Part 1 The hospitalization was involuntary. They interviewed her for about an hour, a process that felt like a clinical excavation, questions designed to map the terrain of a mind that did not know it had drifted, and then they gave her an injection. She woke up the next day without the manic energy that had gripped her. The days in the hospital are a blank to her. She has no memories of them. I have thought about that often: part of how treatment works is by interrupting the very faculty that would remember it. What is lost in those days is not incidental. It is part of what was sick. She went home and was prescribed a cocktail of medications. They did not work well at first, and there is a particular cruelty in that: to survive the episode, survive the hospitalization, and then find that the treatment is also a struggle. Her parents watched her refuse the medications and then offered homeopathy as an alternative. I have thought about whether that was the right choice. I think it was ...
There is a deceptively simple question hiding inside genomic sequence analysis: When is a genome random?
At first glance, the answer seems obvious. Take a genome, shuffle the nucleotides, and compare the original sequence to the shuffled one. If the genome contains patterns that disappear upon shuffling, then the genome must contain order.
But this turns out to be much more complicated.
A genome can look random at one scale and highly structured at another. It can have almost no obvious local pattern while exhibiting correlations thousands of bases apart. It can have heterogeneous nucleotide composition without having a characteristic length scale. And a sequence can differ dramatically from an independently generated random sequence simply because it has GC bias, dinucleotide preferences, repeats, or other mundane statistical properties.
So before asking whether a genome is ordered, we have to ask a more uncomfortable question:
Random relative to what?
I recently went looking through some of the literature on this problem, and a small cluster of papers from the 1990s and 2000s provides a fascinating introduction to the subject.
The first surprise: DNA isn't simply random noise
One of the foundational papers is Wentian Li, Thomas Marr, and Kunihiko Kaneko's 1994 review, "Understanding Long-range Correlations in DNA Sequences."
The central observation is that genomic sequences can contain long-range correlations.
In an ordinary short-memory stochastic process, if I tell you what happened at position (i), that information rapidly becomes irrelevant as I move farther away:
$$
[
C(r) \rightarrow 0
]
$$
fairly quickly as the distance (r) increases.
DNA can behave differently.
Statistical analyses had suggested correlations extending over surprisingly large distances. Rather than having a single characteristic scale, genomic sequences appeared to contain structure across a mixture of length scales. This could produce the famous (1/f)-like spectral behavior seen in some analyses of DNA.
The important conceptual point is that this does not mean that a nucleotide at position 1 somehow "knows" what nucleotide will occur 10,000 bases later.
Correlation is not telepathy.
It means that the statistical properties of the sequence at one location contain information about statistical properties elsewhere in the sequence.
Li and colleagues also proposed an interesting possible source: evolutionary processes such as duplication and rearrangement can themselves generate long-range structure. In their expansion-modification model, relatively simple duplication and mutation processes were sufficient to produce sequences with (1/f)-like spectra.
That is an important lesson.
Non-random structure does not necessarily require a sophisticated biological program.
A relatively simple generative process can produce surprisingly complicated statistical behavior.
Enter statistical physics
The next paper I would put on the reading list is Buldyrev and colleagues' 1998 review, "Analysis of DNA sequences using methods of statistical physics."
This paper takes the idea of genomic correlations and pushes it into statistical mechanics.
The authors analyzed large collections of DNA sequences and reported evidence for remarkably long-range correlations in genes containing noncoding regions. They found that correlations could extend over thousands of base pairs, while coding regions did not show the same kind of long-range behavior.
One of the tools discussed is detrended fluctuation analysis (DFA), which is useful because genomic sequences are not stationary in the simple statistical sense. Their nucleotide composition can drift as we move along the chromosome.
That distinction matters enormously.
Imagine a sequence in which the first half is 70% GC and the second half is 30% GC.
You could calculate a correlation statistic and discover "structure."
But perhaps what you have actually discovered is simply a large-scale change in composition.
The statistical physics literature therefore forced researchers to confront an awkward fact:
A genome can be non-random because it is correlated, because it is heterogeneous, or because both are happening simultaneously.
And those are not necessarily the same thing.
Entropy enters the picture
This brings us to one of my favorite papers in this group:
Román-Roldán, Bernaola-Galván and Oliver (1998), "Sequence Compositional Complexity of DNA through an Entropic Segmentation Method."
