# RecBaselines2023: a new dataset for choosing baselines for recommender
models

Veronika Ivanova<EMAIL_ADDRESS>0009-0009-6864-392X National
Research University Higher School of EconomicsMyasnitskaya Ulitsa,
20MoscowRussian Federation101000 ,  Oleg Lashinin<EMAIL_ADDRESS>0000-0001-8894-9592 Tinkoff2-Ya Khutorskaya Ulitsa, 38A, bld. 26MoscowRussian
Federation117198 ,  Marina Ananyeva<EMAIL_ADDRESS>0000-0002-9885-2230
National Research University Higher School of EconomicsMyasnitskaya Ulitsa,
20MoscowRussian Federation101000 and  Sergey Kolesnikov<EMAIL_ADDRESS>0000-0002-4820-987X Tinkoff2-Ya Khutorskaya Ulitsa, 38A, bld. 26MoscowRussian
Federation117198

###### Abstract.

The number of proposed recommender algorithms continues to grow. The authors
propose new approaches and compare them with existing models, called
baselines. Due to the large number of recommender models, it is difficult to
estimate which algorithms to choose in the article. To solve this problem, we
have collected and published a dataset containing information about the
recommender models used in 903 papers, both as baselines and as proposed
approaches. This dataset can be seen as a typical dataset with interactions
between papers and previously proposed models. In addition, we provide a
descriptive analysis of the dataset and highlight possible challenges to be
investigated with the data. Furthermore, we have conducted extensive
experiments using a well-established methodology to build a good recommender
algorithm under the dataset. Our experiments show that the selection of the
best baselines for proposing new recommender approaches can be considered and
successfully solved by existing state-of-the-art collaborative filtering
models. Finally, we discuss limitations and future work.

recommender systems, dataset, baselines, evaluation

††conference: ECIR 2023; April 02, 2023; Dublin, Ireland

##  1\. Introduction

There is an increasing number of publications in the field of recommender
systems. Authors need to evaluate the performance of the proposed model
against reference models to demonstrate its efficiency. Reference models are
usually referred to as baselines. However, there are no rigid guidelines that
define a comprehensive list of essential baselines. Inaccurate selection of
baselines can lead to incorrect conclusions about the performance of the
proposed model. Subsequent papers (Ferrari Dacrema et al., 2021) on the
reproducibility and progress of existing work have demonstrated this fact. For
example, in two recent papers (Lin, 2019; Yang et al., 2019), the authors
report that for a particular information retrieval task, some non-neural
methods outperform recent neural methods. In 2016, Kharazmi et al. (Kharazmi
et al., 2016) examined previous work on IR and found a tendency to select weak
baselines for comparative experiments. In the field of recommender systems,
the empirical analyses of session-based recommendation papers showed that
sometimes almost trivial methods can outperform the latest neural methods
(Ludewig and Jannach, 2018; Ludewig et al., 2019). One of the reasons for such
disappointing performance of novel models is the poor choice of baselines,
which gives the illusion of better results. Another consequence of not
choosing appropriate baselines for a new algorithm is that the proposed paper
may be rejected (Bedi et al., 2022). Thus, the choice of baselines is
currently one of the major issues in recommender systems research (Ferrari
Dacrema et al., 2019; Beel et al., 2016).

With accurate baseline selection, the development of recommender systems can
progress more quickly. Both researchers and practitioners are faced with an
increasing number of models to select for their experiments in order to
consider relevant baselines. However, the number of baselines included in the
paper is limited for the following reasons. First, more baselines require more
computational time. The recent success of deep learning forces the inclusion
of complicated algorithms as baselines. Therefore, some researchers cannot
afford to choose effective hyperparameters sufficiently (Ferrari Dacrema et
al., 2019). Second, some papers with new recommender algorithms do not have
the source code of the implementation (Ferrari Dacrema et al., 2019). This may
lead to poor performance of third party implementations (Petrov and Macdonald,
2022; Ferrari Dacrema et al., 2019). Finally, due to space limitations, a
paper cannot include too many baselines. Thus, it is common practice to study
the performance of only 3-7 baselines against the newly proposed method. The
problem of selecting a few relevant items from a large set is a well-known
task and can be solved by recommender systems (Bollen et al., 2010). To the
best of our knowledge, there is no open source dataset that can be used as a
basis for developing the recommender baseline suggestion system.

It is important to note that the baseline recommendations can be applied to
other areas of machine learning, such as natural language processing, computer
vision, time series prediction, and others. However, in this paper we focus
only on recommender systems.

In this paper we describe a process for collecting a novel dataset called
RecBaselines2023. It can be considered as a classical dataset with
interactions between papers and baselines. In addition, we present the results
of experiments that have been performed on RecBaselines2023. Our results show
potential advantages of our experiments and open new research directions.
Specifically, the main contribution of this paper can be listed as follows:

  * •

We have created a new open source dataset called RecBaseline2023111We are
releasing an online version of the dataset:
https://github.com/fotol1/recbaselines2023.  for selecting baselines for
experiments on recommender models. We examined 1009 papers for the collection
process. After preprocessing, RecBaseline2023 contains information on 363
baselines used in 903 articles published between 2010 and 2022. We also
provide a data collection procedure and descriptive statistics.

  * •

We discuss that the problem of baseline selection can be solved by
collaborative filtering approaches. We then compare the baseline ranking
quality of seven state-of-the-art top-N recommender models on RecBaseline2023.
The results show that this problem can be effectively solved by selected
algorithms.

  * •

We describe a scenario where a partial list of baselines needs to be
completed. The list is given to collaborative filtering approaches that
recommend baselines based on the list of methods already used. Some other
possible use cases of RecBaseline2023 are also mentioned.

##  2\. Related work

The problem of choosing baselines for research experiments in machine learning
is not well studied. A similar problem is citation recommendations. This
direction aims at suggesting other papers to cite.

The two main classes of citation recommendations are content-based and
collaborative filtering (Beel et al., 2016). The content-based methods use
textual elements such as abstract and title or metadata elements such as
authors. In (Bhagavatula et al., 2018), the authors proposed a content-based
approach requiring only textual features and collected the OpenCorpus dataset
of 7 million articles. The literature graph was created in (Ammar et al.,
2018) using nodes for articles, authors and scientific concepts. Collaborative
filtering methods are based on comparing similarities between articles. Liu et
al. (Liu et al., 2015) measured the cosine similarity of article vectors and
created article vectors based on co-occurrence in the same citation list. The
same concept was used by Haruna et al. (Haruna et al., 2017). However, they
considered the references and citations of the target paper and mined the
hidden associations between them using paper-citation relationships. Later, by
improving the similarity calculation, this approach was further developed in
(Sakib et al., 2020).

Although we know what to cite, it is not clear whether the recommended paper
should be used as a baseline. Therefore, researchers are also working on more
specialised tasks such as tag or baseline recommendations. The task of tag
recommendation has been successfully studied by Wang et al. (Wang et al.,
2013). They used the collaborative topic regression model. The authors sampled
items from the CiteULike dataset, including abstracts, titles and tags for
each article. Bedi et al. (Bedi et al., 2022) introduced the task of
identifying the papers used as baselines in a given scientific article. The
author formulated it as a reference classification problem on a developed ACL
anthology corpus dataset, where about 2000 papers were selected and manually
annotated.

However, research article datasets are not specifically designed for the task
of selecting baselines for recommender system experiments. We hope that our
dataset will help to fill this gap and provide researchers with a practical
approach to selecting baseline models for their research.

