#  Could the “Wow” signal have originated from a stochastic repeating beacon?

David Kipping1 and Robert Gray2
1Dept. of Astronomy, Columbia University, 550 W 120th Street, New York NY
10027
2Gray Data Consulting, 3071 Palmer Square, Chicago, IL 60647 E-mail:
<EMAIL_ADDRESS>
(Accepted . Received ; in original form )

###### Abstract

The famous “Wow” signal detected in 1977 remains arguably the most compelling
SETI signal ever found. The original Big Ear data requires that the signal
turned on/off over the span of ${\sim}3$ minutes (time difference between the
dual antennae), yet persisted for 72 seconds (duration of a single beam
sweep). Combined with the substantial and negative follow-up efforts, these
observations limit the allowed range of signal repeat schedules, to the extent
that one might question the credibility of the signal itself. Previous work
has largely excluded the hypothesis of a strictly periodic repeating source,
for periods shorter than 40 hours. However, a non-periodic, stochastic
repeater remains largely unexplored. Here, we employ a likelihood emulator
using the Big Ear observing logs to infer the probable signal properties under
this hypothesis. We find that the maximum a-posteriori solution has a
likelihood of 32.3%, highly compatible with the Big Ear data, with a broad 2
$\sigma$ credible interval of signal duration $72$ secs$<T<77$ mins and mean
repeat rate $0.043$ days${}^{-1}<\lambda<59.8$ days-1. We extend our analysis
to include 192 hours of subsequent observations from META, Hobart and ATA,
which drops the peak likelihood to 1.78%, and thus in tension with the
available data at the 2.4 $\sigma$ level. Accordingly, the Wow signal cannot
be excluded as a stochastic repeater with available data, and we estimate that
62 days of accumulated additional observations would be necessary to surpass 3
$\sigma$ confidence.

###### keywords:

extraterrestrial intelligence — methods: statistical

††pubyear: 2022††pagerange: Could the “Wow” signal have originated from a
stochastic repeating beacon?–1

##  1 Introduction

In 1977, the Ohio State University Radio Observatory recorded a strong (30
$\sigma$), narrowband ($<10$ kHz) signal near the 21 cm hydrogen line (Kraus,
1979). The signal, discovered during a search for extraterrestrial
intelligence (SETI), bore a remarkable resemble to the properties expected
from an alien transmitter, leading the volunteer analyst Jerry Ehman to
famously annotate “Wow!” on the computer printout (Ehman, 1998). The time-
dependent intensity of the emission matched the antenna pattern signature of a
transiting celestial source (Gray & Marvel, 2001), unlike what would be
expected for interference. This behaviour, combined with its narrowband nature
and location near 21 cm, leads to a trifecta of features long predicted from a
genuine SETI signal (Cocconi & Morrison, 1959), causing Wow to be one of the
few astronomical observations that have bled into popular culture.

Although the field was observed many times, Wow was only seen once during the
entire baseline of observations from the Ohio State University Radio
Observatory, known as the “Big Ear” (Kraus, 1979). Further, Wow only occurred
in one beam of a dual-beam transit antenna system (Dixon & Cole, 1977),
implying that the signal switched on/off in the brief interval between the two
beams sweeping over the source ($\sim 3$ minutes). However, Wow appears
consistent with being continuously “on” during the 72 seconds of one beam
passing over the source.

Despite numerous attempts to replicate the detection of Wow (Gray, 1994; Gray
& Marvel, 2001; Gray & Ellingsen, 2002; Harp et al., 2020), the signal has not
been observed again. The single-beam detection and lack of any repetition
(especially with greater sensitivity, longer observations and broader spectral
coverage), ostensibly places considerable pressure on the credibility of the
Wow signal111For example, in an interview with the
http://www.bigear.org/wow.htm in 1994, Jerry Ehman remarked “We should have
seen it again when we looked for it 50 times. Something suggests it was an
Earth-sourced signal that simply got reflected off a piece of space debris.”.
However, a recent Bayesian formulation of one-off events showed that at least
an order-of-magnitude more data than that used in the original discovery is
often required to place statistically significant tension on a signal’s
credibility (Kipping, 2021).

Motivated by that work, we here explore a rigorous statistical approach to
address: i) how likely is it that Wow repeats but has simply been missed by
follow-up efforts to date, and ii) if so, what are the most likely properties
of the signal under this scenario?

There are two basic ways that Wow could remain a genuine SETI signal yet
remain consistent with all observations to date. It could either be a non-
continuous emission source and/or a continuous source that drifts in
frequency. Regarding the latter possibility, Wow was detected in channel 2 of
50 by Big Ear (Kraus, 1979), each with a 10 kHz band. Given that the signal
appears/disappears over a 3 minute interval, a drift rate of 20 kHz/3 minutes
$\simeq 100$ Hz s (i.e. $\sim 100$ nHz) could conceivably explain this222Such
a signal would drift 7.2 kHz during the 72 second duration of a single-beam,
and thus would also plausibly be confined to a single 10 kHz channel.. We note
that this would be considered at the upper fringes of plausibility for a
genuine SETI signal (Sheikh et al., 2019); for example the Earth’s fractional
drift rate from orbital and rotational motion is three orders-of-magnitude
smaller. Further, subsequent observations by Gray & Ellingsen (2002) used a
band five times wider than than of Big Ear (500 kHz) spanning six 14-hour
continuous blocks, yet reported no Wow-like repetition.

A non-continuous signal is more interesting and might be expected from an
economics perspective, as suggested by Benford, Benford, & Benford (2008). A
strictly periodic repeater might be expected if the source behaved akin to a
lighthouse, perhaps as a continuous source rotating across the sky or
programmed to emit only periodically in our direction. Gray & Ellingsen (2002)
and Harp et al. (2020) excluded strictly periodic repeaters with periods below
40 hours and longer timescales become increasingly improbable to fortuitously
allow the Wow signal being detected in the first place. To date, there has
been no investigation of a non-periodic, stochastic repeater though. A
stochastic repeater could be attractive from the perspective of avoiding
sampling holes. For example, signals that repeat once per 24 hours both risk
evading our observing cadence as well being immediately suspicious as spurious
even if recorded. An intermittent signal could also manifest due to
scintillation within the interstellar medium, in particular for sources $>100$
parsecs away (Cordes, Lazio, & Sagan, 1997). The underlying cause of the
intermittency is not of principal interest here, rather we consider that such
intermittency is feasible and the compatibility of existing observations with
such a scenario has not been extensively studied for Wow.

In this work, we investigate this possibility by evaluating the plausibility
of said hypothesis and the corresponding required signal properties. Although
Kipping (2021) presents a framework for such an analysis, that work is only
valid when the data are homogenous, regular and no substantial changes to the
observing instruments nor strategy are implemented. In order to account for
the sparse, irregular sampling of Big Ear, as well as the subsequent
observations, we require a more detailed approach that emulates the detection
efficiency of the instrument for various proposed signal forms. To this end,
Section 2 presents such an emulator, which is then used in Section 3 to infer
a posterior distribution for the signal form. We extend this analysis in
Section 4 to include observations besides from that of Big Ear, and conclude
in Section 5.

##  2 The likelihood of a Wow-like signal

###  2.1 Model for the Signal(s)

Searches for strict periodicity can be readily achieved using classic
periodogram techniques. Stochastic repeaters, in contrast, may evade detection
using such techniques. Accordingly, we here seek to test the specific
hypothesis of a stochastic repeater.