Instead of asking only whether bases are correlated, the authors ask how the genome can be divided into regions with different compositions.
They introduce sequence compositional complexity (SCC), an entropy-based measure designed to deal with heterogeneous, nonstationary DNA sequences.
The idea is wonderfully intuitive.
Suppose a sequence is compositionally uniform:
$$
[
\text{ATCGATCGATCGATCG...}
]
$$
There isn't much reason to divide it into different compositional domains.
Now imagine:
$$
[
\underbrace{\text{GC-rich region}}{\text{domain 1}}
\quad
\underbrace{\text{AT-rich region}}{\text{domain 2}}
\quad
\underbrace{\text{GC-rich region}}_{\text{domain 3}}
]
$$
There is structure here even if we don't have a nice periodic pattern.
SCC attempts to quantify this structure by finding statistically significant compositional boundaries and examining the resulting organization across different scales. The authors found higher SCC in sequences with long-range correlations and higher SCC in noncoding than coding sequences.
This is a subtle but important shift.
We're no longer asking:
"Does the genome have a pattern?"
We're asking:
"How much structure is present, and at what scales?"
That is a much more interesting question.
But here comes the real problem: what counts as random?
This is where Messer and Arndt's 2006 paper, "CorGen: measuring and generating long-range correlations for DNA sequence analysis," becomes particularly relevant.
Their observation is almost painfully obvious once you see it:
If real genomic DNA contains long-range correlations, then comparing it against ordinary IID random DNA is not necessarily a fair comparison.
An IID model might look something like
$$
[
P(A)=P(C)=P(G)=P(T)=\frac14
]
$$
with each position generated independently.
But real DNA violates many of those assumptions.
So imagine that your statistic detects something very strongly in a genome.
You compare the genome against IID random DNA and obtain:
$$
[
p < 10^{-20}.
]
$$
Wonderful!
Except perhaps your statistic is simply detecting GC-content correlations.
In that case, you've demonstrated that genomes aren't IID random sequences.
That isn't exactly a shocking biological discovery.
Messer and Arndt developed CorGen to measure long-range correlations and generate random DNA sequences possessing the same correlation parameters. Their paper explicitly argues that genomic long-range correlations should be incorporated into appropriate null models.
This is the point at which the problem becomes much more interesting.
The hierarchy of random sequences
There isn't really one thing called "random DNA."
There is a hierarchy of increasingly constrained null models.
At the bottom might be:
1. IID random DNA
Every nucleotide is independently drawn:
$$
[
P(X_i=x)=p_x.
]
$$
No memory.
No correlations.
No structure beyond the prescribed nucleotide frequencies.
2. Mononucleotide-matched random DNA
Now preserve:
$$
[
P(A),P(C),P(G),P(T)
]
$$
but destroy the ordering.
This controls for nucleotide composition.
3. Dinucleotide-matched sequences
Now preserve frequencies such as
$$
[
P(AA),P(AC),P(AG),\ldots
]
$$
while randomizing higher-order structure.
This controls for short-range dependencies.
4. (k)-mer-matched sequences
Preserve increasingly long words:
$$
[
P(X_1X_2\ldots X_k).
]
$$
Now you're controlling for increasingly local sequence structure.
5. Correlation-matched sequences
Preserve properties such as the long-range autocorrelation or power-law scaling.
This is the territory explored by CorGen.
At this point, a sequence that still looks unusual has survived a much more formidable null model.
And that changes the scientific claim completely.
The genome may have many scales of organization
A recent review of genome compositional structure makes this broader picture particularly clear. Statistical-physics approaches have revealed long-range correlations and scale-dependent organization, but genomic structure also contains recognizable characteristic features such as genes, exons, introns, transposable elements, and compositional domains.
In other words, the genome seems to occupy an interesting middle ground.
It isn't simply:
$$
[
\text{random noise}
]
$$
but neither is it:
$$
[
\text{perfect periodic order}.
]
$$
Instead, it contains structure at many scales.
That is precisely why the old language of "random versus ordered" can become misleading.
A sequence can be:
random at one scale,
structured at another,
approximately stationary over one region,
highly heterogeneous over another,
locally constrained,
globally correlated.
The genome is not a neat little crystal.
It is more like a statistical landscape.
