neighboring halos.
The amplitude starts to decrease significantly for $N_{rand}\lesssim 80$. At
$N_{rand}=10$ the amplitude is decreased by roughly $30$%, while we still find
a clear signal.

The amplitude of $\eta_{J}$ is mostly negative, indicating that the halos’
angular momenta tend to be aligned perpendicular to the vector pointing
towards neighboring halos. We see no clear dependence of $\eta_{J}$ on
$N_{rand}$, as we saw it for $\eta_{A}$. A potential explanation for hat
finding could be that the dispersion of the particles’ angular momentum
directions is low for the halos used in our test, such that the average does
not vary much across different random subsets of particles. Another
interesting result is that the absolute values of $\eta_{J}$ are significantly
lower than those of $\eta_{A}$. The lower amplitude of $\eta_{J}$ implies that
blue galaxies, which are aligned with $\bf J$ in our simulation, will always
show a significantly weaker IA signal than red galaxies, which are aligned
with $\bf A$, unless we set a much stronger misalignment for red than for blue
galaxies.

###  A.2 Mass and redshifts dependence

Figure 19: 3D halo alignment statistics of the halo major axis, minor axis and
angular momentum in the MICE simulation (top, central and bottom panels
respectively). Results are shown for halos in the four mass ranges that are
indicated on the top. The limits at
$log_{10}(M_{h})=(11.47,11.77,12.37,12.97)\
[log_{10}([h^{-1}\textrm{M}_{\odot})]$ correspond to $N_{p}=(10,20,80,320)$
particles per halo. Solid, dashed-dotted and dashed lines show results at the
redshifts $z=0.2$, $0.7$ and $1.2$ respectively.

We consolidate this last point by comparing the amplitudes of $\eta_{J}$ with
those of $\eta_{A}$ as well as $\eta_{C}$ for halos in four mass samples at
three different redshifts of the MICE light-cone in Fig. 19. For $\eta_{A}$ we
find positive amplitudes for all mass and redshift samples. The amplitudes are
increasing with the halo mass and decrease with redshift. Overall the redshift
dependence is relatively weak compared to the mass dependence, in particular
for low halo masses. This finding explains why the large scale amplitudes of
$w_{g+}$ in Fig. 10 and the corresponding $A_{1}$ parameter in Fig. 15 are
decreasing more strongly with redshift for luminous than for dim samples,
since the luminous galaxies reside in more massive halos than dim galaxies.
The fact that $\eta_{A}$ increases more strongly with halo mass than with
redshifts may be partially explained by the mass dependent noise on the halo
orientations, causing a weaker alignment for lower halo masses. However, the
resolution effects on $\eta_{A}$ that we show in Fig. 18 are weaker than the
mass dependence in Fig. 19. The results for $\eta_{C}$ are similar to those
$\eta_{A}$, while the amplitudes are negative since $\bf{C}\perp\bf{A}$.

The results for $\eta_{J}$ line up with the results from Fig. 18 as the
amplitude is weaker than for $\eta_{A}$ and $\eta_{C}$ for all redshift and
mass samples. A similar finding has been reported by Forero-Romero et al.
(2014), who compare the alignment amplitude of halo shapes and angular momenta
with the large-scale structure in a cosmological simulation. This consolidates
our expectation that blue galaxies in our simulation would be weakly aligned
compared to red galaxies, even when setting a lower misalignment for blue
galaxies than used in our current model. It is further interesting to note
that $\eta_{J}$ shows a more complex dependence on mass and scale than
$\eta_{A}$ and $\eta_{C}$. For massive objects we find the amplitude to switch
from negative to positive for increasing scales, indicating a change in the
orientations of $\bf J$ from perpendicular to parallel with respect to
neighboring halo directions. Another sign flip occurs at small scales
($r\lesssim 20\ h^{-1}\textrm{Mpc}$) between low and high mass samples. Such a
flip has also been found in other simulation based studies of halo spin
alignment (e.g. Aragon-Calvo & Yang, 2014, Forero-Romero et al., 2014, Lee &
Libeskind, 2020). However, a detailed discussion of these findings is beyond
the scope of this work.