##  3\. Dataset

Collection. We added several common recommendation tasks to our dataset,
including the traditional top-n, next-item and next-basket recommendations.
These tasks were used as the basis for the data collection. For each task
there are well-established and highly cited baselines, some of which are
listed in Table 1. Note that there are no strict guidelines in recommender
systems research as to which baselines should be used for each of the above
tasks. Therefore, we cannot guarantee that other algorithms cannot be used to
complement the list of commonly used algorithms. However, the approaches
listed in Table 1 have many citations, which is appropriate for the starting
point of data collection.

Table 1. The table shows the corresponding highly cited baselines in each of
the three recommender tasks. We use these as a starting point for our data
collection. Citation counts are valid as of 27 March 2023.

Recommender task | Mandatory baselines (Citation counts)
---|---
conventional top-N | BPR-MF (5297) (Rendle et al., 2012), WMF (3688) (Hu et al., 2008), MultVAE (844) (Xu et al., 2021), LightGCN (1245) (Shen et al., 2021a)
next-item | GRU4Rec (2152) (Hidasi et al., 2015), SASRec (1146) (Kang and McAuley, 2018), BERT4Rec (836) (Sun et al., 2019)
next-basket | TIFUKNN (56) (Hu et al., 2020), RepeatNet (189) (Ren et al., 2019)

Table 2. An example of a row in the RecBaselines2023 dataset. This file is
called before_preprocessing.csv and is available online.

Column | Description | Example
---|---|---
Paper_id | Unique paper identification | 12
URL | WEB link to a paper | arxiv.org/pdf/1809.07053.pdf
Title | Paper title |  NAIS: Neural Attentive Item Similarity Model for Recommendation
Year | A year of publishing | 2018
Baselines | List of used recommender algorithms | MF;MLP;FISM;NAIS

Table 3. Examples of different names for the same algorithms Approach | Occurring names
---|---
POP | MostPop, Popular, TopPopular
BPR-MF (Rendle et al., 2012) | BPR, BPR-MF, MF-BPR
NCF (He et al., 2017) | NeuCF, NeuMF, NCF
Mult-VAE (Liang et al., 2018) | Mult-VAE, Multi-VAE, VAE-CF

200920112012201320152016201720182019202020212022$0$$100$$200$Number of papers
(a) Distribution of the number of papers by the year of publication.

34567891011121314151629$0$$50$$100$$150$Number of baselinesNumber of papers
(b) Distribution of the number of papers over the number of algorithms per one
paper.

Figure 1. Dataset statistics

To collect our dataset, we took the following steps:

  1. (1)

For each model from Table 1, we obtain the list of papers that cited the model
in Google Scholar (Jacsó, 2005). If a paper included experiments with the
model, we included it. We did not include papers with experiments on related
problems (such as link prediction or matrix completion, explanation
generation). In addition, papers without experiments are not included in the
dataset. Note that a paper could cite more than one baseline model of Table 1.
Duplicate papers were later filtered out of the dataset during pre-processing.
Once we had gone through all the citations of models in Table 1, we continued
to process citations of papers that had already been added. This was all done
manually by the authors of the paper over the period of one month.

  2. (2)

Information about each paper collected to build our dataset is presented in
Table 2. Each row contains a paper id, URL, paper title, year of publication
and a list of recommender models used. The URL and year of publication are
taken from the Google Scholar page, while the paper title and list of
baselines are taken from the paper itself.

After removing duplicates, we obtained the dataset with 1009 papers and 2187
baselines. A large number of baseline models were only included in one or two
papers.

Preprocessing. A number of steps were taken to preprocess the data for future
research:

  1. (1)

In some papers, popular models are presented under different names. This is
most likely due to space limitations or different names for algorithms that
were even proposed in the original paper. For example, the authors of the
article Neural Collaborative Filtering (NCF) (He et al., 2017) used a
different name, NeuMF, in their experiments. As a result, the cited articles
include both NCF and NeuMF. We try to condense common cases and list them in
Table 3. To resolve this inconsistency, we have replaced the multiple names of
a model with a single option.

  2. (2)

Some papers modify methods slightly and report different variations of the
same methods. For example, the authors introduce three new loss functions in
(Gao et al., 2022) and apply them to different methods such as NeuMF (He et
al., 2017), CML (Hsieh et al., 2017) and LightGCN (Shen et al., 2021a).
Considering three losses for each of the three models makes our dataset more
sparse. To avoid this problem, the preprocessed version of RecBaselnies2023
contains only the main algorithms without any specified modifications.

  3. (3)

To replace rare baselines and papers with extremely few baselines, we then
iteratively filtered the dataset until there were only papers with three or
more baselines and each baseline was present in three or more papers. The
resulting statistics for the filtered dataset can be found in Table 4.

Table 4. Descriptive statistics of RecBaselines2023 dataset. Stage | # papers | # models | # interactions | density
---|---|---|---|---
Collection | 1009 | 2188 | 7748 | 0.3%
Preprocessing | 903 | 363 | 5467 | 1.6%

Statistics. We briefly present some statistics from the collected dataset. The
main characteristics such as number of papers, number of models, number of
interactions and density are presented in Table 4.

BPRGRU4RECFPMCPOPSASRECNEUMFNARMCASERLIGHTGCNITEMKNN$150$$200$$250$$300$$350$Number
of papers Figure 2. Distribution of the number of papers for the top-10 most
popular baselines, included in our dataset.

20092011201320152016201720182019202020212022$0$$20$$40$$60$$80$Number of
papers (a) BPRMF

201720182019202020212022$0$$20$$40$$60$$80$$100$Number of papers (b) GRU4REC

201320152016201720182019202020212022$0$$20$$40$Number of papers (c) FPMC

2016201720182019202020212022$20$$40$$60$Number of papers (d) POP

20182019202020212022$0$$20$$40$$60$Number of papers (e) SASREC

201720182019202020212022$0$$20$$40$$60$Number of papers (f) NEUMF

20172019202020212022$0$$10$$20$$30$$40$$50$Number of papers (g) NARM

20182019202020212022$10$$20$$30$$40$Number of papers (h) CASER

202020212022$0$$20$$40$$60$$80$Number of papers (i) LIGHTGCN

200920112013201720182019202020212022$0$$10$$20$$30$$40$Number of papers (j)
ITEMKNN

2019202020212022$10$$20$$30$$40$$50$Number of papers (k) SRGNN

2019202020212022$0$$20$$40$Number of papers (l) NGCF

2019202020212022$15$$20$$25$$30$$35$Number of papers (m) STAMP

2019202020212022$0$$10$$20$$30$$40$Number of papers (n) BERT4REC

2016201720182019202020212022$5$$10$$15$Number of papers (o) MF

Figure 3. The figures show the distribution over the years of the number of papers in which one of the top-15 most popular baselines was included. Table 5. Performance comparison on RecBaselines2023. The best value is in bold. Model | R@10 | R@20 | N@10 | N@20 | M@10 | M@20
---|---|---|---|---|---|---
Random | 0.045 | 0.08 | 0.029 | 0.029 | 0.008 | 0.008
BPRMF (Rendle et al., 2012) | 0.2281 | 0.353 | 0.134 | 0.134 | 0.035 | 0.035
MostPop | 0.312 | 0.339 | 0.138 | 0.138 | 0.035 | 0.035
MF2020 (Rendle et al., 2020) | 0.348 | 0.446 | 0.227 | 0.227 | 0.067 | 0.067
EASER (Steck, 2019) | 0.397 | 0.549 | 0.243 | 0.243 | 0.069 | 0.069
NeuMF (He et al., 2017) | 0.42 | 0.513 | 0.252 | 0.252 | 0.073 | 0.073
Slim (Ning and Karypis, 2011) | 0.446 | 0.576 | 0.264 | 0.264 | 0.078 | 0.078
VAECF (Liang et al., 2018) | 0.455 | 0.603 | 0.264 | 0.264 | 0.075 | 0.075
$RP3_{\beta}$ (Paudel et al., 2016) | 0.473 | 0.607 | 0.303 | 0.303 | 0.088 | 0.088

Figure 1 and Figure 2 represent three distributions of the dataset: the
distribution of the number of papers over the year they were published, the
distribution of the number of papers over the number of algorithms included in
a paper, and the distribution of the number of papers for the top 10 most
popular baselines included in our dataset. The earliest publication date of a
paper is 2009, the number of papers remains relatively small and only exceeds
10 in 2017. Then the number of papers increases significantly from year to
year. As can be seen in Figure 1, a typical number of baselines included is
between 3 and 8. Therefore, the algorithms used to recommend baselines for
recommender systems have to work with a small number of available
interactions.