And this is where things get interesting for me
Reading this literature changes the question I would want to ask of a new mathematical representation of DNA.
Suppose we map the four nucleotides into some mathematical structure.
Perhaps a group.
Perhaps a vector space.
Perhaps a representation with an associated spectrum.
Perhaps we calculate an entropy, an energy, an eigenvalue distribution, or some other statistic.
And suppose the statistic distinguishes real genomic sequences from shuffled sequences.
That is interesting.
But it isn't yet enough.
The crucial experiment becomes:
$$
[
\text{Does the distinction survive stronger null models?}
]
$$
For example:
$$
[
\text{IID}
\rightarrow
\text{mononucleotide matched}
\rightarrow
\text{dinucleotide matched}
\rightarrow
k\text{-mer matched}
\rightarrow
\text{correlation matched}
\rightarrow
\text{genomic sequence}.
]
$$
Each arrow removes another possible explanation.
If a mathematical representation only distinguishes genomes from IID sequences, it may simply be detecting ordinary sequence statistics.
If it distinguishes genomes from dinucleotide-preserving randomizations, that is more interesting.
If it survives (k)-mer controls, more interesting still.
And if it survives null models that reproduce conventional long-range correlation structure, then we have to start asking a substantially deeper question:
What statistical property of genomic sequences is being detected that conventional representations are missing?
That is a much more exciting problem than simply asking whether DNA is random.
Order is not the opposite of randomness
There is another conceptual point buried in these papers that I find particularly appealing.
We often imagine a spectrum:
$$
[
\boxed{\text{random}}
\qquad\longleftrightarrow\qquad
\boxed{\text{ordered}}
]
$$
But biological sequences don't necessarily live on that line.
A sequence can have high entropy and still possess correlations.
A sequence can have strong correlations without being periodic.
A sequence can be heterogeneous without having a characteristic scale.
And a sequence generated by a relatively simple evolutionary process can develop complicated statistical structure.
So perhaps the more useful picture is:
$$
[
\boxed{\text{simple randomness}}
]
[
\downarrow
]
[
\boxed{\text{local statistical structure}}
]
[
\downarrow
]
[
\boxed{\text{long-range correlations}}
]
[
\downarrow
]
[
\boxed{\text{multi-scale organization}}
]
$$
while recognizing that these categories can overlap.
The interesting object isn't necessarily "order."
It is structure in the probability distribution over sequences.
After reading these papers, I think the most productive formulation is not:
Is the genome random?
It is:
What statistical structure remains after we account for everything we already know how to explain?
That is a much harder question.
It also gives us a very clean experimental philosophy.
Start with a genomic sequence (S).
Construct increasingly sophisticated null models:
$$
[
S_0,S_1,S_2,\ldots,S_n
]
$$
where each successive model preserves more of the conventional statistical structure of the genome.
Then ask whether some statistic
$$
[
T(S)
]
$$
continues to distinguish the real genome from those nulls.
If
$$
[
T(S_{\text{genome}})
\approx
T(S_{\text{null}})
]
$$
once the obvious statistics have been controlled for, the proposed phenomenon may simply be another manifestation of known genomic structure.
But if
$$
[
T(S_{\text{genome}})
\gg
T(S_{\text{null}})
]
$$
even after increasingly stringent controls, then we have something worth understanding.
Not necessarily a new biological mechanism.
Not necessarily a hidden code.
But potentially a new statistical description of genomic organization.
And that, to me, is the more interesting prize.
The papers
Li, Marr & Kaneko (1994)
"Understanding Long-range Correlations in DNA Sequences"
A foundational review of long-range correlations, (1/f)-like spectra, multiple length scales, and possible evolutionary mechanisms.
Buldyrev et al. (1998)
"Analysis of DNA sequences using methods of statistical physics"
A statistical-physics treatment of genomic correlations, including DFA and the striking difference between coding and noncoding sequences reported at the time.
Román-Roldán, Bernaola-Galván & Oliver (1998)
"Sequence Compositional Complexity of DNA through an Entropic Segmentation Method"
Introduces SCC, an entropy-based approach to detecting compositional structure across scales.
Messer & Arndt (2006)
"CorGen: measuring and generating long-range correlations for DNA sequence analysis"
Perhaps the most important paper here for thinking about null models: random sequences should reproduce relevant genomic correlations if we're going to use them as controls.