In what follows, we assume that the source maintains a uniform rate of signal
production over any fixed interval of time within the baseline of the Big Ear
observations. This essentially corresponds to a kind of Copernican Principle
in time - we assume that the Big Ear observations do not occupy a special
point in time when the signal source changed its emission mode (for example).
Indeed, this assumption demands that the mean number of signals produced per
unit time (in our direction) is the same when Big Ear began observing the
source, as when it finished. This defines a Poisson process, for which the
interval between individual signals will follow an exponential distribution.
Accordingly, the $(k+1)^{\mathrm{th}}$ signal will occur at a time

$\displaystyle t_{\mathrm{signal},k+1}\sim
t_{\mathrm{signal},k}+\mathcal{E}[\lambda],$ (1)

where $\mathcal{E}[\lambda]$ is an exponential distribution of mean $\lambda$.
For a given choice of $\lambda$, we thus begin our simulation by iteratively
populating a list of signal mid-times using Equation 1, where we start with
$t_{\mathrm{signal},0}=0$ but ultimately discard this initial entry.

The signal is also characterised by a duration. In what follows, we assume
that the signal is a top-hat persisting for a time $T$ (with no variability),
since this is both consistent with the Wow signal and represents the simplest
possible model. If $t_{\mathrm{signal},k+1}-t_{\mathrm{signal},k}<T$, the two
signals are effectively merged into a longer continuous one.

###  2.2 Model for the Observations

The observations are modelled using the observing logs of the Wow field from
Big Ear itself. In September 1985, co-author Robert Gray visited OSU where the
logs were reviewed and noted. Print outs were pulled from the OSU archives by
Marc Abel, who located runs from 122 days spanning August 1977 to March 1984.
Data near the Wow locale was examined for each day, with the locale defined as
near RA 19 17 in 1977, and RA 19 22 and 19 25 subsequently, since a horn
squint correction had presumably been applied by 1978.

Excluding observations more than 20’ north or south of the Wow declination, or
days with bad data (e.g. clock off, printer failure, power failure), we find
90 useful visits, which are listed in Table 1.

These dates are used to define a set of representative observations in our
simulation, with the first observation occurring at $t_{\mathrm{obs},1}=0.5$
days, and subsequent observations offset in time from this reference point
according to their observation date. Each observation in fact a pair of
measurements, since the Big Ear horns are physically separated and thus a sky
source passes over each. Accordingly, we define the first of these two
occurring at a mid-time of $t_{\mathrm{obs},l}$ (for the $l^{\mathrm{th}}$
observation) and the next occurring at $t_{\mathrm{obs},l}+\Delta t$, where
$\Delta t=172.37$ seconds represents the temporal offset.

Each of the pair of observations is modelled as lasting for 72 seconds. For a
“detection” to occur, we define that a signal must be throughout the entire
duration of this 72-second window. We thus cycle through each observation
($90\times 2$ in total) and query whether any of the generated signals lead to
a detection. After cycling through all observations, we tally up the total
number of detections. Illustrative examples of both Wow-like and non Wow-like
signals are depicted in Figure 1.

Figure 1:  a) An annotated example of a Wow-like signal detection. The
observational window is shown in blue, and the signal in red. Here, the signal
is fully detected but only in one of the windows, not the other, just like
Wow. b) Examples of non-detections, either because of too many (top) or too
few (bottom) signal overlaps with the observing windows.

###  2.3 The Likelihood of a Wow-Like Signal

The process of generating a set of injected signals, comparing them to our
simulated observations and tallying up the number of detections defines a
single realisation. Since the signal generation process is stochastic, a
statistical interpretation requires repeating many realisations for a given
choice of $T$ and $\lambda$. In practice, we approach this by creating a
“while” loop that generates many realisations and continues until exactly one
detection is made - since this is what was found for the Wow signal. The
greater the number of realisations necessary, the less likely it is that the
particular choice of $T$ and $\lambda$ could reproduce Wow. Naively, one might
reason that if it took $C$ realisations to get a Wow-like signal, then the
probability of $T$ and $\lambda$ producing Wow is $1/C$.

For computational efficiency, we cap the maximum number of realisations in our
while loop at 1000 and flag such cases appropriately. Although this single
while loop provides a first estimate of $1/C$ probability, it is inherently
uncertain given the fact only a single loop was executed. Thus, we wish to
repeat the loop $W$ times, in order to improve our confidence in the estimated
probability.

Since our while loop represents a binary outcome of fixed probability, then
$C$ represents the number of Bernoulli trials needed to get one success,
defining a geometric distribution i.e. $C\sim\mathcal{G}[p]$, where
$\mathcal{G}$ represents the geometric distribution and $p$ is the probability
value for this choice of $T$ and $\lambda$. We now wish to infer $p$ from a
sequence of $C$ evaluations emerging from our while loops, which form a vector
with indices $C_{w}$ of length $W$. To accomplish this, we note that

$\displaystyle\mathrm{Pr}(\\{C_{1},C_{2},...,C_{W}\\}|p)$
$\displaystyle=(1-p)^{C_{\mathrm{tot}}}p^{W},$ (2)

where $C_{\mathrm{tot}}\equiv\sum_{w=1}^{W}C_{w}$. To infer
$\mathrm{Pr}(p|\\{C_{1},C_{2},...,C_{W}\\})$ (the posterior distribution of
$p$), we need to use Bayes theorem. Adopting a uniform prior on $p$ for
simplicity, the posterior is simply the normalised likelihood given by

$\displaystyle\mathrm{Pr}(p|\\{C_{1},C_{2},...,C_{W}\\})$
$\displaystyle=\frac{\Gamma[2+W+C_{\mathrm{tot}}](1-p)^{C_{\mathrm{tot}}}p^{W}}{\Gamma[W+1]\Gamma[C_{\mathrm{tot}}+1]}.$
(3)

Differentiating, this has a mode (i.e. maximum a-posteriori, MAP) value of

$\displaystyle\bar{p}$ $\displaystyle=\frac{W}{W+C_{\mathrm{tot}}},$ (4)

which is close to our original intuition of $1/C_{\mathrm{tot}}$ for $W\sim
1$. However, this process also allows us to define the variance to be

$\displaystyle\sigma_{p}^{2}$
$\displaystyle=\frac{(W+1)(C_{\mathrm{tot}}+1)}{(2+W+C_{\mathrm{tot}})^{2}(3+W+C_{\mathrm{tot}})}.$
(5)

Using these results we can say that after $W$ while loops, each of which took
$C_{w}$ realisations to obtain a Wow-like signal, we infer the MAP likelihood
of the choice of $T$ and $\lambda$ to be $\bar{p}$ with a precision
(fractional error) of $\bar{p}/\sigma_{p}$. We thus continue through the while
loops until the precision falls below 10% - which we qualify as a “good”
estimate of $\bar{p}$.

###  2.4 Likelihood Emulator

From before, $\bar{p}$ represents the maximum a-posteriori probability of
obtaining a Wow-like signal with the 90 days of Big Ear data, given a choice
of $\lambda$ and $T$. In moving towards inferring $\lambda$ and $T$, $\bar{p}$
can be considered to be the (most probable) likelihood of obtaining the data
$\mathcal{D}$ (here defined as a single detection in the 90 pairs of
observations) given $\lambda$ and $T$. Thus, we define

$\displaystyle\mathrm{Pr}(\mathcal{D}|\lambda,T)$
$\displaystyle=\frac{W}{W+C_{\mathrm{tot}}}.$ (6)

To make progress, we need to be able to vary $\lambda$ and $T$ and evaluate
likelihoods quickly, to exploit Bayesian inference methods. Given the
computational expensive nature of even a single likelihood evaluation
(sometimes taking many hours), we elected to build a grid of $\lambda$ and $T$
values across which we could pre-calculate the likelihoods and then use them
to build a likelihood emulator at intermediate values. To this end, we set up
a $M=100$ square grid of plausible values for each. The grid was chosen to be
log-uniformly spaced from $\lambda_{\mathrm{min}}$ to $\lambda_{\mathrm{max}}$
and $T_{\mathrm{min}}$ to $T_{\mathrm{max}}$, such that