Figure 3 describes the distribution of the number of papers over the years for
each of the top 15 popular baselines from the RecBaselines2023 dataset. The
most popular models are BPR, GRU4Rec, LightGCN, NeuMF and others. These models
were used as starting points for the collection of other papers. Therefore,
they are represented in the dataset in large numbers.

##  4\. Collaborative Filtering for Baseline Selection

Baseline selection can be solved by collaborative filtering (CF) algorithms.
For example, the following definition was given in (Schafer et al., 2007).

###### Definition 4.0.

Collaborative filtering is the process of filtering or ranking items using the
opinions of other people.

We can replace the word ”items” with ”baselines” and the word ”people” with
”researchers”. This definition then provides a justification for the use of
the technique. In addition, scientists’ ”opinions” are often motivated by
several reasons. The first is the desire to compare the new algorithm with the
best-known or best-performing approaches. The second is to include models
based on the same idea. For example, the authors of (Shen et al., 2021b)
compare their graph-based model with 4 baselines, 3 of which are also graph-
based. These or other reasons explain the choice of models from a large number
of options.

Therefore, researchers and practitioners may be interested in baseline
recommendations based on a partial list of algorithms already in use.
Hopefully, this can be done by applying approaches in the inductive scenario
(Schnabel et al., 2022). These approaches do not have ID-based user embeddings
(Shen et al., 2021b; Xu et al., 2021; Wu et al., 2022; Ananyeva et al., 2022).
They understand user interests based on the set of interactions. Therefore, we
can easily adopt such techniques for suggesting baselines based on a partial
list of already included methods.

##  5\. Experiments

We have experimented with collaborative filtering approaches on the Top-N
recommendation task on the RecBaselines2023 dataset. Our experiments aim to
answer the following question: ”What is the performance of different state-of-
the-art collaborative filtering approaches on the RecBaselines2023 dataset?”

Models. We included popular approaches of different types: simple random and
MostPop; matrix factorisation based BPRMF (Rendle et al., 2012), MF2020; item
based EASE (Steck, 2019), SLIM (Ning and Karypis, 2011); graph based $RB3$
(Paudel et al., 2016), VAE based MultiVAE (Xu et al., 2021). According to
(Anelli et al., 2022; Ferrari Dacrema et al., 2019), such models are very
strong CF-based baselines.

Metrics. Standard quality ranking metrics are chosen, namely Recall@K, NDCG@K
and MAP@K.

Experiment settings. To provide reproducible experiments we use Elliot (Anelli
et al., 2021) similar to (Anelli et al., 2022). This framework allows
experiments to be fully described in a configuration file. This file is
available online 222https://github.com/fotol1/recbaselines2023 and
hyperparameter ranges can be found there. The total number of hyperparameters
set for each model is 20.

Evaluation protocol. All interactions are divided into train/valid/test
splits. The valid split is used for early stopping and hyperparameter
selection. The final quality is estimated on the test split. All papers
published before 2021 are used for training. In addition, 80 $\%$ of the
interactions for papers published in 2021 and 2023 are used for training, and
the remaining 20 $\%$ are used for validation and test, respectively.

Results. To investigate our question, we report quality metrics for different
approaches in Table 5. As we can see, the best model $RP3_{\beta}$ is two
times better than MostPop’s recommendations. This shows that there are not
many universal baselines in recommendations, and that researchers choose
baselines carefully. Surprisingly, the best model, $RP3_{\beta}$, has a
Recall@20 of 0.6. This means that we can find more than half of the hidden
baselines in lists of length 20.

Figure 4. The scientist chooses baselines for his experiments. Three baselines
have already been chosen. He or she can now pass the three approaches to one
of the collaborative filtering algorithms under consideration. As a result,
two more baselines are suggested as additional approaches to include in the
paper. Based on historical data, these recommended baselines may also be
chosen by other researchers.

##  6\. Selecting baselines for partial lists

This section describes one possible way of using RecBaselines2023. The Figure
4 represents the main idea. A scientist has invented a new recommendation
algorithm and wants to compare it with other work. For example, a new approach
was inspired by two methods a and b. So they are automatically included in the
experiments of the new paper. In addition, the researchers know that a current
state-of-the-art algorithm is a model c. So it should also be considered.
Given the set of three baselines $\\{\textbf{a},\textbf{b},\textbf{c}\\}$, he
or she can run the set in one of the adapted collaborative filtering
approaches. This will return a list of recommended baselines. The choice of
these is consistent with the historical data represented in RecBaselines2023.

Table 6. Examples of recommendations based on partial lists of items.

Input items | SLIM | EASE | $RP3_{\beta}$
---|---|---|---
GRU4REC | CASER, SASREC | SASREC, CASER | MIND, DIEN
GRU4REC, SASREC, BERT4REC | CASER, TISASREC | BPR, CASER | TISASREC, JODIE
GRU4REC, SASREC, BERT4REC, TISASREC | CASER, S3REC | BPR, CASER | LESSR, CHORUS
PINSAGE | GCMC, CMN | NGCF, GCMC | CMN, GCMC
VAECF | TRANSCF, LRML | LIGHTGCN, TRANSCF | SGL, TRANSCF
RIPPLENET | PER, CKE | CKE, PER | PER, LIBFM
LIGHTGCN, SGL, VAECF | TRANSCF, LRML | NGCF, BPR | TRANSCF, SBPR

In Table 6 we demonstrate recommendations for some sub-lists of baselines. We
use SLIM, EASE and $RP3_{\beta}$ as recommender models because they are item-
based models that can make predictions based on any input list of items. The
first three examples emulate iterative updates to the next-item recommender
set of baselines. The next examples demonstrate recommendations based on a
single baseline using different frameworks. As we can see, SLIM and
$RP3_{\beta}$ are flexible in changing recommendations as new next-item models
are added. When we provide only one element of a particular framework, our
models recommend baselines using similar frameworks. For example, RippleNet
(Wang et al., 2018) is a knowledge-based model. If someone includes RippleNet
in their experiments, our models will suggest including other knowledge-based
approaches such as PER (Yu et al., 2014), CKE (Wang et al., 2019).

##  7\. Limitations and future work

Our work has some limitations. In this section we will discuss them and show
possible ways to overcome them.

Firstly, the published version of the dataset will become obsolete. We will
publish regular updates. In addition, if authors of newly proposed methods
want to add their work, we can do this quickly in the repository via a pull
request.

Secondly, The dataset may contain misspellings or other errors, with the
presence or absence of some baselines in the included works. We have tried to
do our best, and have double-checked the interactions several times. If you
find any errors, please contact us via issues on Github.