##  Appendix B Color, magnitude and redshift distributions

###  B.1 BOSS LOWZ

In Fig. 20 we compare the distribution of $m_{r}$, $c_{\parallel}$ and
$c_{\perp}$ (defined in Equation (21)) of galaxies in MICE to observations
from the BOSS LOWZ Data Release $11$
141414https://data.sdss.org/sas/dr11/boss/lss/ in the combined Southern and
Northern galactic pole, focusing on the SM16 redshift range. We find a
reasonable agreement between the observed and simulated distributions.
However, when applying the LOWZ cuts on MICE with $\Delta m_{r}=0$ we find a
$\sim 16\%$ lower number density than in the observations. The latter is
obtained from the number of LOWZ Data Release 12 galaxies in the SM16 redshift
range, $N_{g}=249938$ and the corresponding effective area of $A=8,579\
deg^{2}$, given in Table $2$ of Reid et al. (2016), which leads to
$n_{g}=N_{g}/A_{eff}=29.97$. We therefore adjust the cuts on $m_{r}$ in MICE
$\Delta m_{r}=0.085$, which results in a $\sim 0.2\%$ agreement in the
observed number density, with $154617$ galaxies in the $5156.62\ deg^{2}$
octant of the MICE simulation.

Note that the LOWZ catalog used by SM16 is slightly reduced in size and
density compared to the original LOWZ sample since only galaxies for which
three different shape measurements were available were used in that analysis.
As a result dim objects were excluded from their analysis and the average
brightness is increased to some degree. One can therefore expect the
clustering amplitude in the LOWZ sample to be slightly increased. Including
the exact same selection effects in the mock construction is not feasible as
the MICE simulation does not include the relevant observational systematics.
However, we do not expect these effects to be critical, since the clustering
amplitude in our mock is in good agreement with the observations (Fig. 3).

Figure 20: LOWZ selection cuts in BOSS and MICE (left right respectively) on
the apparent SDSS $r$-band magnitude $m_{r}$ and the color cuts $c_{\perp}$
and $c_{\parallel}$, as defined in Equation (20). The $m_{r}$ cut for MICE is
slightly shifted to dimmer magnitudes in order to match the BOSS LOWZ number
density, as indicated by the red dotted line in the top right panel.

The selection of luminosity sub-samples in the mock BOSS LOWZ sample from MICE
is illustrated in the top panel of Fig. 21, where we show the joint magnitude-
redshift distribution. The bottom panel of the same figure compares the
redshift distribution of the sub-samples in MICE LOWZ with the observed
redshift distribution from the BOSS DR11. We find that the redshift
distribution of the MICE sub-samples L1-L3 are consistent with the
observations, while the dimmest sub-sample L4 has a strong overabundance of
galaxies around $z\sim 0.25$. We have validated that this overabundance is
present at all angular positions. It might thus result from an interplay of
the HOD-SHAM methodology used for assigning magnitudes and colors to galaxies
and the LOWZ sample selection, leading to a preferred selection of objects at
that redshift. Further investigations are needed to fully understand this
effect. However, we do not expect this overabundance to significantly affect
our measurements of the projected correlations $w_{gg}$ and $w_{g+}$ since it
is isotropic and therefore taken into account in the random catalogs used in
the estimators (see Section II).

Figure 21: Top: Absolute SDSS r-band magnitude versus redshift of galaxies in
the mock BOSS LOWZ catalog. The LOWZ sample is split into four luminosity sub-
samples (L1-L4, from bright to faint) by quantiles, following SM16. The sample
limits in redshift and magnitude are shown as dashed vertical and horizontal
lines respectively. Bottom: Redshift distribution of the MICE luminosity
samples in the same color coding as in the top panel. The black line shows the
observed distribution of the entire LOWZ sample from BOSS DR11.

###  B.2 DES-like samples

We verify the selection of the DES-like catalog in MICE by comparing the
distributions of apparent magnitudes in the $r$, $i$ and $z$ DES broads from
the DES Y3 data and the remapped MICE photometry (described in Section III.2)
in Fig. 22. We find a reasonable agreement between MICE and DES Y3, with MICE
containing a slightly longer tail toward bright objects. The DES Y3
distributions in that figure employ the fiducial quality cuts described in
Gatti & Sheldon et al., (2021), plus an extra cut removing objects in
Metacalibration with magnitude $i>23$ to enable a closer comparison with the
MICE photometry, which is limited at that magnitude151515Note that the $i<23$
cut does not affect the comparison to COSMOS data in Fig. 1, since this is
based on MICE galaxies in an area, which is complete down to $i<24$..