$\displaystyle T_{i}=\exp\Big{[}\log T_{\mathrm{min}}+(i-1)\frac{\log
T_{\mathrm{max}}-\log T_{\mathrm{min}})}{M-1}\Big{]},$ $\displaystyle
L_{j}=\exp\Big{[}\log\lambda_{\mathrm{min}}+(j-1)\frac{\log\lambda_{\mathrm{max}}-\log\lambda_{\mathrm{min}})}{M-1}\Big{]},$
(7)

where $i$ and $j$ are the integer indices running from 1 to $M$. We set
$T_{\mathrm{min}}=72$ seconds, which defines the minimum allowed duration
given that the signal shape appears consistent with a continuous emission. For
$T_{\mathrm{max}}$, we chose a maximum at least two orders of magnitude larger
and specifically adopted $T_{\mathrm{max}}=e^{-1}$ days. For
$\lambda_{\mathrm{min}}$ and $\lambda_{\mathrm{max}}$, we chose to bracket
them three orders of magnitude either side of $1/T_{\mathrm{max}}$, giving
$\lambda_{\mathrm{min}}=e^{-4}$ days-1 and $\lambda_{\mathrm{max}}=e^{2}$
days-1.

Our later parameter explorations found that this initial grid did not support
certain regions of high likelihood space and needed to be expanded. To
accomplish this, we simply increased the indices $i$ and $j$ beyond 100.
Specifically, we ran up to $i=150$ and $j=160$. This means that in practice
the maximum allowed $T$ value is not in fact $T_{\mathrm{max}}$, but rather
$T_{\mathrm{max}}^{\prime}$, and similarly for $\lambda$, where

$\displaystyle T_{\mathrm{max}}^{\prime}$ $\displaystyle=\exp\Big{[}\log
T_{\mathrm{min}}+\frac{149}{99}(\log T_{\mathrm{max}}-\log T_{\mathrm{min}}),$
$\displaystyle\lambda_{\mathrm{max}}^{\prime}$ $\displaystyle=\exp\Big{[}\log
T_{\mathrm{min}}+\frac{159}{99}(\log T_{\mathrm{max}}-\log T_{\mathrm{min}}),$
(8)

which evaluates to $T_{\mathrm{max}}^{\prime}=7.97$ days and
$\lambda_{\mathrm{max}}^{\prime}=280.4$ days-1.

Since our grid is regular and evenly spaced (in log space), it is amenable to
interpolation. To this end, we use a bi-cubic spline interpolation of the
grid. As shown in Figure 2, this leads to an accurate and quick emulator
across the high likelihood region of interest.

Figure 2:  Comparison of our grid of individual likelihood evaluation (left)
and the emulator used for the MCMC using interpolation (right). We colour code
the points/regions by their likelihood value, with the highest region
exceeding 10%.

##  3 The a-posteriori “Wow” signal properties

###  3.1 MCMC

Having established a likelihood emulator supported over the range
$T_{\mathrm{min}}\leq T<T_{\mathrm{max}}^{\prime}$ and
$\lambda_{\mathrm{min}}\leq\lambda<\lambda_{\mathrm{max}}^{\prime}$, we now
turn to inferring the joint posterior distribution of $T$ and $\lambda$, given
the data $\mathcal{D}$. This can be achieved through Bayes theorem:

$\displaystyle\mathrm{Pr}(T,\lambda|\mathcal{D})\propto\mathrm{Pr}(\mathcal{D}|T,\lambda)\mathrm{Pr}(T,\lambda).$
(9)

For the prior, $\mathrm{Pr}(T,\lambda)$, we assume that the priors are
independent and follow log-uniform distributions over their supported range.
This is most easily achieved by simply walking in $\log T$ and $\log\lambda$
space directly, which is also convenient as this is the native sampling the
grid used for the likelihood emulator. The limits on the walkers are set to
that of our likelihood grid, since outside this region we cannot reliably
emulate the likelihood.

We sample the joint posterior distribution using Markov Chain Monte Carlo
algorithm and Metropolis walkers, calculating the posterior probability as a
product of our likelihood emulator and the prior (which is uniform in this
parameterisation). Each walker completes 110,000 accepted steps, burning off
the first 10,000. The starting point for the chain is randomly drawn from the
joint prior. We repeat for 100 walkers and combine the final results, with a
thinning of a factor of 10 to leave us with $10^{6}$ posterior samples. Chains
and posterior distributions were inspected to ensure convergence and good
mixing.

###  3.2 Analysis

The joint posterior distribution is depicted in Figure 3. We find that the
marginalised duration distribution peaks close to the lower boundary
condition. Since MCMC samplers cannot make jumps below the prior minimum
threshold, a slightly positively skewed posterior is expected even when the
result is consistent with the minimum. Accordingly, we here conclude that
there is no clear lower limit on the signal duration, besides from the
constraint that it appears consistent with being fully-on during the window in
which it was caught (i.e. $T>72$ seconds). However, we find that long duration
signals are disfavoured, which can be understood by the fact that they are
increasingly unlikely to have only been detected once during the ensemble of
Big Ear observations. Indeed, from this, we can place an upper limit that
$T<77$ minutes to $2$ $\sigma$ confidence (95.45%).

Figure 3:  Triangle plot of the posterior distribution for the Wow signal
properties. The joint posterior (bottom-left) naturally has a similar shape to
the likelihood (Figure 2) given our diffuse prior, but the contours now
represent the a-posteriori credible intervals rather than likelihood
confidence limits. Contours are separated by 0.25 $\sigma$.

For $\lambda$, the mean rate of signal emission, we find a broad but peaked
posterior with a maximum at $\lambda=2.01$ days-1 (i.e. twice per day). We can
see that $\lambda$ is inversely correlated to $T$ but compresses in a narrow
region at low $T$. This “fine-tuning” reduces the marginalised posterior
density in this region, and ultimately gets truncated by the lower boundary on
$T$. Very frequent repeat times are only compatible with the data at extremely
short durations, in order to explain the lack of additional detections.
Together, these constraints imply that $\lambda$ is bound within the interval
$0.047$ days-1 to $64.2$ days-1 to 2 $\sigma$.

##  4 Accounting for additional follow-up

###  4.1 Follow-up Observations

Of course, besides from the original Big Ear observations, there have been
several efforts to re-observe the field and look for repetitions, without
success. In Gray (1994), the field was observed using the ultranarrowband
Harvard/Smithsonian-META system (Horowitz et al., 1986), tracking for up to 4
hour at a time. Gray & Marvel (2001) performed the second attempt with the
Very Large Array, tracking the field for no more than 22 minutes a time. Gray
& Ellingsen (2002) performed far longer observations with the University of
Tasmania’s Hobart 26 m radio telescope, tracking the field for 14 hours
continuously on six separate days. Finally, the Allan Telescope Array was used
by Harp et al. (2020), accumulating over 100 hours on target for 30 to 180
minutes each time. In all cases, we note that the observations were
sufficiently sensitive to have recovered a repeat of the Wow signal to high
confidence.

###  4.2 Incorporating the Hobart Constraints

Amongst the four follow-up efforts highlighted above, the Gray & Ellingsen
(2002) data with Hobart is most amenable to incorporating into our analysis,
since the observing window is approximately the same each time (14 hours) and
only six sets were taken. The dates of the observations are provided in Gray &
Ellingsen (2002) and thus we could easily imagine directly extending our
likelihood formalism to include these dates. However, a much simpler approach
would be to ignore the dates and rather just assume that each 14-hour window
is an independent and fair sample over all time within which no detection was
made. For a $F=14$ hour block, the probability of a Poisson process not
yielding a single success equals $e^{-F\lambda}$, and to evade detection in
six independent draws would be $(e^{-F\lambda})^{6}=e^{-6F\lambda}$. This
serves as an additional likelihood term which we can append to the Big Ear
likelihood emulator. With this simple change, we repeated the MCMC analysis
using an otherwise identical setup.