Finally, There are some challenges in publishing baseline recommendations. For
example, some of the baselines presented have been used in previous work.
However, the latest state-of-the-art approaches replace them. The models
considered are not sensitive to this fact. We argue that this problem exists
for other datasets as well. It has been shown in (Neophytou et al., 2022) that
recommending the most recent films can improve quality even for the simple
MostPop method. Nevertheless, the practical application can be modified and
the most recent baselines with high relevance scores can be treated as more
suitable. We leave this as future work.

##  8\. Conclusion

This paper investigates the problem of recommending baselines for experiments.
We have collected an open source dataset RecBaselines2023, which describes
baseline models used for comparative experiments in papers on different types
of recommender systems. It consists of 903 papers and 363 baseline models,
with 5467 interactions between them. The dataset includes interactions between
papers and baseline models, and additional data about each paper, such as web
link to a paper, paper title, and year of publication. RecBaselines2023 can be
used by researchers to properly compile the baseline list for their
experiments. The dataset will be updated as new papers are published. We have
used collaborative filtering techniques to identify the best algorithms based
on incomplete lists of previously included baselines. Our experiments with
hidden predictions of recommender baselines show that state-of-the-art
collaborative filtering techniques can successfully perform this task. We hope
that our dataset can open up new lines of research.

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# Large scale FRET simulations reveal the control parameters of phycobilisome
light harvesting complexes

Emma Joy Dodson1 1Department of Plant and Environmental Science, The Alexander
Silberman Institute of Life Sciences, The Hebrew University in Jerusalem,
Jerusalem, Israel  Nicholas Werren2 1Department of Plant and Environmental
Science, The Alexander Silberman Institute of Life Sciences, The Hebrew
University in Jerusalem, Jerusalem, Israel  Yossi Paltiel3 1Department of
Plant and Environmental Science, The Alexander Silberman Institute of Life
Sciences, The Hebrew University in Jerusalem, Jerusalem, Israel  Erik
Gauger2,* 1Department of Plant and Environmental Science, The Alexander
Silberman Institute of Life Sciences, The Hebrew University in Jerusalem,
Jerusalem, Israel  Nir Keren1,* 1Department of Plant and Environmental
Science, The Alexander Silberman Institute of Life Sciences, The Hebrew
University in Jerusalem, Jerusalem, Israel

###### Abstract

Phycobilisomes (PBS) are massive structures that absorb and transfer light
energy to photochemical reaction centers. Among the range of light harvesting
systems, PBS are considered to be excellent solutions for absorption cross-
sections but relatively inefficient energy transferring systems. This is due
to the combination of a large number of chromophores with intermediate
coupling distances. Nevertheless, PBS systems persisted from the origin of
oxygenic photosynthesis to present day cyanobacteria and red algae, organisms
that account for approximately half of the primary productivity in the ocean.
In this study we modeled energy transfer through subsets of PBS structures,
using a comprehensive dynamic Hamiltonian model. Our approach was applied,
initially, to pairs of phycobilin hexamers and then extended to short rods. By
manipulating the distances and angles between the structures we could probe
the dynamics of exciton transfer. These simulations suggest that the PBS
chromophore network enhances energy distribution over the entire PBS structure
– both horizontally and vertically to the rod axis. Furthermore, energy
transfer was found to be relatively immune to the effects of distances or
rotations, within the range of intermediate coupling distances. Therefore, we
suggest that the PBS provides unique advantages and flexibility to aquatic
photosynthesis.

##  1 Introduction

In the photosynthetic process, light-harvesting pigment-protein complexes
absorb light energy and transfer it to photosystems, where photochemical
reactions occur [1]. While the structure of the photosystems is highly
conserved in evolution, light-harvesting is done by a broad and diverse array
of complexes [2]. One of the most prevalent light-harvesting systems is the
phycobilisome (PBS), present in cyanobacteria and red algae. It is composed of
soluble proteins anchored to the photosynthetic membrane surface and
chromophores – tetrapyrrole molecules that covalently attach to conserved
sites on the proteins [3]. The spectral properties of the chromophores are
determined by their chemical nature and by their interaction with proteins
[4]. The basic building blocks of PBS structures are phycobilin chromophore
binding proteins which assemble into hexamers (see Figure 1 for hexamer
visualization). These hexamers either organize to form the core of the PBS
antenna or stack into rod structures with the help of mostly unpigmented
linker proteins [4]. Antenna rods channel excitonic energy to the PBS core for
transfer to the photochemical reaction centers embedded in the thylakoid
membranes [5].

The efficiency of energy transfer through light harvesting systems is a major
determinant of the overall efficiency of the photosynthetic process. Among
light harvesting systems, the PBS are not considered to be very efficient. The
number of chromophores in a single antenna system can reach well above 1000
[6], creating one of the largest known absorption cross-sections for the
photosynthetic unit [7]. According to FRET random walk principles, the
diffusion length of an exciton increases with the number of pigments in a
photosynthetic unit [8]. However, experimental results of energy transport in
photosystems give a more nuanced picture wherein efficiency and yield are
demonstrated to be constrained by the physical configuration of the system and
its surroundings [9]. The chromophores of PBS are relatively far apart,
generating a network of intermediately coupled chromophores. Furthermore, PBS
rods can reach four hexamers in length, while FRET random walk calculations
demonstrate that an antenna rod extending beyond three hexamers in length will
exceed the limit for maintaining energy-efficient transfer [10]. However, it
is important to note that high efficiency should not be considered a goal in
and of itself for photosynthesis in natural environments [11]. Large dynamic
range and robustness of energy transfer can be equally or even more important
for the fitness of a photosynthetic organism.

Over the past few decades, different approaches have been applied to model and
simulate excitation energy transfer (EET) of the PBS [12, 13, 14, 15, 16]. In
these studies, models of protein hexamers and short rods were used to simulate
energy flow with calculations based on Förster coupling between the transition
dipoles of the interacting chromophores as well as their spectral properties.
While the authors of these studies were able to propose pathways for energy
transfer in the system, the simulations were limited due to missing structural
details of intact PBS systems, particularly the exact conformation of hexamer
stacking along with the orientation of the rods relative to each other. In
recent years, intact PBS structures have been resolved. The first intact
structures were from red algae [6, 17]. These structures are massive,
containing 14 rods and 1598 light-absorbing chromophore molecules. More
recently, the Kerfeld lab resolved the structure of a cyanobacterial PBS
containing four Orange Carotenoid Proteins (OCP) [18]. These structures
provide a good deal of here-to-unknown structural information and a basis for
understanding changes in PBS efficiency, through energy transfer simulations
[18]. However, these simulations did not calculate dynamic quantum effects.

In this work, we present a framework for modelling energy transport with a
quantum model. By combining structural PBS data with a comprehensive
Hamiltonian model, we capture exciton dynamics relevant to fundamental
biological processes. The model described here utilises pairwise, distance-
dependent dipole-dipole coupling (Förster) terms. Using this approach, we were
able to efficiently run large-scale calculations to simulate the distribution
and efficiency of PBS energy transfer which provide insight into the design
principles that govern this immensely complex photosynthetic antenna system.

