# Symbiotic Message Passing Model for Transfer Learning between Anti-Fungal
and Anti-Bacterial Domains ††thanks: ∗Corresponding author.
<EMAIL_ADDRESS>
Ronen Taub  Dept. of Physiology,
Biophysics & Systems Biology
Faculty of Medicine
Technion
Haifa, Israel
Tanya Wasserman  Dept. of Physiology,
Biophysics & Systems Biology
Faculty of Medicine
Technion
Haifa, Israel
Yonatan Savir∗ Dept. of Physiology,
Biophysics & Systems Biology
Faculty of Medicine
Technion
Haifa, Israel

###### Abstract

Machine learning, and representation learning in particular, has the potential
to facilitate drug discovery by screening billions of compounds. For example,
a successful approach is representing the molecules as a graph and utilizing
graph neural networks (GNN). Yet, these approaches still require experimental
measurements of thousands of compounds to construct a proper training set.
While in some domains it is easier to acquire experimental data, in others it
might be more limited. For example, it is easier to test the compounds on
bacteria than perform in-vivo experiments. Thus, a key question is how to
utilize information from a large available dataset together with a small
subset of compounds where both domains are measured to predict compounds’
effect on the second, experimentally less available domain. Current transfer
learning approaches for drug discovery, including training of pre-trained
modules or meta-learning, have limited success. In this work, we develop a
novel method, named Symbiotic Message Passing Neural Network (SMPNN), for
merging graph-neural-network models from different domains. Using routing new
message passing lanes between them, our approach resolves some of the
potential conflicts between the different domains, and implicit constraints
induced by the larger datasets. By collecting public data and performing
additional high-throughput experiments, we demonstrate the advantage of our
approach by predicting anti-fungal activity from anti-bacterial activity. We
compare our method to the standard transfer learning approach and show that
SMPNN provided better and less variable performances. Our approach is general
and can be used to facilitate information transfer between any two domains
such as different organisms, different organelles, or different environments.

###### Index Terms:

Drug Discovery, Graph Neural Networks, Transfer Learning, Antibiotics,
Antifungal Drugs

##  I Introduction

One of the promising future trajectories of drug discovery is harnessing the
power of machine learning to screen for billions of compounds in silico [6, 7,
8, 9, 10]. The main advantage of this approach is that it requires a
reasonable experimental effort to produce a training set based only on the
order of thousands of compounds. One of the key challenges in applying machine
learning for drug discovery is the representation of the molecule. The classic
approach is to define a feature vector composed of properties based on
chemical properties, such as competition of functional groups [11, 12, 13].
Once the molecule is represented as a feature vector, the machine can learn
which features are associated with some function. In this approach, the
definition of the features representing the molecule requires extensive
knowledge and prior knowledge in biochemistry. Representation learning, a new
approach in which the machine learns not only what are the features that are
relevant to a function but also learns the relevant features that represent
the data has emerged as a powerful tool [14].

A successful approach for representing molecules is to represent them as a
graph and utilize graph neural networks (GNN) such as graph convolutional
network (GCN) [15, 16, 17] and directed-message passing neural network
(D-MPNN) [18, 19]. It has several variations and implementations for drug
discovery. In practice, these methods encode the spatial information of the
atomic structure graph into the atoms’ hidden state, resulting in fixed-size
encodings. These encodings are then fed into a binary classifier, to produce
the final prediction.

(a)

(b)

(c)

(d)

Figure 1: (a) Details the settings of the targeted transfer learning problem.
$E$ and $Y$ denote two domains to which we want to provide predictions, e.g.
two organisms, two organs, two environments, etc. In our case, we focus on
bacteria and yeast as the two domains. Group $A$ marks the available
compounds, while group $C$ marks the compounds we wish to infer from $A$ with
the help of $B$. $B$ is the intersection where the properties are known for
both $E$ and $Y$. (b) The dual-step transfer-learning routine. Where an
untrained model is optimized for observations from $A$, and then optimized
again for observations from $B$. The model’s architecture does not vary
between the steps in a significant manner. Subfigures (c) and (d) illustrate
our method. In the second optimization step, we added a new model instance,
which communicates with the first, converged model $M_{A}$ creating a
symbiosis. The manner in which the two models communicate is illustrated in
Subfigure (d). Which is a message-passing scheme, operating at the atoms-
level.

One of the main challenges of applying the machine learning approach is the
need for a training set that usually contains $10^{3}-10^{4}$ compound
measurements. While in some domains we have high availability of data,
resulting in diverse, rich training sets in others it might be more limited
due to, for example, experimental challenges. For example, it is much easier
to measure the effect of compounds on microorganisms in high-throughput
compared to cell-line measurements, or in-vivo experiments. Thus, a key
question is to what degree can we utilize information from a large available
dataset together with a small subset of compounds where we have both domains
to predict compounds’ effect on the second, less experimentally available
domain (Fig. 1(a)). These domains are not limited to two different organisms
but could be relevant when considering different organs, different
environments, different ages, etc.

A possible solution for this challenge is Transfer Learning [2]. In transfer
learning, the model is first optimized on the other-domain dataset. And then,
in a secondary optimization process, it is optimized over the available data
from the relevant domain. Modified versions of the default transfer-learning
scheme, have been shown to work for drug discovery, resulting in improved
predictions [3, 4, 5]. Another common transfer-learning scheme is Meta-
Learning [4], where only a subset of the other-domain data is used, leaving
out data samples that do not contain mutual information or create conflicts
during secondary training.

In this work, we propose a novel method to transfer information from one GNN
model to another. The emphasis in our method is on the symbiosis between the
fused model-parts, rather than on the data-sources. Hence, it can also be
combined with the Meta-Learning framework. Our underlying idea is to add new
trainable components to the model in the secondary optimization, in a unique
manner. Hence, the new components are independent of the other domain raw
data. By doing so, we relax the implicit restriction to the environment of the
initial condition, derived by the first optimization in the other domain.
Meaning, the model can infer new rules, which are unrelated to the first
domain.

We demonstrate how our method can facilitate the prediction of antifungal
activity, the effect of compounds on the viability of the budding yeast
S.cerevisiae, from antibacterial activity on the bacteria E.coli. Both E.coli
and S.cerevisiae are common model organisms for a variety of conditions. While
E.coli is a prokaryote, S.cerevisiae is a eukaryote. Therefore, it provides a
good example of two domains in which the interplay between compounds’ effects
on each domain is not trivial. We compared the different transfer-learning
methods and show that our method provides better results in term of AUROC and
that it is closer to Pareto optimality in terms of F1-score.