Compositional Structure of the Genome: A Review (2023)
A useful modern perspective connecting these older statistical-physics ideas to genome segmentation, compositional domains, long-range correlations, and multi-scale genome organization.
If I had to reduce the whole literature to one sentence, it would be this:
The difficult part isn't demonstrating that genomes aren't random. The difficult part is determining what kind of non-randomness you're actually seeing.
At first glance, the answer seems obvious. Take a genome, shuffle the nucleotides, and compare the original sequence to the shuffled one. If the genome contains patterns that disappear upon shuffling, then the genome must contain order.
But this turns out to be much more complicated.
A genome can look random at one scale and highly structured at another. It can have almost no obvious local pattern while exhibiting correlations thousands of bases apart. It can have heterogeneous nucleotide composition without having a characteristic length scale. And a sequence can differ dramatically from an independently generated random sequence simply because it has GC bias, dinucleotide preferences, repeats, or other mundane statistical properties.
So before asking whether a genome is ordered, we have to ask a more uncomfortable question:
Random relative to what?
I recently went looking through some of the literature on this problem, and a small cluster of papers from the 1990s and 2000s provides a fascinating introduction to the subject.
The first surprise: DNA isn't simply random noise
One of the foundational papers is Wentian Li, Thomas Marr, and Kunihiko Kaneko's 1994 review, "Understanding Long-range Correlations in DNA Sequences."
The central observation is that genomic sequences can contain long-range correlations.
In an ordinary short-memory stochastic process, if I tell you what happened at position (i), that information rapidly becomes irrelevant as I move farther away:
$$
[
C(r) \rightarrow 0
]
$$
fairly quickly as the distance (r) increases.
DNA can behave differently.
Statistical analyses had suggested correlations extending over surprisingly large distances. Rather than having a single characteristic scale, genomic sequences appeared to contain structure across a mixture of length scales. This could produce the famous (1/f)-like spectral behavior seen in some analyses of DNA.
The important conceptual point is that this does not mean that a nucleotide at position 1 somehow "knows" what nucleotide will occur 10,000 bases later.
Correlation is not telepathy.
It means that the statistical properties of the sequence at one location contain information about statistical properties elsewhere in the sequence.
Li and colleagues also proposed an interesting possible source: evolutionary processes such as duplication and rearrangement can themselves generate long-range structure. In their expansion-modification model, relatively simple duplication and mutation processes were sufficient to produce sequences with (1/f)-like spectra.
That is an important lesson.
Non-random structure does not necessarily require a sophisticated biological program.
A relatively simple generative process can produce surprisingly complicated statistical behavior.
Enter statistical physics
The next paper I would put on the reading list is Buldyrev and colleagues' 1998 review, "Analysis of DNA sequences using methods of statistical physics."
This paper takes the idea of genomic correlations and pushes it into statistical mechanics.
The authors analyzed large collections of DNA sequences and reported evidence for remarkably long-range correlations in genes containing noncoding regions. They found that correlations could extend over thousands of base pairs, while coding regions did not show the same kind of long-range behavior.
One of the tools discussed is detrended fluctuation analysis (DFA), which is useful because genomic sequences are not stationary in the simple statistical sense. Their nucleotide composition can drift as we move along the chromosome.
That distinction matters enormously.
Imagine a sequence in which the first half is 70% GC and the second half is 30% GC.
You could calculate a correlation statistic and discover "structure."
But perhaps what you have actually discovered is simply a large-scale change in composition.
The statistical physics literature therefore forced researchers to confront an awkward fact:
A genome can be non-random because it is correlated, because it is heterogeneous, or because both are happening simultaneously.
And those are not necessarily the same thing.
Entropy enters the picture
This brings us to one of my favorite papers in this group:
Román-Roldán, Bernaola-Galván and Oliver (1998), "Sequence Compositional Complexity of DNA through an Entropic Segmentation Method."
Instead of asking only whether bases are correlated, the authors ask how the genome can be divided into regions with different compositions.
They introduce sequence compositional complexity (SCC), an entropy-based measure designed to deal with heterogeneous, nonstationary DNA sequences.
The idea is wonderfully intuitive.
Suppose a sequence is compositionally uniform:
$$
[
\text{ATCGATCGATCGATCG...}
]
$$
There isn't much reason to divide it into different compositional domains.