##  2 Model

###  2.1 Physical model

The approach outlined in the following section is similar to previous
approaches used to model light-harvesting antennae and Iron-stress-induced-A
proteins [19, 5, 20, 21, 22, 23]. We reduce each chromophore in PBS down to an
optical dipole located at the centre of mass from which we orientate the
respective transition dipole vector. These values are influenced by their
position in the photosystem and the corresponding protein environment. We
therefore consider three different species of chromophore in this system:
$\alpha$-84, $\beta$-84, and $\beta$-155. The resulting description of the
aforementioned hexamer structures in PBS is broken down into a collection of
optical dipoles captured by the following Hamiltonian:

$\hat{H_{S}}=\sum^{N}_{j=1}\omega^{s}_{j}\hat{\sigma}_{j}^{z}+\sum^{N}_{j,k=1}J_{j,k}(\textbf{r}_{j,k})\left(\hat{\sigma}_{j}^{+}\hat{\sigma}_{k}^{-}+\hat{\sigma}_{j}^{-}\hat{\sigma}_{k}^{+}\right)~{},$
(1)

where the first term is the bare Hamiltonian of the chromophore sites, with
species-dependent transition frequency $\omega^{s}_{j}$ at the $j$th site, and
the second describes exciton hopping between the $j$th and the $k$th site,
captured by the raising and lowering operators $\hat{\sigma}_{j}^{-}$ and
$\hat{\sigma}_{k}^{-}$. The strength of these hopping terms, the resonant
Förster interactions, are given by $J_{j,k}(\textbf{r}_{j,k})$:

$J_{j,k}(\textbf{r}_{j,k})=\frac{1}{4\pi\epsilon_{0}}\bigg{(}\frac{\textbf{d}_{j}\cdot\textbf{d}_{k}}{|\textbf{r}_{j,k}|^{3}}-\frac{3(\textbf{r}_{j,k}\cdot\textbf{d}_{j})(\textbf{r}_{j,k}\cdot\textbf{d}_{k})}{|\textbf{r}_{j,k}|^{5}}\bigg{)}~{},$
(2)

for dipoles of oscillator strengths $\textbf{d}_{j}$ and $\textbf{d}_{k}$ and
separated by the distance $\textbf{r}_{j,k}$ [21]. Being closely spaced
relative to the wavelength of optical photons, the chromophores can be assumed
to interact collectively with a shared optical bath [24]; this coupling is
captured by the optical interaction Hamiltonian [25],

$\hat{H}_{I,opt}=\sum^{N}_{k=1}\textbf{d}_{k}\hat{\sigma}_{k}^{x}\otimes\sum_{p}f_{p}(\hat{a}_{p}+\hat{a}_{p}^{{\dagger}})~{},$
(3)

where $\hat{a}_{p}^{({\dagger})}$ is the annihilation (creation) operator for
the $p$th optical mode and $f_{p}$ is the coupling strength between the $p$th
mode and the chromophores. Furthermore, the dipoles are each assumed to be
coupled to identical local phonon baths. Thus, we consider a vibrational
interaction Hamiltonian of the form:

$\hat{H}_{I,vib}=\sum^{N}_{k=1}\hat{\sigma}_{k}^{z}\otimes\sum_{q}g_{q}\left(\hat{b}_{k,q}+\hat{b}_{k,q}^{{\dagger}}\right)~{},$
(4)

where $g_{q}\equiv g_{k,q}$ and $\hat{b}_{k,q}^{({\dagger})}$ are the coupling
strength and annihilation (creation) operator for the $q$th phonon mode with
the $k$th dipole, respectively. Lastly, the optical and phonon environmental
modes are governed by the following Hamiltonian:

$\hat{H}_{f}=\sum_{p}\omega_{p}\hat{a}^{{\dagger}}_{p}\hat{a}_{p}+\sum_{q,k}\tilde{\omega}_{q,k}\hat{b}^{{\dagger}}_{q,k}\hat{b}_{q,k}~{},$
(5)

where $\omega_{p}$ and $\tilde{\omega}_{q,k}$ are the frequencies of the $p$th
photon modes and $q$th phonon modes at the $kth$ dipole. The total Hamiltonian
is then given by,

$\hat{H}=\hat{H}_{S}+\hat{H}_{I,vib}+\hat{H}_{I,opt}+\hat{H}_{f}~{},$ (6)

describing the dynamics of chromophores as a collection of dipoles in
interaction with each other, a shared optical bath, and individual local
phonon baths. We make the approximation that the interaction terms are weak
and can therefore be used in a perturbative expansion wherein we truncate to
second-order terms [25, 23]. The resulting Redfield quantum master equation
describes the evolution of the density matrix $\rho$ of only the chromophore
system:

$\frac{\partial\rho}{\partial
t}=-i\left[\hat{H},\rho\right]+\Gamma_{opt}\tilde{\mathcal{D}}_{opt}[\hat{\sigma}_{x}]+\Gamma_{vib}\tilde{\mathcal{D}}_{vib}[\hat{\sigma}_{z}]~{}+\Gamma_{rad}\bar{\mathcal{D}}_{rad}[\hat{\sigma}^{-}_{t}]~{},$
(7)

where $\Gamma_{vib}$ and $\Gamma_{opt}$ are the relaxation rates associated
with the vibrational and optical baths, respectively. As equations (4) and (3)
take the form $\hat{H}=\hat{A}\otimes\hat{B}$ then the terms
$\tilde{\mathcal{D}}_{opt}$ and $\tilde{\mathcal{D}}_{vib}$ are the non-
secular Redfield dissipators for the photon and phonon fields, taking the
form:

$\tilde{\mathcal{D}}=\sum_{n,m}\Gamma_{nm}\left(\omega_{m}\right)\left(A_{m}\left(\omega_{m}\right)\rho(t)A^{{\dagger}}_{n}\left(\omega_{n}\right)\right.\\\
\left.-A^{{\dagger}}_{n}\left(\omega_{n}\right)A_{m}\left(\omega_{m}\right)\rho(t)+h.c.\right)~{},$
(8)

where $A_{n}$ are the system operators associated with the dissipative process
and the prefactors $\Gamma_{nm}(\omega)$ are environment correlation functions
which depend on the relevant microscopic interactions with each environment
[25]. As each vibrational environment is considered to be identical each
correlation function is of the form:

$\Gamma_{nm}(\omega)=\frac{1}{2}\gamma_{nm}(\omega)+iS_{nm}(\omega).~{}$ (9)

We choose flat spectral density in order to construct the following rates for
the vibrational dissipators;

$\gamma^{vib}_{nm}\left(\omega\right)=\kappa^{vib}_{nm}\left(1+n_{vib}(\omega)\right),~{}$
(10)

whereas for the optical dissipators;

$\gamma^{opt}_{nm}\left(\omega\right)=\kappa^{opt}_{nm}\left(1+n_{opt}(\omega)\right),~{}$
(11)

where $\kappa^{vib}_{nm}$ is a phenomenologically motivated prefactor chosen
to reflect the timescale of the phonon rates and $\kappa^{opt}_{nm}$ is the
spontaneous emission rate associated with the transition between the $n$th and
$m$th energy eigenstates defined by the equation;

$\kappa^{opt}_{nm}=\frac{\omega_{nm}^{3}\textbf{d}_{n}\cdot\textbf{d}_{m}}{3\pi\epsilon_{0}\hbar
c^{3}}.~{}$ (12)

The functions $n_{vib}(\omega)$ and $n_{opt}(\omega)$ are the Bose-Einstein
factors associated with the vibrational and optical baths respectively, each
at the appropriate temperature of the associated environment. The final term
of Eq. (7), $\bar{\mathcal{D}}_{rad}[\hat{L}]$, is a Lindblad dissipator term
describing radiative recombination, where $\Gamma_{rad}$ is the rate of this
process, and takes the standard form:

$\bar{\mathcal{D}}(\rho)=\sum_{k}\hat{\sigma}^{-}_{k}\rho\hat{\sigma}^{+}_{k}-\frac{1}{2}\\{\hat{\sigma}^{+}_{k}\hat{\sigma}^{-}_{k},\rho\\}~{},$
(13)

for the lowering (raising) operators $\hat{\sigma}^{-}_{k}$
($\hat{\sigma}^{+}_{k}$) acting upon the $k$th site. We use Equation 7 to
resolve the system dynamics, in contrast to our previous work in Ref. [23]
which analysed equilibrium properties for the same type of model applied to
the IsIA complex. This allows us to here investigate dynamic energy transport
processes emerging from a physically-motivated model following the absorption
of a photon.