The updated chain found that the maximum likelihood dropped from 32.3% using
Big Ear alone, to 3.27%, thus placing tension on the stochastic repeater
hypothesis at the 2.1 $\sigma$ level. This peak occurs at
$\hat{\lambda}=0.233$ days-1 and $\hat{T}=485$ s. Evaluating the likelihood of
this position using the Big Ear likelihood emulator alone (ignoring the Hobart
data) yields 7.40%. Thus, the likelihood shifts from 7.40% to 3.27% - a factor
of 0.442 of the original value, which of course simply equals
$e^{-6F\hat{\lambda}}$. Using this point of highest likelihood, we can test
the accuracy of our assumption of independence.

To this end, we used the dates of the Hobart observations and generated 1000
Poisson processes with a mean rate $\hat{\lambda}$ and duration $\hat{T}$ and
counted how many detections Hobart would have seen. From this, we measure 439
zero-count cases - consistent with the real observations of Gray & Ellingsen
(2002). Thus, we find that a more realistic calculation that accounts for the
specific timings of the observations yields a Hobart-likelihood at
$\hat{\lambda}$ and $\hat{T}$ of $(0.439\pm 0.016)$ \- a value fully
consistent with the much simpler independence assumption. On this basis, we
argue that our simplifying assumption is justified and produces an accurate
final posterior.

From our revised joint posterior, we obtain credible intervals of
$\lambda=[0.021,0.441]$ d-1 and $T=[130\,\mathrm{s},2.2\,\mathrm{d}]$.

###  4.3 Extending to Other Observations

Our approach for incorporating the Hobart observations can be extended to
other data sets too. Recall that for Hobart, the additional likelihood term
was $(e^{-F\lambda})^{6}=e^{-6F\lambda}$, since 6 sets of $F$ baselines were
taken. This is equivalent to adding a single penalty of $6F$ duration (i.e. 84
hours). Accordingly, even if the other observations did not use the same
observational window each, we can still include them by simply counting up the
total observing time of each (on the Wow field).

From Gray (1994), we find 8 hours on each of the two possible Wow positions.
Gray & Marvel (2001) dwelled no longer than 22 minutes on the field and used a
variety of observing modes, and thus we elected to not include these data as
their temporal constraining power is small compared to Hobart. In contrast,
the ATA 100 hour campaign from Harp et al. (2020) more than doubles the
baseline and thus provides a key constraint. Together then, we add a third
likelihood term equal to $e^{-G\lambda}$, where $G=108$ hours.

With this change, the final maximum likelihood is 1.78% (2.37 $\sigma$),
occurring at $\hat{T}=659$ s and $\hat{\lambda}=0.121$ days-1. The final joint
posterior is plotted in Figure 4, where we report a revised 2 $\sigma$
credible interval of $T=[130\,\mathrm{s},1.53\,\mathrm{d}]$ and
$\lambda=[0.021,0.397]$ d-1. In practice, the joint posterior is not
significantly changed by the inclusion of the other observational constraints.

Figure 4:  Same as Figure 3, but the MCMC now includes additional constraints
from the Harvard/Smithsonian-META system (Gray, 1994), the University of
Tasmania’s Hobart 26 m radio telescope (Gray & Ellingsen, 2002) and the Allan
Telescope Array (Harp et al., 2020).

##  5 Discussion

Our work has used a likelihood emulator derived from the Big Ear observing
logs to investigate the properties of the Wow signal, under the assumption it
is a stochastic repeater. Here, we discuss the implications of this result,
and validity of our underlying assumption.

We’ll begin with the latter. It must be acknowledged that have no evidence
that the Wow signal is a repeating signal. Indeed, the lack of any repeat is
what makes the signal so tantalising and ultimately motivates our study to
investigate how viable the repeating hypothesis is. However, even if the
signal is non-repeating, it is technically consistent with our formalism in
the limiting case that the mean repeat time tends to infinity (i.e.
$\lambda\to 0$). With just a single pulse produced over all cosmic time, it’s
incredibly improbable that the Big Ear would have coincidentally been looking
at the correct place at the correct moment to catch it. Indeed, our analysis
with the Big Ear data alone has a marginalised $\lambda$ posterior density
that tends to 0 in the limit of $\lambda\to 0$.

Using just the Big Ear data, strict periodic signals are challenging to
constrain since the field was observed briefly just twice per day, meaning
that periodic signals can easily fall out of phase with an initial detection.
This possibility was investigated in Gray & Ellingsen (2002), although the
duration of the signal was fixed to three scenarios of $T=72$ s, $T=144$ s and
$T=288$ s, rather than freely fitted as done in this work. Gray & Ellingsen
(2002) report that periods of below one day had a $>5$% likelihood of
reproducing the observations, peaking at 35% for signals of approximately an
hour period.

To compare to our stochastic repeater analysis, we find a peak likelihood of
32.3% for a signal of duration $\hat{T}=181$ s and mean rate
$\hat{\lambda}=3.69$ days-1 ($\sim 6.5$ hours mean repeat time). Thus, the
stochastic assumption can be seen to allow for even higher compatibility with
the original data, although the timescales involved of the best fitting
signals are not grossly different. This of course is not surprising given the
inherently more “slippery” nature of a stochastic process.

Of course, follow-up observations have placed much stronger constraints on
strictly periodic signals. Gray & Ellingsen (2002) completely excludes strict
periodicity for periods less than 14 hours. Going further, Harp et al. (2020)
provides the tightest constraints to date, finding that periods from 0 to 40
hour would have been detected in 99% of their realisations.

As shown in Section 4, including these additional observations into our
analysis provides tighter constraints on the joint posterior. Most
importantly, the inclusion of this data reduces the maximum likelihood from
32.3% with Big Ear alone to 1.78% (2.37 $\sigma$). Thus, the data collected to
date are surprising under the stated hypothesis, but not at the level
typically needed to reject a hypothesis ($>3$ $\sigma$ or more). We repeated
the follow-up MCMC but adding $X$ hours of simulated null results to
investigate how much data would be needed to reach a 3 $\sigma$ threshold.
Iterating, we found that 62 days of accumulated additional observations would
be needed333Note that this timescale is insensitive to the priors used since
it represents simply the maximum likelihood value.. On this basis, we argue
that the stochastic repeater hypothesis is not yet excluded as an explanation
for the Wow signal, and yet more follow-up would be necessary.

More broadly, our likelihood emulation approach provides an example of dealing
with sparse, irregular observations to conduct inference of SETI signals of
interest. Indeed this approach could be readily applied to other
technosignature data sets, such as the Breakthrough Listen radio search (Price
et al., 2020; Sheikh et al., 2021) or even non-radio searches such as in the
optical (e.g. see Maire et al. 2020). It is emphasised that such an approach
could also be used for targets without any signals detected, in order to
derive robust upper limits on a hypothesised signal’s properties (be it
periodic or not).

## Acknowledgments

Co-author Robert Gray tragically passed away during this work. RG devoted
himself to pursuing the Wow signal throughout his career, both in science and
public discourse. His insights and contributions to this enigma are
unparalleled and were pivotal to this paper. DK is grateful for his
generosity, time and contributions to this study, which it is hoped live up to
his high standards of inquiry and curiosity.