##  II Methods and Training Data

###  II-A Training Data

For this paper, we have utilized two datasets. The first dataset is a public
one [1], containing the effect of 2336 FDA-approved compounds on the bacteria
E.coli. From this collection, 120 compounds exhibited antibiotic activity. The
second dataset was produced in our lab. We have used a similar FDA-approved
compounds library containing 1467 compounds. The number of shared compounds,
between the two libraries, is 824. We measured the response of S.cerevisiae by
incubating them with the drugs and monitoring their growth. The yeast were
added to each well with a final $OD_{600}$ of $0.05$. Initial $OD_{600}$ was
measured using the Infinite M200 Pro plate reader (Tecan), after which the
plates were sealed with an AeraSealTM (Excel Scientific) and incubated in a
Multitron Pro shaker-incubator (Infors HT) at 30°C, 1000 rpm and 70% humidity.
Final $OD_{600}$ was measured after $16h$. We have defined compounds as having
an anti-fungal activity if they reduced the yeast growth significantly,
compared with control growth, p-cal$<$0.05. In total, there are 43 positively-
labeled compounds with anti-fungal activity. Hereafter, for simplicity, we
will refer to E.coli by the term ’bacteria’, and to S.cerevisiae by the term
’yeast’.

###  II-B Message Passing Neural Network

Message Passing Neural Network (MPNN) is a term for GNN classifier
architecture, that consists of a GNN encoder, aggregation operator, and
predictor functional [1]. In our work, we followed the implementation
available in the code environment, distributed in the Chemprop package (which
is the code environment for [1] and available on GitHub). According to this
implementation, the graph encoder operation can be presented by the following
system:

$\vec{m}_{i\rightarrow
j}^{(0)}=ReLU\left(W_{I}\cdot\begin{pmatrix}\begin{pmatrix}\vec{a}_{j}\end{pmatrix}\\\
\begin{pmatrix}\vec{b}_{i\rightarrow j}\end{pmatrix}\end{pmatrix}\right)\quad
if\quad\vec{b}_{i\rightarrow j}\neq 0$ (1) $\vec{m}^{(t+1)}_{j\rightarrow
i}=ReLU\left(\vec{m}^{(0)}_{j\rightarrow
i}+W_{H}\cdot\sum_{k=1}^{N_{A}}\begin{pmatrix}\begin{pmatrix}\vec{m}^{(t)}_{k\rightarrow
j}\end{pmatrix}\\\ \begin{pmatrix}\vec{b}_{k\rightarrow
j}\end{pmatrix}\end{pmatrix}\right)$ (2)
$\vec{e}_{i}=\sum_{j=1}^{N_{a}}\vec{m}^{(T)}_{j\rightarrow i}$ (3)
$\vec{\alpha}_{i}=ReLU\left(W_{O}\cdot\begin{pmatrix}\begin{pmatrix}\vec{e}_{i}\end{pmatrix}\\\
\begin{pmatrix}\vec{a}_{i}\end{pmatrix}\end{pmatrix}\right)$ (4)

where $\vec{a}_{i}\in\mathrm{R}^{d_{a}}$ is the feature vector of the $i^{th}$
atom, containing the relevant chemical properties for each atom. Similarly,
$\vec{b}_{i\rightarrow j}\in\mathrm{R}^{d_{b}}$ is the feature vector of the
bond going from the $i^{th}$ atom to the $j^{th}$ atom. Equation (eqn. (1)) is
the initialization step, projecting the atoms features and connected bonds
onto the hidden feature space. Where $W_{I}\in\mathrm{R}^{(d_{a}+d_{b})\times
d_{h}}$ is the projection matrix. Equation (eqn. (2)) represent the recursive
encoding operator. In each iteration $t$, for each bond, the states of the
neighbor atoms (of the bond’s source-atom) are summed and linearly transformed
with $W_{h}\in\mathrm{R}^{d_{h}\times d_{h}}$ (in order to update the
corresponding bond-message). After $T$ iterations, all the resulted messages
are summed in (eqn. (3)), to produce encodings at the atoms-level. Finally,
$\alpha_{i}$ is the final encoding of the $i^{th}$ atom after the external
stimulation, described in (eqn. (4)).

###  II-C Transfer Learning

Transfer Learning is a methodology to transfer gathered intelligence,
implicitly contained within one model, to another model. When having two
groups of datasets, the common approach for transfer learning is a dual-step
procedure, where the model is first trained on the knowledge of larger group
$A$ and then trained again on the partial knowledge of group $B$ (Fig. 1(b)).
We used this approach to benchmark our method. In our case, we first trained
on the bacteria dataset and than re-trained the model using different sets
with different number of compounds, that are shared between the bacteria and
yeast dataset. The size of the sets goes from 100 to 824 in jumps of 100.

###  II-D SMPNN: Symbiotic Message Passing Neural Network

The motivation for the SMPNN model is to create a beneficial symbiosis between
model-parts trained on data from group $A$, and model-parts trained on
available data from group $B$. This way, we have unique model parameters
supporting the domain knowledge in group $A$ (i.e., optimized on), unique
model parameters supporting the domain knowledge in group $B$, and unique
model parameters that support both. While in regular transfer-learning methods
and or meta-learning, we have only the latter. Hence, the hybrid model
provides the maximum flexibility to fit observed data (1(c))

Our basic model instance carries the operation described by equations (eqn.
(1) - (4)). The MPNN architecture contains a graph-encoder, an aggregation
operator, and a fully-connected network (as for binary classifier). After the
aggregation operator, the observability of the encoder state is limited, since
it combines all the atoms’ encodings. Before the aggregation operator and past
the encoder, different converged MPNN instances produce different features for
each atom, creating a synchronization problem. Therefore, we decided to
connect two different MPNN pipelines, at the encoder level.

In Fig. 1(d), there is an illustration of the manner in which we connect two
GNN encoders. We route the messages from each atom in the enriched model to
its mirror atom in the less-enriched model. We have decided to not pass the
messages directly, but rather linearly project them with projection matrix
$W_{S}\in\mathrm{R}^{d_{h}\times d_{h}}$. The reasoning behind this is to
allow the second pipeline to maintain a different hidden-state space (feature
space) from the first pipeline converged space. The modified versions of
equations (eqn. (1) - (4)) equal:

$\widetilde{\vec{m}}_{i\rightarrow
j}^{(0)}=ReLU\left(\widetilde{W}_{I}\cdot\begin{pmatrix}\begin{pmatrix}\vec{a}_{j}\end{pmatrix}\\\
\begin{pmatrix}\vec{b}_{i\rightarrow j}\end{pmatrix}\end{pmatrix}\right)\quad
if\quad\vec{b}_{i\rightarrow j}\neq 0$ (5) $\vec{m}_{i\rightarrow
j}^{(0)}=ReLU\left(W_{I}\cdot\begin{pmatrix}\begin{pmatrix}\vec{a}_{j}\end{pmatrix}\\\
\begin{pmatrix}\vec{b}_{i\rightarrow j}\end{pmatrix}\end{pmatrix}\right)\quad
if\quad\vec{b}_{i\rightarrow j}\neq 0$ (6)
$\widetilde{\vec{m}}^{(t+1)}_{j\rightarrow
i}=ReLU\left(\widetilde{\vec{m}}^{(0)}_{j\rightarrow
i}+\widetilde{W}_{H}\cdot\sum_{k=1}^{N_{A}}\begin{pmatrix}\begin{pmatrix}\widetilde{\vec{m}}^{(t)}_{k\rightarrow
j}\end{pmatrix}\\\ \begin{pmatrix}\vec{b}_{k\rightarrow
j}\end{pmatrix}\end{pmatrix}\right)$ (7) $\displaystyle
m^{(t+1)}_{j\rightarrow i}=ReLU\large\Bigg{(}\normalsize m^{(0)}_{j\rightarrow
i}+W_{H}\cdot\sum_{k=1}^{N_{A}}\begin{pmatrix}\begin{pmatrix}\vec{m}^{(t)}_{k\rightarrow
j}\end{pmatrix}\\\ \begin{pmatrix}\vec{b}_{k\rightarrow
j}\end{pmatrix}\end{pmatrix}$
$\displaystyle+W_{S}\cdot\widetilde{W}_{H}\cdot\sum_{k=1}^{N_{A}}\begin{pmatrix}\begin{pmatrix}\widetilde{\vec{m}}^{(t)}_{k\rightarrow
j}\end{pmatrix}\\\ \begin{pmatrix}\vec{b}_{k\rightarrow
j}\end{pmatrix}\end{pmatrix}\large\Bigg{)}\normalsize$ (8)
$\vec{e}_{i}=\sum_{j=1}^{N_{a}}\vec{m}^{(T)}_{j\rightarrow i}$ (9)
$\vec{\alpha}_{i}=ReLU\left(W_{O}\cdot\begin{pmatrix}\begin{pmatrix}\vec{e}_{i}\end{pmatrix}\\\
\begin{pmatrix}\vec{a}_{i}\end{pmatrix}\end{pmatrix}\right)$ (10)

where $\widetilde{W}_{I},\widetilde{W}_{H}$ are the weights from the converged
model in the ’transfer-from’ domain (i.e., where the training data is
available). The matrices $W_{I},W_{H},W_{O}$ are the new trainable component
added to assemble the hybrid model. The equations (eqn. (5),(7)) maintain the
original message passing scheme of the ’transfer-from’ model. While equation
(eqn. (II-D)), is where the transferring occurs, with matrix $W_{S}$.

To summarize the entire method, we detail the full-training process with the
following recipe:

  1. 1.

Train MPNN instance over the ’transfer-from’ enriched data according to the
original labels.

  2. 2.

Construct SMPNN instance according to equations (eqn. (5) - (10)).

  3. 3.

Initialize the weights of the fully-connected network binary classifier of
SMPNN, to be the same ones as in the converged MPNN.

  4. 4.

Initialize the weight matrices $\widetilde{W}_{I},\widetilde{W}_{H}$ to be
$W_{I},W_{H}$ from converged MPNN.

  5. 5.

Train the SMPNN over the available subset of examples from the ’transfer-to’
targeted domain.

##  III Results

In this work, we focused on evaluating the effectiveness of transfer learning
between the bacteria domain $E$ and the yeast domain $Y$. To do this, we
divided the data populations into 3 groups: $E,E\cap Y,Y/\\{E\cap Y\\}$. Group
$A$, the ’transfer-from’ population contains the compounds in $E$ with labels
that denote their antibacterial activity. Group B, which is the training set
for the second optimization has the compounds from $E\cap Y$ with antifungal
activity. Test group C has the compounds in $Y/\\{E\cap Y\\}\\}$. In numbers,
$|E|=2336$ compounds, $|E\cap Y|=824$ compounds, $|Y/\\{E\cap Y\\}|=643$
compounds. To test the effect of adding more compounds we used intersection
subsets with increasing numbers. $B_{N}$ is a subset of $B$ that contains $N$
examples from $B$. We examined various $B_{N}$ groups for various values of
$N$, to observe in what manner the transfer-learning improves with more data
available for the second training. More specifically, we randomly selected
$B_{N}$ out of the intersection population (at each time), with the condition
$B_{N}\subset B_{N+1}$ (for the same random-generator-seed).

Figure 2: The effect of transfer learning dataset size on the AUROC (area
under the ROC curve). A model was trained only on the compounds with known
effects on bacteria ($A$ in Fig. 1(a)) and then was retrained using a subset
of compounds to which both the bacteria and yeast effect are known ($A\cap B$
in Fig. 1(a)). The AUROCs are the result of predicting antifungal activity on
unseen compounds ($C$ in Fig. 1(a)). The x-axis is T the number of compounds
in retrain set $B_{N}$. ’No-Transfer’ denotes training only occurs only on
$B_{N}$ (without the initial training on set $A$). ’Dual’ and ’Symbiotic’ are
the two examined transfer-learning methods. Each dot represents a different
random choice of the $B_{N}$ compounds. The lines are the averages.

First, to estimate the interplay between the bacteria domain $E$ and the yeast
domain $Y$, we trained the model only with the bacteria ground-truth labels
and tested on the yeast set, that is training on $A$ and testing on $C$. The
AUROC (area under the ROC curve) received for this model is $0.73$. To make
sure that this results is indeed related to the information encapsulated
within the bacteria ground-truth label and not the result of training on the
large chemical space, we tested the results of training on a similar number of
compounds but with random ground-truth labels. We randomized the labels 20
times to get a representative population. The distribution of the AUROC in
this case is in the range of $0.58\sim 0.63$. This indicate that the
antibacterial data indeed carries information for the antifungal activity.

Next, tested the effect of using transfer-learning, that is using also ground-
truth information from the yeast measurements, and evaluate the improvement
delta when using SMPNN over the common transfer-learning (1(b)). For this we
compared SMPNN with two alternatives: training only on $B_{N}$ (i.e., not
using bacteria domain knowledge, ’No-Transfer’), and training on $A$ first and
then training again on $B_{N}$ for the same model (i.e., ’Dual-Training’). We
gradually changed the value of $N$ to observe the manner in which the
performance transition. The results are presented in Fig. 2. One can see that
for every value of $N$, it applies that the AUROC statistics of SMPNN is
better than the AUROC statistics for ’Dual-Training’, and the AUROC statistics
for ’Dual-Training’ is better than the AUROC statistics when not performing
transfer-learning at all. Interestingly, the AUROC of SMPNN is always greater
than $M_{A}$, but that is not the case always for ’Dual-Training’. This
implies, that there might be conflicts during the second training, when using
the same model. Another interesting fact, is that the AUROC for SMPNN in the
case of $B_{100}$ is better than the results for ’Dual-Training’ and no-
transfer-learning in the case of $B_{900}$.

Figure 3: Swarmplot with boxplot of improvement delta in AUROC between the
symbiotic model (i.e. SMPNN) and the traditional dual-training scheme. Each
dot represents a different random choice of the $B_{N}$.

Fig. 3 illustrated the improvement of using in SMPNN in terms of AUROC. The
improvement delta increases with $N$. Meaning, the marginal value of
additional data, is greater with the Symbiotic model. Hence, the results for
$B_{800}$ should contain the best improvement delta. In numbers, the
improvement delta when using SMPNN is $8\%$ on average with respect to ’Dual-
Training’, and $33\%$ on average with respect to not performing transferring
learning at all (for $B_{800}$). Besides the improvement median performance,
using SMPNN yields a more reproducible cross validation. There is a $58\%$
decrease decrease in the SMPNN standard-deviation compared with the ’Dual-
Training’ method, and $69\%$ decrease compared with no transfer learning.