Now imagine:
$$
[
\underbrace{\text{GC-rich region}}{\text{domain 1}}
\quad
\underbrace{\text{AT-rich region}}{\text{domain 2}}
\quad
\underbrace{\text{GC-rich region}}_{\text{domain 3}}
]
$$
There is structure here even if we don't have a nice periodic pattern.
SCC attempts to quantify this structure by finding statistically significant compositional boundaries and examining the resulting organization across different scales. The authors found higher SCC in sequences with long-range correlations and higher SCC in noncoding than coding sequences.
This is a subtle but important shift.
We're no longer asking:
"Does the genome have a pattern?"
We're asking:
"How much structure is present, and at what scales?"
That is a much more interesting question.
But here comes the real problem: what counts as random?
This is where Messer and Arndt's 2006 paper, "CorGen: measuring and generating long-range correlations for DNA sequence analysis," becomes particularly relevant.
Their observation is almost painfully obvious once you see it:
If real genomic DNA contains long-range correlations, then comparing it against ordinary IID random DNA is not necessarily a fair comparison.
An IID model might look something like
$$
[
P(A)=P(C)=P(G)=P(T)=\frac14
]
$$
with each position generated independently.
But real DNA violates many of those assumptions.
So imagine that your statistic detects something very strongly in a genome.
You compare the genome against IID random DNA and obtain:
$$
[
p < 10^{-20}.
]
$$
Wonderful!
Except perhaps your statistic is simply detecting GC-content correlations.
In that case, you've demonstrated that genomes aren't IID random sequences.
That isn't exactly a shocking biological discovery.
Messer and Arndt developed CorGen to measure long-range correlations and generate random DNA sequences possessing the same correlation parameters. Their paper explicitly argues that genomic long-range correlations should be incorporated into appropriate null models.
This is the point at which the problem becomes much more interesting.
The hierarchy of random sequences
There isn't really one thing called "random DNA."
There is a hierarchy of increasingly constrained null models.
At the bottom might be:
1. IID random DNA
Every nucleotide is independently drawn:
$$
[
P(X_i=x)=p_x.
]
$$
No memory.
No correlations.
No structure beyond the prescribed nucleotide frequencies.
2. Mononucleotide-matched random DNA
Now preserve:
$$
[
P(A),P(C),P(G),P(T)
]
$$
but destroy the ordering.
This controls for nucleotide composition.
3. Dinucleotide-matched sequences
Now preserve frequencies such as
$$
[
P(AA),P(AC),P(AG),\ldots
]
$$
while randomizing higher-order structure.
This controls for short-range dependencies.
4. (k)-mer-matched sequences
Preserve increasingly long words:
$$
[
P(X_1X_2\ldots X_k).
]
$$
Now you're controlling for increasingly local sequence structure.
5. Correlation-matched sequences
Preserve properties such as the long-range autocorrelation or power-law scaling.
This is the territory explored by CorGen.
At this point, a sequence that still looks unusual has survived a much more formidable null model.
And that changes the scientific claim completely.
The genome may have many scales of organization
A recent review of genome compositional structure makes this broader picture particularly clear. Statistical-physics approaches have revealed long-range correlations and scale-dependent organization, but genomic structure also contains recognizable characteristic features such as genes, exons, introns, transposable elements, and compositional domains.
In other words, the genome seems to occupy an interesting middle ground.
It isn't simply:
$$
[
\text{random noise}
]
$$
but neither is it:
$$
[
\text{perfect periodic order}.
]
$$
Instead, it contains structure at many scales.
That is precisely why the old language of "random versus ordered" can become misleading.
A sequence can be:
random at one scale,
structured at another,
approximately stationary over one region,
highly heterogeneous over another,
locally constrained,
globally correlated.
The genome is not a neat little crystal.
It is more like a statistical landscape.
And this is where things get interesting for me
Reading this literature changes the question I would want to ask of a new mathematical representation of DNA.
Suppose we map the four nucleotides into some mathematical structure.
Perhaps a group.
Perhaps a vector space.
Perhaps a representation with an associated spectrum.
Perhaps we calculate an entropy, an energy, an eigenvalue distribution, or some other statistic.
And suppose the statistic distinguishes real genomic sequences from shuffled sequences.