This work was enabled thanks to supporters of the Cool Worlds Lab, including
Mark Sloan, Douglas Daughaday, Andrew Jones, Elena West, Tristan Zajonc, Chuck
Wolfred, Lasse Skov, Graeme Benson, Alex de Vaal Mark Elliott, Methven Forbes,
Stephen Lee, Zachary Danielson, Chad Souter, Marcus Gillette, Tina Jeffcoat,
Jason Rockett, Scott Hannum, Tom Donkin, Andrew Schoen, Jacob Black, Reza
Ramezankhani, Steven Marks, Philip Masterson, Gary Canterbury, Nicholas
Gebben, Joseph Alexander, Mike Hedlund, Dhruv Bansal, Jonathan Sturm, Rand
Corporation, Ian Attard & Leigh Deacon.

## Data Availability

The code used and results generated by this work are made publcily available
at this URL.

## References

  * Benford, Benford, & Benford (2008) Benford J., Benford G., Benford D., 2008, arXiv, arXiv:0810.3964
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  * Dixon & Cole (1977) Dixon R. S., Cole D. M., 1977, Icar, 30, 267. doi:10.1016/0019-1035(77)90158-0
  * Ehman (1998) Ehman, J. R., 1998, “The Big Ear Wow! Signal: What We Know and Don’t Know About It after 20 Years”, OSURO working paper
  * Gray (1994) Gray R. H., 1994, Icar, 112, 485. doi:10.1006/icar.1994.1199
  * Gray & Marvel (2001) Gray R. H., Marvel K. B., 2001, ApJ, 546, 1171. doi:10.1086/318272
  * Gray & Ellingsen (2002) Gray R. H., Ellingsen S., 2002, ApJ, 578, 967. doi:10.1086/342646
  * Harp et al. (2020) Harp G. R., Gray R. H., Richards J., Shostak G. S., Tarter J. C., 2020, AJ, 160, 162. doi:10.3847/1538-3881/aba58f
  * Horowitz et al. (1986) Horowitz P., Matthews B. S., Forster J., Linscott I., Teague C. C., Chen K., Backus P., 1986, Icar, 67, 525. doi:10.1016/0019-1035(86)90129-6
  * Kipping (2021) Kipping D., 2021, MNRAS, 504, 4054. doi:10.1093/mnras/stab1129
  * Kraus (1979) Kraus J., 1979, CosSe, 1, 31
  * Maire et al. (2020) Maire J., Wright S. A., Werthimer D., Antonio F. P., Brown A., Horowitz P., Lee R., et al., 2020, SPIE, 11454, 114543C. doi:10.1117/12.2562786
  * Price et al. (2020) Price D. C., Enriquez J. E., Brzycki B., Croft S., Czech D., DeBoer D., DeMarines J., et al., 2020, AJ, 159, 86. doi:10.3847/1538-3881/ab65f1
  * Sheikh et al. (2019) Sheikh S. Z., Wright J. T., Siemion A., Enriquez J. E., 2019, ApJ, 884, 14. doi:10.3847/1538-4357/ab3fa8
  * Sheikh et al. (2021) Sheikh S. Z., Smith S., Price D. C., DeBoer D., Lacki B. C., Czech D. J., Croft S., et al., 2021, NatAs, 5, 1153. doi:10.1038/s41550-021-01508-8

Table 1:  Summary of OSU reobservations of the Wow locale, 1977-1984. Data comes from printouts pulled by Marc Abel around September 1985, and then filtered against bad data (e.g. clock errors, printer failures, etc) by co-author Robert Gray. Wow detection date is highlighted in bold.  Date | Days Passed | Date | Days Passed
---|---|---|---
13-Aug-1977 | 0 | 22-Jan-1983 | 1988
14-Aug-1977 | 1 | 22-Jan-1983 | 1988
15-Aug-1977 | 2 | 23-Jan-1983 | 1989
17-Aug-1977 | 4 | 24-Jan-1983 | 1990
16-Sep-1977 | 34 | 25-Jan-1983 | 1991
17-Sep-1977 | 35 | 28-Jan-1983 | 1994
18-Sep-1977 | 36 | 29-Jan-1983 | 1995
19-Sep-1977 | 37 | 2-Feb-1983 | 1999
20-Sep-1977 | 38 | 2-Feb-1983 | 1999
21-Sep-1977 | 39 | 4-Feb-1983 | 2001
22-Sep-1977 | 40 | 5-Feb-1983 | 2002
23-Sep-1977 | 41 | 6-Feb-1983 | 2003
24-Sep-1977 | 42 | 7-Feb-1983 | 2004
25-Sep-1977 | 43 | 12-Feb-1983 | 2009
26-Sep-1977 | 44 | 14-Feb-1983 | 2011
27-Sep-1977 | 45 | 15-Feb-1983 | 2012
28-Sep-1977 | 46 | 16-Feb-1983 | 2013
29-Sep-1977 | 47 | 17-Feb-1983 | 2014
30-Sep-1977 | 48 | 18-Feb-1983 | 2015
4-Oct-1977 | 52 | 19-Feb-1983 | 2016
30-Oct-1977 | 78 | 20-Feb-1983 | 2017
10-Apr-1978 | 240 | 21-Feb-1983 | 2018
11-Apr-1978 | 241 | 22-Feb-1983 | 2019
17-Apr-1978 | 247 | 23-Feb-1983 | 2020
24-Apr-1978 | 254 | 24-Feb-1983 | 2021
25-Apr-1978 | 255 | 25-Feb-1983 | 2022
26-Apr-1978 | 256 | 26-Feb-1983 | 2023
27-Apr-1978 | 257 | 27-Feb-1983 | 2024
28-Apr-1978 | 258 | 28-Feb-1983 | 2025
1-May-1978 | 261 | 1-Mar-1983 | 2026
2-May-1978 | 261 | 7-Mar-1983 | 2032
28-Aug-1978 | 380 | 8-Mar-1983 | 2033
29-Aug-1978 | 381 | 9-Mar-1983 | 2034
30-Aug-1978 | 382 | 12-Mar-1983 | 2037
31-Aug-1978 | 383 | 13-Mar-1983 | 2038
1-Sep-1978 | 384 | 14-Mar-1983 | 2039
2-Sep-1978 | 385 | 20-Mar-1983 | 2045
3-Sep-1978 | 386 | 21-Mar-1983 | 2046
4-Sep-1978 | 387 | 23-Mar-1983 | 2048
2-Jan-1983 | 1968 | 26-Mar-1983 | 2051
3-Jan-1983 | 1969 | 28-Mar-1983 | 2053
4-Jan-1983 | 1970 | 29-Mar-1983 | 2054
7-Jan-1983 | 1973 | 26-Apr-1983 | 2082
13-Jan-1983 | 1979 | 9-May-1983 | 2095
15-Jan-1983 | 1981 | 6-Dec-1984 | 2672


11institutetext: Paolo Rissone 22institutetext: Isabel Pastor 33institutetext:
Felix Ritort 44institutetext: Facultat de Fisica, Universitat de Barcelona,
Carrer de Martí i Franqués 1, Barcelona 08028, Spain 55institutetext: Felix
Ritort 66institutetext: Institut de Nanociència i Nanotecnologia (IN2UB),
Facultat de Fisica, Universitat de Barcelona, Carrer de Martí i Franqués 1,
Barcelona 08028, Spain, 66email<EMAIL_ADDRESS>
# Unraveling RNA by Mechanical Unzipping

Paolo Rissone  Isabel Pastor  Felix Ritort

###### Abstract

We review the basic concepts and tools for mechanically unzipping RNA hairpins
using force spectroscopy. By pulling apart the ends of an RNA molecule using
optical tweezers, it is possible to measure the folding free energy at varying
experimental conditions. Energy measurements permit us to characterize the
thermodynamics of RNA hybridization (base pairing and stacking), the dynamics
of the formation of native and kinetic (intermediates and misfolded) molecular
states, and interactions with metallic ions. This paper introduces basic
concepts and reviews recent developments related to RNA force thermodynamics,
native and barrier RNA energy landscapes, and RNA folding dynamics. We
emphasize the implications of mechanical unzipping experiments to understand
non-coding RNAs and RNAs in extreme environments.