Figure 4: Scatter-plot of the $B_{800}$ models, comparing ’No-transfer-
learning’ (blue), ’Dual-Training’ (orange), and ’Symbiotic’ model (green). The
precision and recall values are derived from the predictions of the converged
model (in each point) with respect to the same test, i.e., group $C$. The
contour lines are constant F1 score values. The average point for each method
is the large circle with black-color edges. The blue dashed line is precision
equals to recall.

To analyze the results further, we examined the F1-score metric on the
precision-recall plane. We took the results for $B_{800}$ in the three
methods. As a reminder, $B_{800}$ means that we train on the bacteria dataset
(group A) first. Then we train on the randomly picked 800 molecules from group
$B$, named as $B_{800}$. And finally, test on group $C$ (which are molecules
from the yeast dataset that do not appear in the intersection). The precision
and recall results are displayed in Fig. 4. The fact that the area covered by
’Dual-Training’ is significantly larger than the area for ’Symbiotic’ model.
That is due to the fact that the performance standard deviation in the case of
the ’Symbiotic’ model is significantly lower. It is insightful to consider the
resolution to the tradeoff between precision and recall. The SMPNN model also
provides a more balanced solution to the recall-precision tradeoff. In the
case there is a tradeoff between recall and sensitivity such that both of them
have the same cost, then the SMPNN model is closer to Pareto optimality.

##  IV Conclusions

While previous works in the field of drug discovery, have focused more on the
correct mixture of datasets (either manually or with meta-learning) for
fruitful transfer-learning, our Symbiotic model (SMPNN) puts the focal point
on the manner in which domain models are fused together. To emulate the meta-
learning scheme, we reconstructed the sub-optimality domain (meta-learning
operates on) by randomizing $B_{N}$. The results indicate that SMPNN excels
for all the values of $N$ (in almost all instances of $B_{N}$). We hypothesize
that this is due to the fact that our specific symbiosis allows the model to
learn rules that are not necessarily connected to, or agree with, the first
converged model. This is supported also by the fact that the improvement delta
values (from results) increase with $N$. In general, in the case of transfer
learning, reproducibility of the results, as we mix datasets from different
domains, which are not explicitly linked, is crucial. Our method exhibits
significant improvement in the standard deviation of the performances.

The aim of any drug-discovery model is to provide predictions that have to be
validated experimentally. It is insightful to consider the performance of drug
discovery in terms of compound coverage (recall) and experimental yield
(precision). The number of predicated positive compounds determines the cost
of the experimental validation. Hence, the experimental yield, which is the
ratio between true positive and predicted positive, is desired to receive a
high value (closer to 1). The second important metric is compound coverage,
which is the ratio between true positive and overall positive compounds. This
metric is also desired to be a high number, close to 1. Meaning, we wish to
discover new drugs, and not receive the same already recorded drugs. There is
usually a built-in trade-off between these two metrics. A natural metric that
accounts for both of these quantities is the F1-score. Our results indicate
that, if recall and precision have a tradeoff between them such that they have
similar costs, the Symbiotic model is closer to the Pareto optimal resolution.

While we have demonstrated the value of our approach using two different
organisms, our approach is general and can be used to facilitate information
transfer between any two domains.

## Acknowledgment

This work is supported by the Israeli Ministry of Science and Technology
(MOST) grant #2149.

## References

  * [1] Stokes JM, Yang K, Swanson K, Jin W, Cubillos-Ruiz A, Donghia NM, MacNair CR, French S, Carfrae LA, Bloom-Ackermann Z, Tran VM, Chiappino-Pepe A, Badran AH, Andrews IW, Chory EJ, Church GM, Brown ED, Jaakkola TS, Barzilay R, Collins JJ. A Deep Learning Approach to Antibiotic Discovery. Cell. 2020 Feb 20;180(4):688-702.e13. doi: 10.1016/j.cell.2020.01.021.
  * [2] Weiss, K., Khoshgoftaar, T.M. & Wang, D. A survey of transfer learning. J Big Data 3, 9 (2016). https://doi.org/10.1186/s40537-016-0043-6
  * [3] Li, X., Fourches, D. Inductive transfer learning for molecular activity prediction: Next-Gen QSAR Models with MolPMoFiT. J Cheminform 12, 27 (2020). https://doi.org/10.1186/s13321-020-00430-x
  * [4] Q. Lv, G. Chen, Z. Yang, W. Zhong and C. Y. -C. Chen, ”Meta Learning With Graph Attention Networks for Low-Data Drug Discovery,” in IEEE Transactions on Neural Networks and Learning Systems,https://doi.org/10.1109/TNNLS.2023.3250324.
  * [5] Cai, Chenjing and Wang, Shiwei and Xu, Youjun and Zhang, Weilin and Tang, Ke and Ouyang, Qi and Lai, Luhua and Pei, Jianfeng. (2020). Transfer Learning for Drug Discovery. Journal of Medicinal Chemistry. vol. 63. https://doi.org/10.1021/acs.jmedchem.9b02147
  * [6] Stephenson N, Shane E, Chase J, Rowland J, Ries D, Justice N, et al. Survey of machine learning techniques in drug discovery. Curr Drug Metab 2019;20(3):185–93.
  * [7] Ekins S, Puhl AC, Zorn KM, Lane TR, Russo DP, Klein JJ, et al. Exploiting machine learning for end-to-end drug discovery and development. Nat Mater 2019;18(5):435–41.
  * [8] Zhang L, Tan J, Han D, Zhu H. From machine learning to deep learning: progress in machine intelligence for rational drug discovery. Drug Discov Today 2017;22(11):1680–5.
  * [9] Lavecchia A. Machine-learning approaches in drug discovery: methods and applications. Drug Discov Today 2015;20(3):318–31.
  * [10] Vamathevan J, Clark D, Czodrowski P, Dunham I, Ferran E, Lee G, et al. Applications of machine learning in drug discovery and development. Nat Rev Drug Discov 2019;18(6):463–77.
  * [11] Mauri A, Consonni V, Pavan M, Todeschini R. DRAGON software: An easy approach to molecular descriptor calculations. Match 2006;56(2):237–48.
  * [12] Moriwaki H, Tian YS, Kawashita N, Takagi T. Mordred: A molecular descriptor calculator. J Cheminform 2018;10(1):4.
  * [13] Rogers D, Hahn M. Extended-connectivity fingerprints. J Chem Inf Model 2010;50(5):742–54.
  * [14] Bengio Y, Courville A, Vincent P. Representation learning: A review and new perspectives. IEEE Trans Pattern Anal Mach Intell 2013;35(8):1798–828.
  * [15] Duvenaud D, Maclaurin D, Aguilera-Iparraguirre J, Gómez-Bombarelli R, Hirzel T, Aspuru-Guzik A, et al. Convolutional Networks on Graphs for Learning Molecular Fingerprints.
  * [16] Lin X, Quan Z, Wang Z-J, Ma T, Zeng X. KGNN: Knowledge Graph Neural Network for Drug-Drug Interaction Prediction. 2020.
  * [17] Li B, Wang W, Sun Y, Zhang L, Ali MA, Wang Y. GraphER: Token-Centric Entity Resolution with Graph Convolutional Neural Networks. Proc AAAI Conf Artif Intell 2020;34(05):8172–9.
  * [18] Yang K, Swanson K, Jin W, Coley C, Eiden P, Gao H, et al. Analyzing Learned Molecular Representations for Property Prediction. J Chem Inf Model 2019;59(8):3370–88.
  * [19] Stokes JM, Yang K, Swanson K, Jin W, Cubillos-Ruiz A, Donghia NM, et al. A Deep Learning Approach to Antibiotic Discovery. Cell 2020;180(4):688-702.e13.