That is interesting.
But it isn't yet enough.
The crucial experiment becomes:
$$
[
\text{Does the distinction survive stronger null models?}
]
$$
For example:
$$
[
\text{IID}
\rightarrow
\text{mononucleotide matched}
\rightarrow
\text{dinucleotide matched}
\rightarrow
k\text{-mer matched}
\rightarrow
\text{correlation matched}
\rightarrow
\text{genomic sequence}.
]
$$
Each arrow removes another possible explanation.
If a mathematical representation only distinguishes genomes from IID sequences, it may simply be detecting ordinary sequence statistics.
If it distinguishes genomes from dinucleotide-preserving randomizations, that is more interesting.
If it survives (k)-mer controls, more interesting still.
And if it survives null models that reproduce conventional long-range correlation structure, then we have to start asking a substantially deeper question:
What statistical property of genomic sequences is being detected that conventional representations are missing?
That is a much more exciting problem than simply asking whether DNA is random.
Order is not the opposite of randomness
There is another conceptual point buried in these papers that I find particularly appealing.
We often imagine a spectrum:
$$
[
\boxed{\text{random}}
\qquad\longleftrightarrow\qquad
\boxed{\text{ordered}}
]
$$
But biological sequences don't necessarily live on that line.
A sequence can have high entropy and still possess correlations.
A sequence can have strong correlations without being periodic.
A sequence can be heterogeneous without having a characteristic scale.
And a sequence generated by a relatively simple evolutionary process can develop complicated statistical structure.
So perhaps the more useful picture is:
$$
[
\boxed{\text{simple randomness}}
]
[
\downarrow
]
[
\boxed{\text{local statistical structure}}
]
[
\downarrow
]
[
\boxed{\text{long-range correlations}}
]
[
\downarrow
]
[
\boxed{\text{multi-scale organization}}
]
$$
while recognizing that these categories can overlap.
The interesting object isn't necessarily "order."
It is structure in the probability distribution over sequences.
After reading these papers, I think the most productive formulation is not:
Is the genome random?
It is:
What statistical structure remains after we account for everything we already know how to explain?
That is a much harder question.
It also gives us a very clean experimental philosophy.
Start with a genomic sequence (S).
Construct increasingly sophisticated null models:
$$
[
S_0,S_1,S_2,\ldots,S_n
]
$$
where each successive model preserves more of the conventional statistical structure of the genome.
Then ask whether some statistic
$$
[
T(S)
]
$$
continues to distinguish the real genome from those nulls.
If
$$
[
T(S_{\text{genome}})
\approx
T(S_{\text{null}})
]
$$
once the obvious statistics have been controlled for, the proposed phenomenon may simply be another manifestation of known genomic structure.
But if
$$
[
T(S_{\text{genome}})
\gg
T(S_{\text{null}})
]
$$
even after increasingly stringent controls, then we have something worth understanding.
Not necessarily a new biological mechanism.
Not necessarily a hidden code.
But potentially a new statistical description of genomic organization.
And that, to me, is the more interesting prize.
The papers
Li, Marr & Kaneko (1994)
"Understanding Long-range Correlations in DNA Sequences"
A foundational review of long-range correlations, (1/f)-like spectra, multiple length scales, and possible evolutionary mechanisms.
Buldyrev et al. (1998)
"Analysis of DNA sequences using methods of statistical physics"
A statistical-physics treatment of genomic correlations, including DFA and the striking difference between coding and noncoding sequences reported at the time.
Román-Roldán, Bernaola-Galván & Oliver (1998)
"Sequence Compositional Complexity of DNA through an Entropic Segmentation Method"
Introduces SCC, an entropy-based approach to detecting compositional structure across scales.
Messer & Arndt (2006)
"CorGen: measuring and generating long-range correlations for DNA sequence analysis"
Perhaps the most important paper here for thinking about null models: random sequences should reproduce relevant genomic correlations if we're going to use them as controls.
Compositional Structure of the Genome: A Review (2023)
A useful modern perspective connecting these older statistical-physics ideas to genome segmentation, compositional domains, long-range correlations, and multi-scale genome organization.
If I had to reduce the whole literature to one sentence, it would be this:
The difficult part isn't demonstrating that genomes aren't random. The difficult part is determining what kind of non-randomness you're actually seeing.
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