###### Keywords:

single-molecule spectroscopy — RNA unzipping — out-of-equilibrium
thermodynamics — RNA free-energy landscape

##  1 Introduction

RNA, the genome’s dark matter, directly impacts biological diversity and life
(Darnell 2011; Filipowicz 2022). RNAs can fold into multiple configurations
stabilized by secondary and tertiary structures (Butcher and Pyle 2011;
Herschlag et al. 2018), multivalent cations, and ligands (Pyle 2002; Woodson
2005; Draper et al. 2005; Bowman et al. 2012). The promiscuity of base pairing
and stacking interactions makes RNA a unique biopolymer with many functions,
from information carrier to regulatory and enzymatic activity. RNA exhibits a
significant degree of heterogeneity at the sequence level and the
conformational level (Treiber and Williamson 2001; Russell et al. 2002; Brion
and Westhof 1997; Cruz and Westhof 2009). Upon folding, RNA can form native
and non-native structures (such as misfolded and intermediates) (Woodson
2010), with critical roles at the level of genomic maintenance and the
cellular function (Mattick and Makunin 2006; Aalto and Pasquinelli 2012),
therapeutics (Esteller 2011; Matsui and Corey 2017) and diseases (Jain and
Vale 2017; Blaszczyk et al. 2017; Zhao and Usdin 2021). Although the molecular
forces operating in RNA are known, the role played by disorder at the
structural and functional levels poses severe challenges to the life scientist
who must cope with unprecedented complexity.

In recent years, a knowledge gap has appeared not only at the level of RNA
transcriptomics but also at the level of non-coding RNAs (ncRNAs) and their
remarkable variety of functions in concert with ligands and proteins (Stefani
and Slack 2008; Aalto and Pasquinelli 2012; Mercer et al. 2009; Ma et al.
2013; Fatica and Bozzoni 2014; Statello et al. 2021).

Although the role of many RNAs remains unknown, new RNAs with new
functionalities and structures are being discovered. Besides the much-studied
tRNA, rRNA, microRNA, riboswitches, ribozymes, and artificially evolved RNAs,
novel behaviors have been observed in response to environmental cues such as
temperature (e.g. RNA cold denaturation (Mikulecky and Feig 2002), RNA
thermometers (Loh et al. 2013)), and in concerted action with proteins
(catalytic complexes, chaperones, packaging, condensation, etc.).

Despite the enormous progress in next-generation sequencing and big data
analysis, our current knowledge of RNA diversity is compromised by the limited
accuracy, sensitivity, and specificity of available methods to detect
different RNA conformations across RNA populations. Moreover, determining the
folding pathways and the energetics of the various RNA structures is essential
to understanding RNA function. Single-molecule techniques have represented a
big step in addressing RNA complexity (Ritort 2006; Seidel and Dekker 2007).
Their great sensitivity and accuracy permit us to detect and measure the
folding energies of rarely occurring conformations that escape detection by
the standard bulk methods. Powerful techniques such as single-molecule FRET
(Zhuang 2005; Alemán et al. 2008) and force spectroscopy (Ritchie and Woodside
2015; Bustamante et al. 2021) can monitor RNA conformational transitions in
real-time More recently, solid-state nanopore microscopy for RNA target
detection can analyze thousands of single RNAs without amplification offering
exciting prospects (Henley et al. 2016; Bošković and Keyser 2022).

Compared to DNA, RNA exhibits more complex behavior. The replacement of
deoxyribose for ribose and thymine for uracil makes RNA catalytic due to the
reactive polarizable 2’-OH group of ribose. Ribose also induces large changes
at the level of base stacking interactions between contiguous bases. In their
double-stranded forms, nucleic acids (NA) form distinct right-handed double
helices, B-form and A-form. Although DNA can adopt both A-form and B-form, RNA
is mainly found in A-form with only a few rare exceptions (Shi et al. 2003).
In DNA, bases are mainly parallel to the helical plane with an interphosphate
distance of 3.4Å. In contrast, RNA bases are tilted by approximately 19
degrees relative to the helical plane and the interphosphate distance is
smaller ($\sim 2.8$Å); These structural differences generate stacking between
inter-strand bases and tighter water molecular bridges between phosphates and
bases in RNA. Overall, base stacking tends to be stronger in RNA than in DNA.
Base stacking is due to the Van der Waals attractive forces of the fluctuating
dipole-dipole interactions between contiguous bases. Much weaker than the
covalent nature of hydrogen bonding, the effect of the latter is minimized
upon secondary structure formation due to the compensation effect of hydrogen
bonding with water. Overall, base stacking and hydrogen bonding contribute
equally to RNA helix stabilization, albeit the $1/r^{6}$ dependence of Van der
Waals forces makes stacking strongly sensitive to the inter-base distance,
$r$. Therefore, RNA structure strongly depends on RNA stacking between intra-
strand and inter-strand bases, making RNA folding prediction a difficult
problem.

Here we briefly review recent discoveries in the thermodynamics of RNA folding
using force spectroscopy studies with laser optical tweezers (LOT). The paper
is organized as follows. Section 2 describes three main experimental
techniques to investigate RNA kinetics and thermodynamics at the single-
molecule level. In Sec. 3 we focus on RNA force spectroscopy, the main
experimental protocols and a few selected exemples. In Sec. 4 we discuss how
RNA unzipping experiments make it possible to derive RNA thermodynamics at
0.1kcal/mol accuracy. Section 5 introduces the concept of the barrier energy
landscape and the importance of stem-loops to stabilize a multiplicity of RNA
kinetic structures. Finally, in Sec. 6 we digress about future perspectives in
single-RNA manipulation.

##  2 The Power of Single RNA Manipulation

Figure 1:  Single-molecule techniques. (A) On-surface-immobilized smFRET
experiment. (B) Nanopore microscopy uses a biological pore (top panel) and a
solid-state pore (bottom panel). (C) Optical tweezers experiments in a double-
trap geometry (left panel and in a single-trap geometry with a micropipette
(right panel).

Single-molecule (SM) experiments permit us to study individual molecules’
behavior and characterize the molecular properties’ heterogeneity. On the
contrary, in conventional bulk experiments, any observation is an average over
a large population of molecules, making it impossible to observe ephemeral or
transient events. During the last decades, the development of single-molecule
techniques opened a new window to the rich phenomenology of RNA molecules. In
particular, three useful single-molecule techniques are single-molecule
fluorescence (Förster) resonance energy transfer (smFRET) (Alemán et al. 2008;
Zhuang 2005; Ray et al. 2018; Chauvier et al. 2019; Ha et al. 1999; Zhao and
Rueda 2009), solid-state nanopore microscopy (Seidel and Dekker 2007; Cui et
al. 2021; Bandarkar et al. 2020; Lee et al. 2021; Bošković and Keyser 2022)
and force spectroscopy using optical tweezers (Woodside et al. 2006;
Bustamante et al. 1991; Ritchie and Woodside 2015; Manosas and Ritort 2005;
Wen et al. 2007; Zhuang 2005). First, we overview these techniques, focusing
next on the optical tweezers experiments.