# Black String Thermal Equilibrium

João Chakrian<EMAIL_ADDRESS>Antônio de Pádua Santos
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal Rural de
Pernambuco, 52171-900 Recife, PE, Brazil

###### Abstract

The black string solution is a model of black holes in (3+1) dimensions that
can be related to the BTZ (2+1) dimensions black hole solution. This object
can be seen as a cylindrical black hole. In this work it is considered the
black string Hawking temperature obtained by the Hamilton-Jacobi method and a
description of a system composed by a black string and a thermal reservoir,
considering the thermal equilibrium state.

Black strings, Hawking radiation

††preprint: APS/123-QED

##  I  Introduction

In General Relativity, the spherically symmetric black hole solutions are
widely studied due to their relevant role in describing various physical
systems of astrophysical and cosmological importance, such as in the
description of the final state of a gravitational collapse of stars. The
interest of describing black holes with different topology is not considered
due the formulation of hoop conjecture, established by Thorne Thorne (1972),
which states that event horizons can be created only if a mass is compressed
into a region with circumference less than $4\pi GM$ in every direction, which
means that only spherically symmetric black holes should exist. However, the
hoop conjecture holds only when cosmological constant is assumed to be zero,
and therefore, if one admits negative cosmological constant, the spacetime
shall become asymptotically Anti-de Sitter. This fact makes possible to
investigate cylindrically symmetric black hole solutions, that is, black
strings.

The black strings thermodynamic properties are known Chakrian and Santos
(2020); Fatima and Saifullah (2012) and in particular the heat capacity
positive behavior is interesting for this paper once heat capacity, and as
consequence the specific heat, is always positive. This fact can be understood
as the property of a system composed by the black string and a thermal
reservoir reach a thermal equilibrium state. In this work, it is analyzed the
black string behavior when thermal equilibrium is considered. It is assumed
that the black string mass per unit of length changes due Hawking radiation
since its temperature (Hawking temperature) is different from the temperature
of the thermal reservoir $T_{R}$. In this conjecture, the black string
temperature can be higher, $T>T_{R}$, or lower $T<T_{R}$ than the reservoir
temperature. The temporal evolution of mass per unit of length is investigated
in this work.

##  II  Black string

The papers Lemos (1995); Lemos and Zanchin (1996a) performed by Lemos were a
breakthrough in the subject of cylindrical black holes since he has shown that
the black strings are predicted when a negative cosmological constant is
considered, and moreover, it is possible Lemos and Zanchin (1996b) to relate
this (3+1) dimensions solution to the BTZ (2+1) dimension solution. In such
initial papers, the black string solution arises from solving the motion
equations that result from the Einstein-Hilbert four dimensions action

$S=\frac{1}{16\pi G}\int\sqrt{-\mathrm{g}}(R-2\Lambda)d^{4}x,$ (1)

where $S$ is the Einstein-Hilbert action in four dimensions, g is the
determinant of the metric tensor, $\textsl{g}=\text{det}(g_{\mu\nu})$, $R$ is
the Ricci scalar, $\Lambda$ is the cosmological constant and $G$ is the
Newtonian gravitational constant. The black string solution, considering
$G=\hbar=c=1$, is given by Lemos and Zanchin (1996a)

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(\alpha^{2}r^{2}-\frac{4M}{\alpha
r}\right)dt^{2}+\left(\alpha^{2}r^{2}-\frac{4M}{\alpha r}\right)^{-1}dr^{2}$
(2) $\displaystyle+$ $\displaystyle r^{2}d\phi^{2}+\alpha^{2}r^{2}dz^{2},$

where $-\infty<t<\infty$, the radial coordinate $0\leq r<\infty$, the angular
coordinate $0\leq\phi<2\pi$, the axial coordinate $-\infty<z<\infty$,
$\alpha^{2}=-\frac{\Lambda}{3}>0$ and $M$ the mass per unit length in the
$z$-direction. This solution represents a static straight black string with
one event horizon at

$r_{+}=\frac{(4M_{+})^{\frac{1}{3}}}{\alpha}.$ (3)

##  III  Temperature

The Hamilton-Jacobi method represents an approach to investigate the quantum
tunneling from scalar field in black string spacetime. This method is based on
the tunneling approach by Parikh and Wilczeck Parikh and Wilczek (2000), the
path integrals method by Srinivasan Srinivasan and Padmanabhan (1999) and can
be found in the references Gohar and Saifullah (2013a, b). We start from the
Klein-Gordon equation on curved spaces for a scalar field $\Phi$

$\mathrm{g}^{\mu\nu}\partial_{\mu}\partial_{\nu}\Phi-\frac{m^{2}}{\hbar^{2}}\Phi=0,$
(4)

where $\mathrm{g}^{\mu\nu}$ is the metric tensor related to the line element
(2) and $m$ is the mass of the field. In order to solve this equation, we use
the ansatz

$\Phi(t,r,\phi,z)=\exp\left\\{\frac{i}{\hbar}S(t,r,\phi,z)\right\\},$ (5)

where $S$ represents the trajectory action associated to the particle
tunneling. We can recognize this solution as the WKB approximation Birrell
_et al._ (1984). This low wavelength solution is justified since the amount of
null geodesics tends to infinity at the event horizon which means that there
is a blueshift Parikh and Wilczek (2000). Substituting (5) into (4) and using
the first term of the expansion of the action in terms of $\hbar$, we find

$\mathrm{g}^{tt}(\partial_{t}S)^{2}+\mathrm{g}^{rr}(\partial_{r}S)^{2}+\mathrm{g}^{\phi\phi}(\partial_{\phi}S)^{2}+\mathrm{g}^{zz}(\partial_{z}S)^{2}+m^{2}=0.$
(6)

The solution to equation (6) can be considered by separation of variables as
follows

$S(t,r,\phi,z)=-Et+W(r)+J_{\phi}\phi+J_{z}z+C,$ (7)

where $E$ is a constant associated to the energy, $W(r)$ is a function to be
determined, $J_{\phi},J_{z}$ are constants associated to angular momentum in
$\phi$ and $z$ direction, respectively, and $C$ is a constant. After
performing an integration on complex plane, the function $W(r)$ is given by

$W_{\pm}(r)=\pm\frac{i\pi E}{3\alpha^{2}r_{+}}$ (8)

where it was used the relation (3) and we remark that $W_{+}=-W_{-}$ . The
probabilities of crossing the event horizon can be written as