Figure 22: Comparison between magnitude distributions on the DES Y3
Metacalibration catalog and the MICE photometry in $riz$ bands. We introduce
an extra apparent magnitude cut of $i<23$ to Metacalibration in this
comparison, matching MICE specifications.

In Fig. 23 we display the $u-r$ color index (as defined in Section III.3)
versus the absolute SDSS $r$-band magnitude for the DES-like redshift samples
from MICE, showing how a significant fraction of central galaxies are defined
as blue according to the color cut used in our modeling, (see Table 2). Fig.
23 further reveals a lack of central galaxies dimmer than $M_{r}\sim-19$. This
cut-off results from the fact that we require halos to have at least $10$
particles. This condition imposes an absolute magnitude limit that affects
only central galaxies as a consequence of the HOD-SHAM model with which
galaxies are generated in MICE 161616Note here that this lack of galaxies in
low mass halos only affects the MICE IA catalog, not the full simulated
catalog, for which FoF halos down to two particles were populated with
galaxies.. Due to the apparent magnitude limit in the DES-like samples of
$i<23$, this artefict affects mainly centrals in the two lowest redshift bins
$1$ and $2$ of the DES-like samples. The $i<23$ cut on the other hand affects
a significant fraction of centrals as well as satellites in the two highest
redshift bins $3$ and $4$.

Figure 23: Normalized distribution of the $u-r$ restframe color index versus
absolute r-band magnitude in four redshift bins of the DES-like samples with
i<23\. Red lines indicate the color cut used to define red and blue galaxies
in the IA modeling.

Both, the $M_{r}\sim-19$ and the $i=23$ cuts, can potentially boost the IA
amplitude, as they remove centrals in low mass halos as well as faint
satellites. Centrals in low mass halos should be weakly aligned due to a
decrease in halo alignment with decreasing mass (Piras et al., 2018), and,
moreover, the noise in measured orientations increases as the number of halo
particles decreases. Satellites with faint absolute magnitudes have more
randomized orientations than bright satellites, according to our semi-analytic
IA model (Section V). Removing such highly randomized objects from the sample
could hence increase the amplitude of the IA statistics. However,
investigating the magnitude of such an increase would require a higher
resolution simulation that is complete below to $i=23$. In this work we
therefore regard the IA signal predicted by MICE for the DES-like samples as
an upper bound for an IA signal that we would measure when including galaxies
in halos with less than $10$ particles and apparent magnitudes below $i=23$.

###  B.3 Volume limited sample selection

Figure 24: Normalized distribution of absolute Subaru $r$-band magnitudes in
COSMOS and MICE for red and blue galaxies. The set of volume limited sub-
samples used for comparing the axis ratio distributions in Fig. 6 are marked
by red rectangles. The drop in the density for galaxies in MICE with
$M_{r}>-20$ results from a lack of host halos with less than $10$ particles
(see discussion Section III.5)

The selection of volume limited samples, used to compare the galaxy axis ratio
distributions in COSMOS and MICE (Section IV) is illustrated in Fig. 24,
separately for red and blue galaxies. Fig. 25 shows the selection of the
volume limited samples that are used to study the redshift, magnitude and
color dependence of the IA statistics in MICE (Section VI.1), separately for
central and satellite galaxies.

Figure 25: Normalized distribution of the $u-r$ restframe color index versus
absolute r-band magnitude in three redshift bins of the MICE simulation. Red
dashed lines indicate the limits of the color-redshift samples used for the
measuring predictions for $w_{m+}$ (Fig. 12).

##  Appendix C Jackknife covariance

Our delete-one jackknife estimates of the covariance matrix of the projected
correlations, described in Section II.1.1 are derived from angular sub-
samples, that are defined as healpix pixels with $N_{side}=8$. The area
covered by the different sub-samples in the MICE octant is shown in Fig. 26.
In Fig. 27 we show examples of the normalized covariance estimated for
$w_{m+}$ measurements in $4$ of our $16$ volume limited samples. We find that
all covariances are dominated by the diagonal elements, while the off-diagonal
elements are noisy, which potentially affects the fits of the NLA and the TATT
model predictions for $w_{m+}$ to the measurements.