###  2.1 smFRET Experiments

smFRET measurements measure conformational changes with a high temporal
resolution (ms) (Joo et al. 2008; Ha 2001). This technique measures the
distance between two fluorophores (donor and acceptor) at different positions
along the biomolecule. The donor is located in the vicinity of the acceptor
(distance less than 10 nm) and excited with an appropriate light wavelength.
Part of the energy emitted by the donor is transferred to the acceptor through
a non-radiative dipole-dipole interaction, causing the acceptor’s emission.
Donor-acceptor energy transfer is a quantum-mechanical effect due to the
overlap between the donor emission spectrum and the acceptor absorption
spectrum. The energy transfer efficiency, defined as the ratio
$I_{A}/(I_{D}+\eta I_{A})$ between light intensity emitted by the donor
$I_{D}$ and the acceptor $I_{A}$ ($\eta$ being a quantum yield correction
factor), depends on the distance between the two fluorophores according to the
equation: $E=1/(1+(R/R_{0})^{6})$, where $R$ is the distance between both
fluorophores and $R_{0}$ is the characteristic distance (Förster distance)
where the efficiency is one-half (Lakowicz 2006).

The experiments can be carried out either on-surface-immobilized (Fig. 1A) or
freely diffusing molecules. The former permits parallelized measurements
during a long time (until photobleaching, typically up to tens of seconds)
allowing the detection of slow conformational changes. smFRET requires
biochemical modifications by attaching fluorophores to the molecule under
study, which may interfere with the (unmodified) molecular behavior (Ha et al.
1999; Zhao and Rueda 2009). In addition, in the case of on-surface-immobilized
smFRET experiments, the perturbations that the surface may exert on the
molecule must be taken into account(Schmitz et al. 2015). There are several
works where smFRET has been key to determining intermediate or misfolded
states in RNA folding (Bartley et al. 2003; Bokinsky et al. 2003; Xie et al.
2004; Alemán et al. 2008; Bokinsky et al. 2003).

###  2.2 Nanopore Microscopy

In recent years, nanopore microscopy has shown to be a promising tool to
address RNA complexity. Nanopores, either solid-state or biological ones, are
extremely sensitive to the sequence and structure of biomolecules. This
technique is based on the current established between two electrodes placed in
different pools connected by a nanometric hole. The pools are filled with a
salt solution and a charged biomolecule is placed in one of them. When a
voltage difference is applied, the flow of ions through the nanopore results
in an electric current. The flow of ions and the electric field between the
electrodes makes the molecules flow through the nanopore. As molecular
translation proceeds, the flow of ions is partially (or fully) obstructed,
resulting in a reduction of the net current. These ion fluctuations depend on
the biomolecule’s properties. Usually, the biomolecule needs to rearrange to
translocate, making nanopore translocation an important tool to screen
different molecular conformations.

Traditionally, RNA translocation experiments use biological pores such as ion
channel proteins (Fig. 1B, top). A widely used model is alpha-hemolysin
(Butler et al. 2007; Sultan and Kanavarioti 2019; Meller et al. 2006) due to
its large stability and small pore diameter that confers much sensitivity to
detect individual nucleotides. Biological pores generally exhibit lower noise.
However, solid-state nanopores (Fig. 1B, bottom) can operate at higher
voltages and bandwidths, making it possible to achieve a better signal-to-
noise ratio, key to detecting translocation events. Moreover, solid-state
nanopores are pore-size and geometry-tunable and more robust than biological
ones.

A key advantage of solid-state nanopore microscopy is that it does not require
chemically modifying the biomolecule. However, there are some disadvantages
such as the limited nanopores reproducibility and the high translocation
speed.

###  2.3 Optical Tweezers

Among all single-molecule force spectroscopy techniques, optical tweezers have
proved to be one of the most powerful for studying the complexity of nucleic
acids, especially in the case of RNA. Laser optical tweezers (LOT) use a
focused laser beam to optically trap a transparent microbead attached to one
end of an RNA molecule. By attaching the other end to a surface, the RNA can
be pulled by displacing the optically trapped bead by moving the laser. LOT
can exert forces in the range of tenths to hundreds of piconewtons measuring
energies with 0.1kcal/mol accuracy. LOT can monitor force and bead position,
detecting conformational transitions with sub-millisecond temporal resolution.

There are different setups to carry out LOT experiments, such as dual-trap
(Fig. 1C, left) and single-trap configurations (Fig. 1C, right). In the dual-
trap configuration, the molecule is tethered between two trapped beads, while
in the single-trap configuration, the molecule is tethered between two beads:
one is optically trapped and the other is held by air suction at the tip of a
micropipette. Usually, the dual-trap setup is more sensitive because the traps
are formed from the same laser and have less drift. The single-trap
micropipette configuration is used in many experiments, such as those
presented in this review.

Independently on the setup, to avoid bead-bead interactions, the RNA molecule
under study is inserted between two molecular linkers or spacers usually DNA-
RNA hybrid handles (Wen et al. 2007; Manosas et al. 2007; White and Visscher
2011; Collin et al. 2005; Liphardt et al. 2001; Onoa et al. 2003; Martinez-
Monge et al. 2022; Wu et al. 2014). As compared with other single-molecule
techniques, the mechanical manipulation of RNAs allows monitoring the
unfolding/folding of individual structural domains of large RNAs and their
folding pathways. Moreover, RNAs can be pulled to detect short-lived
intermediates and misfolded states, which can be modified by the ion
concentration (Na+,Mg+2, etc.) or the temperature. In addition, RNA
manipulation with optical tweezers permits measuring energies with 0.1kcal/mol
accuracy (Severino et al. 2019; Rissone et al. 2022; Martinez-Monge et al.
2022).

##  3 RNA Pulling in a Nutshell

Several pulling protocols can be implemented to study RNAs with optical
tweezers. The most common are force-ramp, hopping, and force-jump experiments.
In the following, we show examples of RNA pulling experiments with LOT.

Figure 2:  Optical tweezers experiments. (A) Force-ramp protocol. The trap-
pipette distance is changed at a constant velocity. The typical FDC obtained
in force-ramp experiments is shown in the bottom panel. (B) Hopping protocols.
Top: PM experiments. The optically trapped bead moves toward or away from the
trap center as the molecule unfolds or folds, respectively. The right panels
show force and extension variation with time (top and bottom, respectively).
Bottom: CFM experiments. The force is kept constant by a feedback protocol
compensating for extension changes of the molecule. The right panels show
force and position variation with time (top and bottom, respectively). (C)
Force jumps protocol. The force or the distance is quickly changed to a
different value. The typical distance versus time signal is reported in the
bottom panel.

In force-ramp protocols, the optical trap is repeatedly moved back and forth
from the micropipette (Fig. 2A, top) with the force steadily increasing and
decreasing, respectively. Upon increasing the force the RNA hairpin switches
from its folded (native) state to the totally unfolded state and vice-versa
upon releasing the force. A plot of the applied force versus the distance
between the trap and the micropipette gives the force-distance curve (FDC)
(Fig. 2A, bottom). In small RNA hairpins (typically a few tens of bases) that
cooperatively fold and unfold, the rips in the FDC indicate unfolding or
folding transitions between the native folded hairpin and the stretched ssRNA
conformation. For longer RNA hairpins (a few hundred bases) the unfolding into
ssRNA is a sequential process in which groups of base-pairs open. In this
case, the FDC exhibits a sawtooth pattern depending on the molecular sequence
(Fig. 5). Pulling experiments allows for the characterization of the
unfolding/folding forces as well as the thermodynamics and kinetics of RNA
molecules (see Sections 4 and 5).