$\displaystyle\Gamma_{\mbox{\scriptsize emission}}$ $\displaystyle=$
$\displaystyle\exp\left\\{-\frac{2}{\hbar}\left[\operatorname{\mathrm{Im}}(W_{+})+\operatorname{\mathrm{Im}}(C)\right]\right\\}$
(9) $\displaystyle\Gamma_{\mbox{\scriptsize absorption}}$ $\displaystyle=$
$\displaystyle\exp\left\\{-\frac{2}{\hbar}\left[\operatorname{\mathrm{Im}}(W_{-})+\operatorname{\mathrm{Im}}(C)\right]\right\\}$
(10)

and by the condition that the probability of entering in the event horizon of
the black string is $100\%$ Gohar and Saifullah (2013a), we determine that
$\operatorname{\mathrm{Im}}(W_{-})=\operatorname{\mathrm{Im}}(C)$ and the
replacement of this result into (9) leads to

$\displaystyle\Gamma_{\mbox{\scriptsize emission}}$ $\displaystyle=$
$\displaystyle\exp\left\\{-\frac{4}{\hbar}\operatorname{\mathrm{Im}}(W_{+})\right\\}$
$\displaystyle\Gamma_{\mbox{\scriptsize emission}}$ $\displaystyle=$
$\displaystyle\exp\left\\{-\frac{4}{\hbar}\frac{\pi
E}{3\alpha^{2}r_{+}}\right\\}.$ (11)

Now, if one compares the emission factor (III) to the canonical ensemble
Boltzmann factor $\exp\left\\{E/T_{H}\right\\}$, the Hawking temperature is
written as, (we set $\hbar=1$)

$T_{H}=\frac{3\alpha^{2}}{4\pi}r_{+}.$ (12)

This result can also be found in the references Fatima and Saifullah (2012);
Cai and Zhang (1996) by a different approach and in the references Gohar and
Saifullah (2013a, b) by using the Hamilton-Jacobi method. It is interesting to
see that the temperature behavior is linear in terms of the event horizon
radius $r_{+}$. If one uses the relation (3) to write this temperature in
terms of mass per unit of length associated to the event horizon radius, one
obtains

$T_{H}=\frac{3\alpha}{2\pi}\left(\frac{M_{+}}{2}\right)^{\frac{1}{3}},$ (13)

which makes it clear that $T_{H}\rightarrow 0$ only if $M_{+}\rightarrow 0$.
From this point, unless stated otherwise we shall consider $T_{H}=T$.

##  IV  Empty Space as Thermal Reservoir

The Stefan-Boltzmann equation can be used to investigate the energy flux due
the difference of temperature of the black string and the empty space as
presented in the reference de Santi and Santarelli (2019). We adapt this
procedure in order to study the temporal evolution of the black string mass
per unit length. The Stefan-Boltzmann relation can be written as

$J=\lambda A(T^{4}-T_{R}^{4})$ (14)

where $T$ is identified, in this work, as the Hawking temperature associated
to the black string, $T_{R}$ is the temperature associated to the reservoir,
$L$ represents the energy flux through the system components, $\lambda$ is the
Stefan-Boltzmann constant: $\lambda=\pi^{2}k_{B}^{4}/60\hbar^{3}c^{2}$ in SI
units and $A$ is the area per unit of length of the black string $A=2\alpha\pi
r_{+}^{2}$ Cai and Zhang (1996). The system proposed is defined as the black
string, in vacuum, with temperature $T_{H}$, we use $T_{R}=0$ and, for this
case, the relation (14) reduces to

$J=\lambda AT^{4}.$ (15)

It is important to point out that it shall be used arbitrary units because, in
the point of view of this work, the general behavior of the results are
interesting and their values could be investigated considering any choice of
units. Then, rewriting the relation (15) using the equation (3) to write
$r_{+}$ in terms of $M_{+}$ and considering $k_{B}=G=c=\hbar=1$, we have

$J=\frac{27\alpha^{3}}{160\pi}M_{+}^{2}.$ (16)

Now, for the investigated system we can consider $J$ as the temporal variation
of energy of the black string and, as consequence, its temporal variation of
the mass per unit length since the energy is modified by emission (or
absorption) of radiation. In this perspective, it is possible to write
$J=-dM_{+}/dt$ and using the relation (16) one finds

$\frac{dM_{+}}{dt}=-\frac{27\alpha^{3}}{160\pi}M_{+}^{2}.$ (17)

Integrating this relation, considering the initial conditions $t_{0}=0$ and
$M_{+}(t_{0})=M_{0}$, as

$\int_{M_{0}}^{M}\frac{dM^{\prime}_{+}}{{M^{\prime}_{+}}^{2}}=-\frac{27\alpha^{3}}{160\pi}\int_{t_{0}=0}^{t}dt^{\prime}$
(18)

we obtain the temporal evolution of the black string mass per unit length
considering the empty space as the thermal reservoir:

$M(t)=\frac{1}{\displaystyle at+\frac{1}{M_{0}}}.$ (19)

Here, $a=\frac{27\alpha^{3}}{160\pi}$. This relation establishes that the
“complete evaporation” ($M\rightarrow 0$) would demand an infinite time. This
fact is physically plausible because if $M\rightarrow 0$, then $T\rightarrow
0$ from relation (13) and $T\rightarrow 0$ should not happen due the third law
of thermodynamics. In other words, it is not possible to the black string
completely evaporate. We remark that this prediction suggests a different
behavior in comparison to the Schwarzchild black hole since for the last one
it is predicted the existence of a lifetime which depends on the initial mass
of the black hole.

Figure 1: Temporal evolution of the mass per unit of length considering the
empty space as thermal reservoir. Here, the parameters are $a=1$ and $M_{0}=2$
expressed in arbitrary units.

The figure 1 presents a plot of the relation (19) which represents the
analytic solution to the equation (17) and also presents a numerical111It was
applied the Classical Runge-Kutta method using the open source software of
mathematics SageMath. All numerical calculations presented in this work were
performed using the mentioned software. solution to such equation. It is
possible to observe the decrease in the value of the mass per unit length.
This fact was expected since the black string temperature is higher than the
empty space temperature $T>T_{R}$ and therefore the black string radiates and
evaporates continuously until the system reach the equilibrium temperature.