It is important to highlight that the model presented in this work does not
explore the limiting case a strong interaction between chromophores and the
surrounding vibrational environment. One could resolve this by moving to the
polaron frame, which leads to an effective rescaling of the transition
frequencies and hopping terms the bare Hamiltonian [26, 22]. However, the
complexity of phycobilisome complexes and the number of sites required for a
meaningful associated transport model makes this computationally demanding.
Furthermore, such a polaron approach would require more detailed knowledge of
the particular spectral density present in PBS. For these reasons, we employ
the presented weak coupling model, which is sufficient to explore the basic
nature of exciton transport in certain physical regimes whilst still providing
a framework upon which more complex approaches can be built.

###  2.2 Structural model

Figure 1: Schematic showing the generation of models to simulate energy
distribution in the PBS antenna system. We started out with the S. elongatus
hexamer (PDB 4H0M), since the hexamer is the base unit of the antenna system.
We then removed the scaffold proteins and left only the chromophores of the
hexamer while maintaining their exact positions, as energy is transferred
between the chromophores of the system. We aligned chromophore templates
generated by Gaussian 09 to the hexamer chromophores according to type to
determine the centers of mass and dipoles for the entire hexamer. Finally, we
removed the chromophore structures and left only the centers of mass (COM) and
dipole moment vectors (white arrows) for input into the simulation.

From an extensive analysis of available structurally-resolved red algae PBS,
it is clear that the hexamer structures throughout the system are uniform and
the stacking of hexamers into rods is repetitive. This means that within an
individual hexamer or rod, there, is little variation in the spacing and
orientation between chromophores. However, the positioning of rods relative to
each other compounds the variability of the structure, affecting spacing and
orientation of inter-rod chromophores. Therefore, in this research we focused
on the transfer of energy between hexamers laterally, rather than within
single rods or hexamers. For our models, we used the solved Synechococcus
elongatus hexamer structure. We chose this hexamer, rather than one of the red
algae rod hexamers, because of the simpler chemical properties of its
chromophores and the smaller number of chromophores per hexamer. Due to the
computational power needed for our simulation, we needed to simplify the
representation of these complex molecules in our models. The dipoles and
centers of mass for each of the three chromophore types were calculated using
Gaussian 09 [27]. This data was used with the model discussed in the preceding
section of the simulation to construct a physically-motivated description of
system dynamics (see Figure 1).

To investigate the effect of distance and orientation on lateral energy
transfer, we placed two parallel copies of the S. elongatus hexamer model as
close together as possible without overlapping. For the aligned model,
$\beta$-155 chromophores from both hexamers were positioned in the inter-
hexamer interface. The rotated model was created by rotating the second
hexamer of the aligned model 65$\circ$ (Figure 2). We ran the simulation for
both models, shifting the parallel hexamers further apart at 10 Å increments
until reaching a final distance of 220 Å between hexamers (Figure 3).

Figure 2: Models of the parallel hexamer pairs in both aligned (left) and
rotated (right) orientations. Red; $\beta-84$ chromophores with a dipole
vector magnitude of 9.836 Debye, blue; $\alpha-84$ chromophores with a dipole
vector magnitude of 12.816 Debye, green; $\beta-155$ chromophores with a
dipole vector magnitude of 10.709 Debye. Hexamer 1 chromophores are labeled
and colored more brightly (on the left side of each model) and hexamer 2
chromophores are colored in a faded shade (on the right side of each model).
In the aligned model, $\beta-155$ chromophores from both hexamers are present
in the interface. In the rotated model, hexamer 2 is rotated 65$\circ$ so that
its $\beta-155$ chromophores are staggered on either side of the interface
(their positions are marked in dotted line boxes).

The rod models were produced by adding a second hexamer below each of the
original hexamers of the aligned and rotated hexamer models (see Figure 4).
The purpose of this was to determine whether lateral energy transfer would be
maintained in the presence of a vertical transfer option. The hexamers were
stacked according to the stacking patterns found in the red algae P. purpureum
PBS, one of the only two publicly available sources for rod structures. A
third rod model was created by aligning the S. elongatus hexamer model to two
hexamers of two neighboring rods of the P. purpureum PBS. This was done to
represent a real-world spatial positioning of rod pairs relative to each
other, in contrast to the very artificial, parallel setups for the other
models used in this study. We used the artificial models to define control
parameters for energy transfer in an intermediate coupled system, then
evaluated the implementation of these parameters in the real-world structure.

Figure 3: Hexamer pair model – Two copies of the S. elongatus hexamer, PDB
4H0M, were were placed in parallel next to each other as close as possible
without overlapping. The hexamers were then moved apart at increments of 10 Å
until reaching a final distance of 220 Å. Simulations were run on each
incremental shift. Purple – hexamer 1; rainbow spectrum – hexamer 2 shifted at
10 Å increments away from hexamer 1. This Figure shows only the original
position of hexamer 2 (red) and the first five shifts that reached a distance
of 50 Å.  Figure 4: Rod pair models. Left - aligned rods, middle - rotated
rods, right - real-world rods. Rod 1 for each pair is colored in cyan, rod 2
is colored in green. The $\beta-$155 chromophores in each model are colored in
purple and shown as spheres. The rest of the system is shown in cartoon
representation.

##  3 Results

###  3.1 Control Parameters for Energy transfer Between Hexamers

To evaluate the effect of distance and orientation on energy transfer, models
were constructed to simulate the lateral transfer of energy between hexamer
pairs. For each simulation, energy transfer was initiated by exciting one of
the 18 chromophores in hexamer 1. Under physiological light conditions the
photon flux density is such that double excitation of a single PBS system are
not expected and we therefore consider only the single excitation manifold
[28, 29]. The subsequent distribution of energy in both hexamer 1 and hexamer
2 is calculated across 100 picoseconds following initial excitation. For these
simulations we set the following phenomenologically-motivated rates:
indicative phonon relaxation rate $\kappa_{vib}$, $1.0$
$\mathrm{p}\mathrm{s}$$-1$), and non-radiative recombination rate
$\Gamma_{rad}$, $0.2$, $\mathrm{n}\mathrm{s}$$-1$ [21, 22, 23]. The latter
rate is chosen such that it competes with the optical rates, on the order of
nanoseconds$-1$ and calculated using $\kappa_{opt}$, whereas the former is set
such that the hierarchy of rates is appropriate, with vibrational rates being
$10^{3}$ faster than the optical rates.

Pymol [30] movies were used to visualize the energy distribution over time.
These movies are provided in supplementary material. Movie1 and Movie2 show
the energy distribution upon initially exciting an $\alpha-$84 chromophore in
the aligned and rotated models, respectively. As can be seen in the movies,
most of the energy of the system is distributed between all the other
$\alpha-$84 chromophores in both aligned and rotated models, although slightly
less is laterally transferred to the $\alpha-$84 chromophores in hexamer 2, as
compared to those in hexamer 1. There is less but still significant energy
transfer to the $\beta-$84 chromophores in hexamer 1, and a very small amount
of transfer to the $\beta-$84 chromophores in hexamer 2. There was no
significant distribution of energy to any of the $\beta-$155 chromophores.