Figure 26: Healpix regions ($N_{side}=8$) in the MICE octant, used for the
jackknife estimation of the covariance for $w_{g+}$ and $w_{m+}$. Figure 27:
Jackknife estimates of the normalized covariance matrix $\hat{C}_{ij}\equiv
C_{ij}/(\sigma_{i}\sigma_{j})$ for $w_{m+}$ measurements in the bins $i$ and
$j$. Results are shown for red galaxies in four of our volume limited samples.

##  Appendix D Scale dependence of TATT parameters

We investigate here how well the NLA and the TATT model predictions for
$w_{m+}$ fit the measurements over different ranges of the transverse distance
scale $r_{p}$. In addition we study how variations in the fitting range affect
the inferred model parameters. For that purpose we perform the Bayesian
parameter inference, described in Section VI.2, for the lower limits of
$r_{p}^{min}\in[1,2,4,8]\ h^{-1}\textrm{Mpc}$ and the upper limits of
$r_{p}^{max}\in[30,60]\ h^{-1}\textrm{Mpc}$. In Fig. 28 we show the
significance of the deviations between $w_{m+}(r_{p})$ measurements and fits,
defined as the absolute difference over the standard deviation, for different
$r_{p}^{min}$ and a fixed value of $r_{p}^{max}=60\ h^{-1}\textrm{Mpc}$.
Results are shown for the same $16$ volume limited sample of red galaxies as
in Fig. 12. Note that we do not investigate results for blue samples, since
their alignment signal is consistent with zero by construction. The results in
Fig. 28 show that the fits of the TATT model do not deviate by more than
$2\sigma$ from the measurements within the scale range over which the fit is
performed. This is true even for $r_{p}^{min}=1\ h^{-1}\textrm{Mpc}$, which we
therefore choose as lower scale cut for the fits shown in Fig. 12. In addition
we find that the fitting performance of the TATT model at large scales
($r_{p}>8\ h^{-1}\textrm{Mpc}$) is only weakly affected by the small scale
cut. The results for the TATT model are contrasted by those for the NLA model.
For this model we find strong deviations of more than $4\sigma$ when fitting
between $1$ and $60\ h^{-1}\textrm{Mpc}$. The fitting performance at large
scales is thereby strongly affected by the lower scale cut, indicating a lack
of flexibility in the model. However, when restricting the lower scale cut to
$r_{p}^{min}=8\ h^{-1}\textrm{Mpc}$ we find the NLA model to fit the
measurements with a similar $2\sigma$ uncertainty as the TATT model. This
scale cut is therefore chosen for the NLA fits shown in Fig. 12. It is further
interesting to note that there is no clear dependence of the fitting
performance on either magnitude or redshift and hence on the amplitude of
$w_{m+}$.

Figure 28: Significance of the deviations between the $w_{m+}$ measurements,
shown in Fig. 12 and fits to the NLA and the TATT model. Results are shown for
$16$ volume limited samples whose redshift and magnitude ranges are indicated
on the top and right respectively. Dashed, dashed-dotted, dashed-double dotted
and dotted lines are results derived for different lower limits of the fitting
range, which are set to $r_{p}^{min}=1,2,4,8\ h^{-1}\textrm{Mpc}$
respectively. These limits are marked by the vertical lines in the
corresponding line types. The upper limit of the fitting range is set to
$r_{p}^{max}=60\ h^{-1}\textrm{Mpc}$.

In Fig. 29 we show the parameters of the TATT model versus the luminosity of
the different volume limited samples in three redshift bins. We find no
significant change of the parameters when changing the upper (lower) limit of
the fiducial fitting range of $1<r_{p}^{max}=60\ h^{-1}\textrm{Mpc}$ to $4$
($30$) $h^{-1}\textrm{Mpc}$. This finding shows that the conclusions drawn
from Fig. 15 are robust towards moderate variations of the chosen fitting
range.