##  V  Cosmic Microwave Background as Thermal Reservoir

After the temperature and density decrease due Universe expansion, the
interactions among photons and hydrogen atoms also decreased. The photons,
earlier with sufficient energy to remove electrons from atoms, became less
energetic and could no longer remove electrons, but moved freely without
interacting with matter. Hence, the Big Bang model predicts the cosmic
radiation background and its energy is associated to the hydrogen atom
ionization energy. Such radiation was detected as black body radiation in
microwave frequency by Arno Penzias and Robert Wilson in 1965 Penzias and
Wilson (1965); Dicke  _et al._ (1965). The current approximated value
associated to the Cosmic Microwave Background - CMB is $T_{\mbox{\scriptsize
CMB}}\approx 2.7255\mbox{K}$. We consider the CMB as the thermal reservoir and
we study the energy exchange in the system. Therefore, it is possible to
associate a thermal current (energy flux) by considering the Stefan-Boltzmann
relation, writing $T_{R}=T_{\mbox{\scriptsize CMB}}$, as

$J=\lambda A\left(T^{4}-T_{\mbox{\scriptsize CMB}}^{4}\right).$ (20)

Performing the procedure adopted in the previous section, the temporal
evolution of the black string mass per unit of length can be determined by the
following differential equation

$\frac{dM_{+}}{dt}=-aM_{+}^{2}+bM_{+}^{\frac{2}{3}},$ (21)

where

$\displaystyle a$ $\displaystyle=\frac{27\alpha^{3}}{160\pi}$ (22)
$\displaystyle b$
$\displaystyle=\frac{4^{\frac{2}{3}}\pi^{3}}{30\alpha}T_{\mbox{\scriptsize
CMB}}^{4}.$

The relation (21), which represents the temporal evolution of the mass per
unit of length considering the CMB as thermal reservoir, can be investigated
for the purpose of obtaining a clue with respect to the mass behavior of the
system. It is possible to calculate an equilibrium mass relation: after some
modifications, we integrate the equation (21) as

$\int
dt^{\prime}=\int\frac{dM^{\prime}_{+}}{-a{M^{\prime}_{+}}^{2}+b{M^{\prime}_{+}}^{\frac{2}{3}}}=-3\int\frac{dx}{ax^{4}-b}$
(23)

where $x={M^{\prime}_{+}}^{\frac{1}{3}}$, $a$ and $b$ are given in equation
(22). Performing the integration, we obtain

$\displaystyle t$ $\displaystyle=$
$\displaystyle\frac{3}{4a^{\frac{1}{4}}b^{\frac{3}{4}}}\left\\{2\arctan\left[\left(\frac{a}{b}\right)^{\frac{1}{4}}M^{\frac{1}{3}}\right]+\ln\left(b^{\frac{1}{4}}+a^{\frac{1}{4}}M^{\frac{1}{3}}\right)\right.$
(24) $\displaystyle-$
$\displaystyle\left.\ln\left(b^{\frac{1}{4}}-a^{\frac{1}{4}}M^{\frac{1}{3}}\right)\right\\}+K$

where $K$ is an integration constant. We notice a change in behavior when the
logarithms arguments are zero, that is,

$\displaystyle b^{\frac{1}{4}}-a^{\frac{1}{4}}M^{\frac{1}{3}}$
$\displaystyle=$ $\displaystyle 0$ (25) $\displaystyle
b^{\frac{1}{4}}+a^{\frac{1}{4}}M^{\frac{1}{3}}$ $\displaystyle=$
$\displaystyle 0.$ (26)

The second relation above results negative mass and for this reason such
relation is not important. However, the first relation provides the mass
associated to the change in system behavior as

$M_{e}=\left(\frac{b}{a}\right)^{\frac{3}{4}}=\frac{16}{27}\left(\frac{\pi
T_{\mbox{\scriptsize CMB}}}{\alpha}\right)^{3}.$ (27)

The mass per unit of length shown in equation (27) represents the mass value
for which the thermal equilibrium of the system is reached. This
interpretation can be confirmed if we calculate what should be the mass
associated with the CMB temperature. Considering the relation (12) applied for
CMB one obtains

$r_{\mbox{\scriptsize CMB}}=\frac{4\pi}{3\alpha^{2}}T_{\mbox{\scriptsize
CMB}}$

and now, considering the relation (3), one obtains the following result which
confirms the interpretation concerning the equation (27)

$M_{\mbox{\scriptsize CMB}}=\frac{16}{27}\left(\frac{\pi T_{\mbox{\scriptsize
CMB}}}{\alpha}\right)^{3}=M_{e}.$

The figure 2 shows a plot of the solution to the equation (21). It can be seen
that the predictions provided by the previous discussion in the section I have
been confirmed. One can see that the mass per unit of length assumes a
constant value, $M=1$ and such fact corroborates two predictions: first, the
mass per unit of length reaches a constant value and it indicates that the
temperature gradient ceases and there is no energy exchange between black
string and CMB anymore; second, the equilibrium mass per unit of length
$M_{e}=1$ confirms relation (27) since, for this plot, it was set $a=b=1$ and
the relation (27) predicts that, for such choice of parameters, $M_{e}=1$.

Figure 2: For the solution presented it is considered $a=b=1$ and some values
for the initial mass per unit of length $M_{0}$ in arbitrary units.

##  VI  Cosmic Microwave Background in Evolution as Thermal Reservoir

The CMB evolves according to the temporal evolution of the universe. We
consider this time evolution divided into three eras: the radiation era, the
matter era, and the dark energy era. Each one represents the era when there is
a predominance of each one of the cosmic fluids (radiation, matter and dark
energy) considering the big bang as the beginning of time. The energy exchange
between the black string and the CMB is investigated by considering that the
temperature $T_{\mbox{\scriptsize CMB}}$ assumes different behaviors for each
era of the Universe. The temperature for each era can be written, in SI units,
as222These relations, presented in SI units are known and their demonstration
can be found in de Santi and Santarelli (2019). In this work, we apply such
relations in arbitrary units to understand the black string temporal
evolution.

$\displaystyle T_{R}$ $\displaystyle=$
$\displaystyle\frac{T_{\mbox{\scriptsize CMB}}}{\displaystyle\left(\frac{32\pi
G}{3}\rho_{R_{0}}\right)^{\frac{1}{4}}t^{\frac{1}{2}}}$ (28) $\displaystyle
T_{M}$ $\displaystyle=$ $\displaystyle\frac{T_{\mbox{\scriptsize
CMB}}}{\left(6\pi G\rho_{M_{0}}\right)^{\frac{1}{3}}t^{\frac{2}{3}}}$ (29)
$\displaystyle T_{\Lambda}$ $\displaystyle=$ $\displaystyle
T_{\mbox{\scriptsize CMB}}\exp\left[-\left(\frac{8\pi
G}{3}\rho_{\Lambda_{0}}\right)^{\frac{1}{2}}t\right]$ (30)

where $\rho_{R_{0}}$,$\rho_{M_{0}}$, $\rho_{\Lambda_{0}}$ represent the
current densities of radiation, matter and dark energy, respectively. Photons,
neutrinos, cosmic microwave background and gravitons are associated to the
radiation era; hadronic and non-hadronic matter are associated to the matter
era; and the dark energy era is associated to dark energy.