Movie3 and Movie4 in the supplementary section show examples of energy
distribution upon initially exciting a $\beta-$84 chromophore in the aligned
and rotated models, respectively. Here we see the opposite effect as we saw
after initially exciting the $\alpha-$84 chromophores. Most of the energy in
the system is distributed between the $\beta-$84 chromophores, and to a lesser
extent to the $\alpha-$84 chromophores. Here also, there is no significant
distribution to the $\beta-$155 chromophores.

It is therefore not surprising that following initial excitation of each of
the $\beta-$155 chromophores (see Movie5 and Movie6 in the supplementary
section for the aligned and rotated models, respectively), energy distribution
is isolated to the other $\beta-$155 chromophores of the system for both
aligned and rotated models. Significantly, when an interface $\beta-$155
chromophores is initially excited in the aligned model, there is localization
of energy. That is, despite the relatively weak coupling inherent to the
system, almost all the energy remains trapped between four $\beta-$155
chromophores.

Figure 5: Heatmap showing the lateral energy transfer from hexamer 1 to
hexamer 2 over time (x-axis) and distance (y-axis) following initial
excitation of each $\alpha-84$ chromophore in hexamer 1. The color bar
measures the amount of energy transferred out of the total energy of the
system. 1.0 signifies all of the total energy, and 0.0 signifies none of the
total energy. Thus, dark blue indicates low energy transfer, while red
indicates high energy transfer.

To break down what we are seeing in the simulation further, the heatmaps in
Figure 5 show the lateral energy transfer from hexamer 1 to hexamer 2 over
time and distance following initial excitation of the $\alpha-$84 chromophores
on hexamer 1. At close proximity, we observe rapid oscillations of lateral
transfer between the hexamers (chains A, C, G, and I in the aligned and
rotated models). Over larger distances, both oscillation frequency and overall
lateral transfer declines. At close proximity in the aligned model, more
energy is transferred to hexamer 2 in the first 50 picoseconds of the
simulation, and less is transferred to hexamer 2 as time continued,
particularly in chains A, C, G, and I. At larger distances, most energy is
transferred to hexamer 2 in the second half of the simulation. Overall, the
patterns of lateral energy transfer are similar in both the aligned and
rotated models, although slightly more energy transferred laterally in the
aligned model.

Figure 6: Heatmap showing the lateral energy transfer from hexamer 1 to
hexamer 2 over time (x-axis) and distance (y-axis) following initial
excitation of each $\beta-84$ chromophore in hexamer 1. The color bar measures
the amount of energy transferred out of the total energy of the system. 1.0
signifies all of the total energy, and 0.0 signifies none of the total energy.
Thus, dark blue indicates low energy transfer, while red indicates high energy
transfer.

Following initial excitation of the $\beta-$84 chromophores on hexamer 1
(Figure 6), lateral energy transfer is also characterized by rapid oscillation
when hexamer 2 was in close proximity to hexamer 1, especially in chains D, F,
J, and L of the aligned model. However, this oscillation is not nearly as
rapid as what was seen for the $\alpha-$84 chromophores. The interior position
of the $\beta-$84 chromophores in the core of the hexamer, in contrast to the
more exterior position of the $\alpha-$84 chromophores, could explain this
reduced short-range oscillation 2. However, there is still very strong long-
range lateral energy transfer to hexamer 2 at much greater distances following
initial excitation of the $\beta-$84 chromophores, compared to what was
simulated for the $\alpha-$84 chromophores.

Figure 7: Heatmap showing the lateral energy transfer from hexamer 1 to
hexamer 2 over time (x-axis) and distance (y-axis) following initial
excitation of each $\beta-155$ chromophore in hexamer 1. The color bar
measures the amount of energy transferred out of the total energy of the
system. 1.0 signifies all of the total energy, and 0.0 signifies none of the
total energy. Thus, dark blue indicates low energy transfer, while red
indicates high energy transfer.

When either of the two hexamer 1 $\beta-$155 interface chromophores are
initially excited (Figure 7), extremely rapid energy oscillation and energy
transfer occurs. This can be seen in chains D and J of the aligned and rotated
models. However, when any of the other four chromophores on hexamer 1 (chains
B, F, H, and L) are initially excited, there was very little energy transfer
at distance 0, although energy transfer does increase between 10 Å and about
40 Å. This is the case in both aligned and rotated models.

After summing the total lateral energy transfer for each distance, we see that
following initial excitation of many of the $\alpha-$84 and $\beta-$155
chromophores, the maximum total energy transfer does not occur when the two
hexamers are at distance 0 (see Figure S1). Rather, the optimal distance for
lateral energy transfer can be as much as 30 Å from the original position.
This shows that while there is a general correlation between distance and
lateral energy transfer (the closer the hexamers, the higher the lateral
energy transfer), lateral energy transfer is somewhat diminished when the
interface chromophores are too close. Further explanation can be found in the
supplementary section.

Overall, we can see from our results that for both aligned and rotated hexamer
models energy transfer is not sensitive to moderate changes in orientation,
provided the positioning of the chromophores in the interface does not result
in strong energy localization. There is a small degree of difference in the
optimal distances for lateral energy transfer between the aligned and rotated
models. There is slightly more energy transfer in the aligned model following
initial excitation of an $\alpha$-84 chromophore, but in general energy
transfer seems to be very robust and flexible, tolerating a fair amount of
variation with relatively constant transfer patterns. Distance obviously
controls the rate of transfer, but the correlation is not trivial. Very short
hexamer-hexamer distances can be detrimental to transfer efficiency due to the
trapping of the energy in local states between chromophore pairs in the
interface.

###  3.2 Control Parameters for Energy transfer Between Rods

While simulating the energy transfer in hexamer pairs gave us a good start in
understanding the control parameters of our system, most of the PBS is made up
of rods along which energy is transferred from the periphery to the core of
the structure. We therefore placed an additional hexamer below each of the
original hexamers of our models to add the possibility of vertical transfer to
our model (see 4). This procedure created simple rod models in both the
aligned and rotated orientations.

We also created a more “real-world” model by locating two neighboring rods in
the publicly-available EM resolved P. purpureum structure (PDB 6KGX) and
aligning our cyanobacteria hexamer to two red algae hexamers in each rod. We
chose to create a homology based model instead of using the P. purpureum
hexamer itself, as the rods of the P. purpureum antenna system bind
phycoerythrobilin or phycourobilin chromophores instead of phycocyanobilin
chromophores. Moreover, the P. purpureum hexamers contain more chromophores
than S. elongatus hexamers, which would have increased the complexity of the
simulation considerably. “Real-world” rods are not parallel as in our aligned
and rotated models, rather they have a staggered orientation (see Figure 4).
As was done with the two hexamer model simulations, each of the 18
chromophores in hexamer 1 were initially excited, and energy distribution
throughout the rod pairs was simulated. Here also, energy distribution movies
were used to visualize the energy distribution in the rod systems.

Movie7, Movie8, and Movie9 in the supplementary section display the
distribution in the system after initially exciting an $\alpha$-84 chromophore
in the aligned, rotated, and real-world models, Movie10, Movie11, and Movie12
show distribution after initially exciting a $\beta-84$ chromophore in the
aligned, rotated, and real-world models, respectively, and Movie13, Movie14,
and Movie15 represent the distribution after initially exciting a $\beta-$155
chromophore in the aligned, rotated, and real-world models, respectively. For
the aligned and rotated models, these movies clearly show lateral energy
transfer patterns that are very similar to those of their hexamer pair
counterparts, despite the presence of a vertical transfer option.