Figure 29:  Marginalized posterior distributions of the TATT model parameters
from fits to $w_{m+}$ measurements in different volume limited samples of red
galaxies in MICE versus each sample’s logarithmic normalized mean $r$-band
luminosity (analogous to Fig. 15). Results are shown for different scale
ranges used for the fits, as indicated in the top left panel in units of
$h^{-1}\textrm{Mpc}$.

## References

  * Abbott et al. (2022) Abbott T. M. C., et al. 2022, Phys. Rev. D, 105, 023520
  * Amon et al. (2022) Amon A., et al. 2022, Phys. Rev. D, 105, 023514
  * Aragon-Calvo & Yang (2014) Aragon-Calvo M. A., Yang L. F., 2014, MNRAS, 440, L46
  * Bate et al. (2020) Bate J., et al. 2020, MNRAS, 491, 4057
  * Bernstein & Jarvis (2002) Bernstein G. M., Jarvis M., 2002, A.J., 123, 583
  * Bertola et al. (1991) Bertola F., Vietri M., Zeilinger W. W., 1991, ApJ, 374, L13
  * Bett (2012) Bett P., 2012, MNRAS, 420, 3303
  * Bhowmick et al. (2019) Bhowmick A. K., et al. 2019, Monthly Notices of the Royal Astronomical Society, 491, 4116–4130
  * Binney (1978) Binney J., 1978, MNRAS, 183, 501
  * Blanton et al. (2003) Blanton M. R., et al. 2003, Astrophys. J. , 592, 819
  * Blazek et al. (2015) Blazek J., Vlah Z., Seljak U., 2015, JCAP, 2015, 015
  * Blazek et al. (2019) Blazek J. A., et al. 2019, Phys. Rev. D, 100, 103506
  * Bridle & King (2007) Bridle S., King L., 2007, New Journal of Physics, 9, 444
  * Carretero et al. (2015) Carretero J., et al. 2015, MNRAS, 447, 646
  * Carretero et al. (2017) Carretero J., et al. 2017, in Proceedings of the European Physical Society Conference on High Energy Physics. 5-12 July. p. 488, doi:10.22323/1.314.0488
  * Catelan et al. (2001) Catelan P., Kamionkowski M., Blandford R. D., 2001, MNRAS, 320, L7
  * Chisari et al. (2015) Chisari N., et al. 2015, MNRAS, 454, 2736
  * Chisari et al. (2017) Chisari N. E., et al. 2017, MNRAS, 472, 1163
  * Codis et al. (2018) Codis S., et al. 2018, MNRAS, 481, 4753
  * Crittenden et al. (2001) Crittenden R. G., et al. 2001, Astrophys. J. , 559, 552
  * Crocce et al. (2015) Crocce M., et al. 2015, MNRAS, 453, 1513
  * Croft & Metzler (2000) Croft R. A. C., Metzler C. A., 2000, Astrophys. J. , 545, 561
  * Dawson et al. (2013) Dawson K. S., et al. 2013, A.J., 145, 10
  * De Vicente et al. (2016) De Vicente J., Sánchez E., Sevilla-Noarbe I., 2016, Monthly Notices of the Royal Astronomical Society, 459, 3078
  * Fang et al. (2017) Fang X., et al. 2017, JCAP, 2017, 030
  * Fang et al. (2020) Fang X., Eifler T., Krause E., 2020, Monthly Notices of the Royal Astronomical Society, 497, 2699–2714
  * Foreman-Mackey et al. (2013) Foreman-Mackey D., et al. 2013, PASP, 125, 306
  * Forero-Romero et al. (2014) Forero-Romero J. E., Contreras S., Padilla N., 2014, MNRAS, 443, 1090
  * Fortuna et al. (2021a) Fortuna M. C., et al. 2021a, MNRAS, 501, 2983
  * Fortuna et al. (2021b) Fortuna M. C., et al. 2021b, A&A, 654, A76
  * Fosalba et al. (2008) Fosalba P., et al. 2008, MNRAS, 391, 435