###  VI.1  Radiation Era

We analyze the energy flux between the black string and the thermal reservoir
by rewriting the Stefan-Boltzmann law for each era as

$J=\lambda A\left(T^{4}-T_{i}^{4}\right),$ (31)

where $A=2\alpha\pi r_{+}^{2}$, $\lambda$ is the Stefan-Boltzmann constant as
introduced previously and $i=R,M,\Lambda$ represents radiation, matter or dark
energy, respectively. Applying the relations (28) and (31) we obtain the
following differential equation in radiation era

$\frac{dM_{+}}{dt}=-aM_{+}^{2}+\frac{b\delta_{R}}{t^{2}}M_{+}^{\frac{2}{3}}$
(32)

where

$\delta_{R}=\frac{3}{\displaystyle 32\pi G\rho_{R_{0}}},$ (33)

$a$ and $b$ are given in relation (22). The solution to the equation (32) is
shown in figure 3 and it is possible to see curves approach a horizontal
asymptote defined by the value of $M_{e}$, the value of mass per unit length
associated to the thermal equilibrium which represents an equilibrium stable
point. The decrease in value of mass per unit length represents the black
string evaporation by Hawking radiation.

Figure 3: Radiation era. For this plot, $a=0.1$, $b=\delta_{R}=0.001$

###  VI.2  Matter Era

Now we consider the relation (31) for the matter era. The temporal evolution
of mass per unit length can be written as

$\frac{dM_{+}}{dt}=-aM_{+}^{2}+\frac{b\delta_{M}}{t^{\frac{8}{3}}}M_{+}^{\frac{2}{3}}$
(34)

where

$\delta_{M}=\frac{1}{\displaystyle\left(6\pi
G\rho_{M_{0}}\right)^{\frac{4}{3}}}.$ (35)

It is possible to observe that, as in the previous cases, the equation (34) is
a non-linear differential equation and the figure 4 describes its numerical
solution. This graph indicates a decrease its mass value in a similar way to
the previous ones. This represents that in the matter era the black string
evaporates and decreases its mass per unit length until the thermal
equilibrium is established. It is important to remark that the decrease of
mass value occurs faster than the other cases. Investigating the horizontal
axis, one can observe, by comparison between 4 and the previous plots, that
the decrease of mass per unit length occurs significantly for values ten times
smaller than the previous cases. This fact indicates that there is an intense
black string evaporation predicted in matter era.

Figure 4: Matter era. For this plot it is considered $a=2.5$, $b=10^{-5}$ and
$\delta_{M}=2\cdot 10^{-4}$.

###  VI.3  Dark Energy Era

We analyze now the dark energy era by applying the equation (31) with
$i=\Lambda$. The non-linear differential equation obtained is

$\frac{dM_{+}}{dt}=-aM_{+}^{2}+be^{-\delta_{\Lambda}t}M_{+}^{\frac{2}{3}}$
(36)

where

$\delta_{\Lambda}=4\left(\frac{8\pi
G}{3}\rho_{\Lambda_{0}}\right)^{\frac{1}{2}}.$ (37)

The numerical solution to the equation (36) is shown in figure 5.

Figure 5: Dark energy era. It is considered $a=0.1$, $b=0.075$ and
$\delta_{\Lambda}=0.025$ and some values for initial mass per unit length
$M_{0}$.

We remember that the choice of parameters is made with the objective of
producing a clear plot. The constants of the problem are expressed in
arbitrary units. The parameters introduced in equation (36) are also written
in arbitrary units. The figure 5 can be understood as a confirmation of the
predictions made in this paper. Since we make the assumption that the heat
capacity of the black rope is a positive quantity, we expect that a state of
thermal equilibrium shall be reached and therefore an equilibrium mass shall
be reached. Considering the figure 5, it is possible to observe that the black
string mass per unit length decreases more slowly than in the previous cases
which means that the evaporation is less intense. If we consider all the three
cases investigated in this section we notice that in the case of matter era,
the black string evaporation is more intense than the others and we observe a
similar behavior when we compare the cases of radiation era, figure 3, and
dark energy era, figure 5.

##  VII Conclusions

This paper represents a conjecture regarding the presence of the black string
in the universe as we know it considering its eras. This conjecture is
interesting to situate the black string model in the temporal evolution of the
universe. In principle, the black string subjected to the universe as we
currently understand it would be an object in thermal equilibrium with the
universe, with respect to current measurements of cosmic microwave background
radiation temperature. This interpretation refers to figure 2 in which it is
clearly seen that as time passes, the mass per unit length assumes a constant
value. This means that the rate of energy exchange between the black string
and the reservoir (CMB) has ceased and thermal equilibrium has been reached.
The study considering the change in reservoir temperature as time passes
performed in sections VI.1, VI.2 and VI.3 shows that the mass values decrease
rapidly indicating a rapid evaporation of the black string as can be seen by
investigating figures 3, 4 and 5. These conjectures are important for an
understanding of the thermal behavior of objects representing singularities
and for using a method for such studies.

###### Acknowledgements.

The authors thank to Brazilian agency CAPES - Coordenação de Aperfeiçoamento
de Pessoal de Nível Superior (Coordination for the Improvement of Higher
Education Personnel) for financial support.

## References

  * Thorne (1972) K. S. Thorne, Freeman, San Francisco (1972).
  * Chakrian and Santos (2020) J. Chakrian and A. d. P. Santos, arXiv preprint arXiv:2012.04614 (2020).
  * Fatima and Saifullah (2012) A. Fatima and K. Saifullah, Astrophysics and Space Science 341, 437 (2012).
  * Lemos (1995) J. S. Lemos, Physics Letters B 353, 46 (1995).
  * Lemos and Zanchin (1996a) J. P. Lemos and V. T. Zanchin, Physical Review D 54, 3840 (1996a).
  * Lemos and Zanchin (1996b) J. P. Lemos and V. T. Zanchin, Physical Review D 53, 4684 (1996b).
  * Parikh and Wilczek (2000) M. K. Parikh and F. Wilczek, Physical Review Letters 85, 5042 (2000).
  * Srinivasan and Padmanabhan (1999) K. Srinivasan and T. Padmanabhan, Physical Review D 60, 024007 (1999).
  * Gohar and Saifullah (2013a) H. Gohar and K. Saifullah, Astrophysics and Space Science 343, 181 (2013a).
  * Gohar and Saifullah (2013b) H. Gohar and K. Saifullah, Astroparticle Physics 48, 82 (2013b).
  * Birrell  _et al._ (1984) N. D. Birrell, N. D. Birrell, and P. Davies,  _Quantum fields in curved space_ , 7 (Cambridge university press, 1984).
  * Cai and Zhang (1996) R.-G. Cai and Y.-Z. Zhang, Physical Review D 54, 4891 (1996).
  * de Santi and Santarelli (2019) N. S. M. de Santi and R. Santarelli, Brazilian Journal of Physics 49, 897 (2019).
  * Note (1) It was applied the Classical Runge-Kutta method using the open source software of mathematics SageMath. All numerical calculations presented in this work were performed using the mentioned software.
  * Penzias and Wilson (1965) A. A. Penzias and R. W. Wilson, The Astrophysical Journal 142, 419 (1965).
  * Dicke  _et al._ (1965) R. H. Dicke, P. J. E. Peebles, P. G. Roll, and D. T. Wilkinson, The Astrophysical Journal 142, 414 (1965).
  * Note (2) These relations, presented in SI units are known and their demonstration can be found in de Santi and Santarelli (2019). In this work, we apply such relations in arbitrary units to understand the black string temporal evolution.