Figure 8: Pymol visualization of energy distribution after initial excitation
of all six $\beta-155$ chromophores in the aligned rod model. Chromophores are
shown as sticks and colored according to type ($\alpha-84$ chromophores in
blue, $\beta-84$ chromophores in red, and $\beta-155$ chromophores in green).
The amount of energy distributed to an individual chromophore is represented
by a sphere, the size of which indicates the highest amount of energy
transferred to that specific chromophore at any point during the
100-picosecond simulation. Sphere colors match chromophore type, except for
the white sphere, which marks the initially excited chromophore. See the
methods section for more details.

To provide an overview of the energy distribution in all the rod systems after
initial excitation of all 18 chromophores in hexamer 1, we created a static
visualization similar to the movies described above. One difference in the
static visualization, however, is that the size of the sphere indicates the
highest amount of energy transferred to a given chromophore at any point
during the 100-picosecond simulation.

Figure 9: Pymol visualization of energy distribution after initial excitation
of all six $\beta-155$ chromophores in the rotated rod model. Chromophores are
shown as sticks and colored according to type ($\alpha-84$ chromophores in
blue, $\beta-84$ chromophores in red, and $\beta-155$ chromophores in green).
The amount of energy distributed to an individual chromophore is represented
by a sphere, the size of which indicates the highest amount of energy
transferred to that specific chromophore at any point during the
100-picosecond simulation. Sphere colors match chromophore type, except for
the white sphere, which marks the initially excited chromophore. See the
methods section for more details.

The most apparent differences in transfer patterns between models can be seen
following initial excitation of all six of the $\beta-$155 chromophores
(Figures 8, 9, and 10). In the aligned models, the energy distribution remains
in the area of the initially excited chromophore, and significant lateral
transfer only occurs when the initially excited chromophore is located in the
inter-rod interface. When this happens, energy is trapped between a few very
local $\beta-$155 chromophores. These patterns play out in the rotated model
as well, but the energy is trapped in the interface to a lesser degree and
spreads out more into rod 2, most likely due to the larger distance between
the $\beta-$155 chromophores in the interface. Significantly, in the real-
world model, there is a much more even distribution between all the
$\beta-$155 chromophores in both rods. This could be because the staggered
orientation causes most of the $\beta-$155 chromophores in the interface to
move slightly further apart, although there are still several interface
$\beta-$155 chromophores that are as close to each other in the staggered
model as they are in the rotated model.

The simulations of short rod structures provide additional insight into energy
transfer patterns. Importantly, both lateral and horizontal transfer are
mostly unaffected by changes in the hexamer unit orientation, provided close
localization of pigment pairs is avoided. The staggered, angled positioning of
the real-world rods seems specifically configured to avoid such localizations.

Figure 10: Pymol visualization of energy distribution after initial excitation
of all six $\beta-155$ chromophores in the real-world rod model. Chromophores
are shown as sticks and colored according to type ($\alpha-84$ chromophores in
blue, $\beta-84$ chromophores in red, and $\beta-155$ chromophores in green).
The amount of energy distributed to an individual chromophore is represented
by a sphere, the size of which indicates the highest amount of energy
transferred to that specific chromophore at any point during the
100-picosecond simulation. Sphere colors match chromophore type, except for
the white sphere, which marks the initially excited chromophore. See the
methods section for more details.

##  4 Conclusions

At first glance, massively pigmented intermediately-coupled PBS systems do not
seem to fit as effective light harvesting systems. However, the simulations
presented in this paper provide additional insights that demonstrate certain
advantages of such structures over tightly-coupled pigment systems. As long as
closely localized hexamer pairs are avoided, as in the “real-world” model,
energy transfer is robust. Energy is distributed relatively evenly both
horizontally and vertically to the rod axis, regardless of the type of
chromophore excited and its location in the structure. A single PBS system is
large enough to provide excition energy to more than one photosystem [31].
Therefore, the ability to evenly distribute energy between rods and over the
entire PBS can play an important biological role. Furthermore, energy transfer
is relatively immune to the effects of distances or rotations, within the
range of intermediate coupling distances. This robustness is critical in the
context of large biological structures operating at room temperature in living
cells. Considering the erratic nature of the light field in marine
environments [32], it is easy to see how the robustness and the ability to
distribute energy over the large volume of the PBS structure can convey
advantages to marine photosynthesis.

##  5 Methods

###  5.1 Static Pymol visualization

The static Pymol visualization was used to create Figures 8, 9, and 10. Each
chromophore was shown as sticks and colored according to type ($\alpha-$84
chromophores in blue, $\beta-$84 chromophores in red, and $\beta-$155
chromophores in green), except for the white sphere, which marks the
chromophore that was initially excited during the simulation.

The amount of energy distributed to each individual chromophore was
represented by a sphere located at its center of mass coordinates. Ranges for
sphere size were given according to the highest proportion of energy that
transferred to a given chromophore at any point during the 100-picosecond
simulation (see Table 1). If the highest amount of energy for a given was less
than .002, no sphere was generated to represent the chromophore.

Proportion of energy | Sphere size in Pymol
---|---
$\geq 0.9$ | 12
$0.8\rightarrow 0.9$ | 11
$0.7\rightarrow 0.8$ | 10
$0.6\rightarrow 0.7$ | 9
$0.5\rightarrow 0.6$ | 8
$0.4\rightarrow 0.5$ | 7
$0.3\rightarrow 0.4$ | 6
$0.2\rightarrow 0.3$ | 5
$0.1\rightarrow 0.2$ | 4
$0.05\rightarrow 0.1$ | 3
$0.025\rightarrow 0.05$ | 2
$0.002\rightarrow 0.025$ | 1
$0.0\rightarrow 0.002$ | 0
Table 1: Chart showing the Pymol sphere size for the proportion of energy
transferred to a chromophore.

###  5.2 Dynamic Pymol visualization

The dynamic Pymol visualization was used to create movies 1-15 in the
supplemental section. The methodology for generating the dynamic visualization
using Pymol was similar to that of the static visualization, except that
instead of using a sphere to represent the largest amount of energy
distributed to each chromophore over the 100 ps simulation, the exact amount
of energy transferred to each chromophore for every 0.1 ps timepoint was
represented by a sphere in a separate video frame (see Table 1). The videos
have a total of 1000 frames showing energy transfer over the course of the 100
ps simulation.

###  5.3 Transition dipole moment calculations

The magnitude and orientation of the transition dipole moment of each
chromophore in the system are used in the Equations (2) and (12) to calculate
the interaction between dipoles. These calculations are necessary for our
simulation of the overall energy flow of the PBS antenna system. The
transition dipole moments for the three protonated phycocyanobilin conformers
were obtained using quantum chemistry calculations. Heavy-atom coordinates
were used from the Protein Data Bank [33] with explicit hydrogen atoms
incorporated in a constrained geometry optimization. Density functional theory
calculations were carried out at the B3LYP level using the 6-31G(d) basis,
with the five lowest electronic excited states computed via time-dependent
methods. All calculations used Gaussian 09 [27].

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## Acknowledgments

NK and EJD were supported by the Israel Science Foundation grant 1182/19 and
the Zelman Cowen Academic Initiatives.

NW and EMG acknowledge funding from the EPSRC grant no. EP/T007214/1.

The authors acknowledge the University of Maryland supercomputing resources
(http://hpcc.umd.edu) made available for conducting the research reported in
this paper.

Special thanks to Prof. Leah Dodson for her contribution and kind support and
to Noa Hazony for her teamwork and technical support.