  * Fosalba et al. (2015a) Fosalba P., et al. 2015a, MNRAS, 447, 1319
  * Fosalba et al. (2015b) Fosalba P., et al. 2015b, MNRAS, 448, 2987
  * Freedman & Diaconis (1981) Freedman D., Diaconis P., 1981, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 57, 453
  * Gatti et al. (2021) Gatti M., et al. 2021, MNRAS, 504, 4312
  * Georgiou et al. (2019) Georgiou C., et al. 2019, A&A, 622, A90
  * Gonzalez et al. (2022) Gonzalez E. J., et al. 2022, arXiv e-prints, p. arXiv:2205.06827
  * Graham & Worley (2008) Graham A. W., Worley C. C., 2008, MNRAS, 388, 1708
  * Griffith et al. (2012) Griffith R. L., et al. 2012, ApJS, 200, 9
  * Guo et al. (2020) Guo R., et al. 2020, Astrophys. J. , 897, 162
  * Handley et al. (2015) Handley W. J., Hobson M. P., Lasenby A. N., 2015, MNRAS, 453, 4384
  * Hartlap et al. (2007) Hartlap J., Simon P., Schneider P., 2007, A&A, 464, 399
  * Häußler et al. (2011) Häußler B., et al. 2011, in Evans I. N., et al. eds, Astronomical Society of the Pacific Conference Series Vol. 442, Astronomical Data Analysis Software and Systems XX. p. 155
  * Heavens et al. (2000) Heavens A., Refregier A., Heymans C., 2000, MNRAS, 319, 649
  * Heymans et al. (2004) Heymans C., et al. 2004, MNRAS, 347, 895
  * Heymans et al. (2006) Heymans C., et al. 2006, MNRAS, 371, 750
  * Heymans et al. (2021) Heymans C., et al. 2021, A&A, 646, A140
  * Hikage et al. (2019) Hikage C., et al. 2019, PASJ, 71, 43
  * Hilbert et al. (2017) Hilbert S., et al. 2017, MNRAS, 468, 790
  * Hirata & Seljak (2003) Hirata C., Seljak U., 2003, MNRAS, 343, 459
  * Hirata & Seljak (2004) Hirata C. M., Seljak U., 2004, Phys. Rev. D, 70, 063526
  * Hirata et al. (2007) Hirata C. M., et al. 2007, MNRAS, 381, 1197
  * Hoffmann et al. (2022) Hoffmann K., et al. 2022, MNRAS, 515, 3603
  * Huang et al. (2018) Huang H.-J., et al. 2018, MNRAS, 474, 4772
  * Huizinga & van Albada (1992) Huizinga J. E., van Albada T. S., 1992, MNRAS, 254, 677
  * Ilbert et al. (2006) Ilbert O., et al. 2006, A&A, 457, 841
  * Ilbert et al. (2009) Ilbert O., et al. 2009, Astrophys. J. , 690, 1236
  * Jarvis et al. (2004) Jarvis M., Bernstein G., Jain B., 2004, MNRAS, 352, 338
  * Joachimi & Schneider (2010) Joachimi B., Schneider P., 2010, A&A, 517, A4
  * Joachimi et al. (2011) Joachimi B., et al. 2011, A&A, 527, A26
  * Joachimi et al. (2013a) Joachimi B., et al. 2013a, MNRAS, 431, 477
  * Joachimi et al. (2013b) Joachimi B., et al. 2013b, MNRAS, 436, 819
  * Joachimi et al. (2015) Joachimi B., et al. 2015, Space Sci. Rev., 193, 1
  * Johnston et al. (2019) Johnston H., et al. 2019, A&A, 624, A30
  * Johnston et al. (2021) Johnston H., et al. 2021, A&A, 646, A147
  * Kiessling et al. (2015) Kiessling A., et al. 2015, Space Sci. Rev., 193, 67
  * Kirk et al. (2015) Kirk D., et al. 2015, Space Sci. Rev., 193, 139
  * Knebe et al. (2008) Knebe A., et al. 2008, MNRAS, 386, L52
  * Knebe et al. (2020) Knebe A., et al. 2020, MNRAS, 495, 3002
  * Krause et al. (2016) Krause E., Eifler T., Blazek J., 2016, MNRAS, 456, 207
  * Krause et al. (2021) Krause E., et al. 2021, arXiv e-prints, p. arXiv:2105.13548
  * Laigle et al. (2016) Laigle C., et al. 2016, ApJS, 224, 24
  * Lambas et al. (1992) Lambas D. G., Maddox S. J., Loveday J., 1992, MNRAS, 258, 404
  * Landy & Szalay (1993) Landy S. D., Szalay A. S., 1993, Astrophys. J. , 412, 64
  * Lee & Libeskind (2020) Lee J., Libeskind N. I., 2020, Astrophys. J. , 902, 22
  * Mandelbaum et al. (2006) Mandelbaum R., et al. 2006, MNRAS, 367, 611
  * Mandelbaum et al. (2011) Mandelbaum R., et al. 2011, MNRAS, 410, 844
  * McEwen et al. (2016) McEwen J. E., et al. 2016, JCAP, 2016, 015
  * Myles et al. (2021) Myles J., et al. 2021, MNRAS, 505, 4249
  * Noerdlinger (1979) Noerdlinger P. D., 1979, Astrophys. J. , 234, 802
  * Okumura & Jing (2009) Okumura T., Jing Y. P., 2009, ApJ, 694, L83
  * Okumura et al. (2009) Okumura T., Jing Y. P., Li C., 2009, Astrophys. J. , 694, 214
  * Piras et al. (2018) Piras D., et al. 2018, MNRAS, 474, 1165
  * Pitié et al. (2005) Pitié F., Kokaram A. C., Dahyot R., 2005, Proceedings of the IEEE International Conference on Computer Vision, 2, 1434
  * Pyne & Joachimi (2021) Pyne S., Joachimi B., 2021, MNRAS, 503, 2300
  * Reid et al. (2016) Reid B., et al. 2016, MNRAS, 455, 1553
  * Reyes et al. (2012) Reyes R., et al. 2012, MNRAS, 425, 2610
  * Rix & Zaritsky (1995) Rix H.-W., Zaritsky D., 1995, Astrophys. J. , 447, 82
  * Rodríguez & Padilla (2013) Rodríguez S., Padilla N. D., 2013, MNRAS, 434, 2153
  * Ryden (2004) Ryden B. S., 2004, Astrophys. J. , 601, 214
  * Samuroff et al. (2019) Samuroff S., et al. 2019, MNRAS, 489, 5453
  * Samuroff et al. (2021) Samuroff S., Mandelbaum R., Blazek J., 2021, MNRAS, 508, 637
  * Sandage et al. (1970) Sandage A., Freeman K. C., Stokes N. R., 1970, Astrophys. J. , 160, 831
  * Schmitz et al. (2018) Schmitz D. M., et al. 2018, JCAP, 2018, 030
  * Secco et al. (2022a) Secco L. F., et al. 2022a, Phys. Rev. D, 105, 023515
  * Secco et al. (2022b) Secco L. F., et al. 2022b, Phys. Rev. D, 105, 103537
  * Sevilla-Noarbe et al. (2021) Sevilla-Noarbe I., et al. 2021, ApJS, 254, 24
  * Singh & Mandelbaum (2016) Singh S., Mandelbaum R., 2016, MNRAS, 457, 2301
  * Singh et al. (2015) Singh S., Mandelbaum R., More S., 2015, MNRAS, 450, 2195
  * Springel (2005) Springel V., 2005, MNRAS, 364, 1105
  * Springel et al. (2001) Springel V., et al. 2001, MNRAS, 328, 726
  * Tallada et al. (2020) Tallada P., et al. 2020, Astronomy and Computing, 32, 100391
  * Tenneti et al. (2015) Tenneti A., et al. 2015, MNRAS, 453, 469
  * Troxel & Ishak (2015) Troxel M. A., Ishak M., 2015, Phys. Rep., 558, 1
  * Troxel et al. (2018) Troxel M. A., et al. 2018, Phys. Rev. D, 98, 043528
  * Velliscig et al. (2015) Velliscig M., et al. 2015, MNRAS, 454, 3328
  * Wei et al. (2018) Wei C., et al. 2018, Astrophys. J. , 853, 25
  * Zehavi et al. (2011) Zehavi I., et al. 2011, Astrophys. J. , 736, 59
  * van den Bosch et al. (2002) van den Bosch F. C., et al. 2002, Astrophys. J. , 576, 21