In hopping experiments, either the trap position or the force is kept
constant, and the jumps of the molecule from the unfolded (folded) to folded
(unfolded) state are monitored over time. There are two kinds of hopping
assays: passive mode (PM) and constant force mode (CFM) experiments. In PM
hopping experiments, the trap position is clamped (Fig. 2B, top), and the RNA
hairpin hops between the unfolded and the folded state with a force jump at
every transition. In PM both the force and the molecular extension change over
time because the trapped bead relaxes to a new position at every RNA hop, and
the force changes accordingly. The PM allows direct monitoring of the
molecular transitions in a narrow range of forces close to the coexistence
force where the RNA equally populates the native and unfolded states. PM is
not suitable for long-time measurements in LOT in the single-trap
configuration due to the uncontrolled movements of the pipette (drift
effects). In contrast, in CFM hopping experiments, the force is kept constant
with a feedback loop (Fig. 2B bottom), and the RNA extension is recorded as a
function of the time. Here, the force is fixed at a value near the RNA
coexistence force, allowing for measuring the transition between its
folded/unfolded states and the lifetime of each state at the studied force.

The force-jump protocol is helpful in characterizing RNA irreversible
processes. In this case (Fig. 2C), the force or the trap position is quickly
changed to a new preset value and kept constant. The RNA lifetime is measured
until a conformational transition is observed. Force-jump is an irreversible
pulling protocol, requiring multiple experiments at different present force
values to derive the unfolding and folding kinetics.

###  3.1 Brief History of RNA Pulling Experiments

Figure 3:  Different results obtained in pulling experiments. (A) P5ab RNA
force-extension curves in 10 mM Mg2 (adapted from (Liphardt et al. 2001)).
Structure of P5ab and P5abc RNAs (inset). (B) 16S ribosomal FDCs, the colors
represent successive pulling for the same molecule. Figure adapted from
(Harlepp et al. 2003). (C) Unfolding (black) and folding (pink) FDCs of the T.
thermophila ribozyme (left panel). The letters indicate different kinetics
barriers. The right panel shows the secondary structure. Figure adapted from
(Onoa et al. 2003).

Since the beginning of the 21st century, LOT experiments have been used in
many works studying the structure and energetics of RNAs and their native,
misfolded, and short-lived intermediate states, and the salt dependency of the
RNA conformation. In (Liphardt et al. 2001), it has been shown for the first
time that by mechanically unzipping RNA hairpins, it is possible to derive its
folding free energy (Fig. 3A). They studied three RNAs: a small hairpin
(P5ab), a molecule with a three-helix junction (P5abc$\Delta$A), and a more
complex molecule (P5abc), which in presence of Mg+2 ions folded into a stable
tertiary structure. In particular, they found that P5abc folded into a stable
tertiary structure through a short-lifetime intermediate.

Mechanical pulling experiments have also been carried out on longer RNAs: in
(Harlepp et al. 2003), it has been studied the 1540 nucleotides 16S ribosomal
RNA from E. coli (Fig. 3B). This large RNA exhibits a surprisingly well-
structured and reproducible unfolding pathway under mechanical stretching.
Similar results have been found in (Onoa et al. 2003) by pulling the L-21
derivative of the Tetrahymena thermophila ribozyme (a 390 nucleotides RNA).
This molecule featured a complex secondary structure with multiple unfolding
intermediate states (Fig. 3C). To identify the different RNA structures, they
measured the number of released base pairs along the FDCs of progressively
large fragments of the RNA molecule. At the same time, they used mutants and
anti-sense oligonucleotides to characterize these RNA structures further and
measure their free energy of formation.

Figure 4:  Different results obtained in pulling experiments. (A) Left: the
two structures proposed for rpsO RNA: double hairpin (top) and pseudoknot
(bottom). Right: FDCs of the double hairpin (HR), pseudoknot (BR), and mixed
transitions (2R). Figure adapted from (Wu et al. 2014). (B) Left: The highly
conserved crystal structure of the RNA three-way junction molecule from T.
thermophilus (Serganov et al. 2001). Right: FDCs for the native and misfolded
structures in 10 mM Mg+2. Figure adapted from (Martinez-Monge et al. 2022).

Another example of a biologically relevant RNA is the study of the operator
rpsO of the RNA gene coding for the S15 subunit of 30S ribosomal protein from
E. coli (Wu et al. 2014). This mRNA can fold into two spontaneously
interchangeable structures: a double hairpin and a pseudoknot (Fig. 4A, left).
Their work showed that the conversion from the double hairpin to the
pseudoknot is stabilized by the interaction between the two hairpins (Fig. 4A,
right). Sequence mutations can modulate the interaction between the two
hairpins.

Finally, in a more recent work (Martinez-Monge et al. 2022) RNA pulling
experiments have been carried out to measure the non-specific and specific
binding energies of magnesium to RNA. The RNA three-way junction (3WJ)
containing the minimal binding site to protein S15 of the ribosomal RNA from
E. coli was studied in monovalent and divalent salt conditions. This molecule
can fold into a native structure that contains the 3WJ motif with specific
Mg+2 binding sites or into a misfolded structure with a double hairpin
structure where the 3WJ and the binding sites have been disrupted (Fig. 4B).
By comparing the free energy of formation of these two structures with and
without magnesium, it has been possible to determine the specific and non-
specific energy contributions of Mg+2 binding to the RNA.

##  4 RNA Energetics at $0.1$ kcal/mol Accuracy

The characterization of RNA thermodynamics is fundamental to understanding the
promiscuity of behaviors observed in RNA, from the multiplicity of native
structures (Gralla and Delisi 1974) to misfolding (Alemany et al. 2012). Only
RNA exhibits such behavior despite DNA and RNA forming double-stranded
helices. DNA unzipping is a fully reversible process (Fig. 5A) over a broad
range of salt conditions and loading rates (Huguet et al. 2010; Bizarro et al.
2012). Instead, RNA unzipping presents strong irreversibility between the
unfolding and refolding FDCs in the same experimental conditions (Fig. 5B). In
this case, transient off-pathway misfolded structures appear during the
unzipping–rezipping process, slowing down the hybridization reaction (Liphardt
et al. 2001; Chen and Dill 2000; Woodson 2010). Characterizing these off-
pathway structures competing with the native stem is a challenging problem
(Rissone et al. 2022).

Figure 5:  Unzipping (red) and rezipping (blue) FDCs measured by pulling a
6.8kbp DNA (A) and a 2kbp RNA (B) hairpins in a 1M NaCl buffer at $25^{\circ}$
C. While DNA unzipping is a reversible process, RNA unzipping shows many off-
pathway out-of-equilibrium states. The resulting hysteresis requires the
computation of an equilibrium FDC (black) to measure the NNBPs’ free energies.

###  4.1 The RNA Free Energy of Formation

The hybridization of the double-stranded helix of nucleic acids (NA) is
governed by the Watson-Crick pairing rules between nucleotides (adenine,
guanine, cytosine, thymine or uracil) from opposite NA strands (Saenger 1984).
These relations account for purine–pyrimidine bonding so that adenine can only
bond to thymine (or uracil in the RNA case) and guanine to cytosine. As
discussed before, the double helix is not only stabilized by base-pairing but
also by the stacking interactions between adjacent nucleotides. To account for
these effects, the energetics of NA formation is usually described using the
nearest-neighbor (NN) model (DeVoe and Tinoco Jr 1962; Breslauer et al. 1986).
According to this model, the base-pairing energy of two complementary bases
depends on the base itself and the first neighbor located in the same strand
along the $5^{\prime}\rightarrow 3^{\prime}$ direction. This gives 16
different possible combinations of nearest-neighbour base-pairs (NNBPs): out
of these, 6 are degenerate, and 2 can be expressed as a linear combination of
the others (circular symmetry) (Huguet et al. 2017), leaving with only 8
independent parameters. The ten RNA NNBP are denoted as $\rm
XY/\overline{X}\,\overline{Y}$ (x-label in Fig. 7A) where $\rm X,Y=A,C,G,U$,
and $\rm\overline{X}(\overline{Y})$ is the complementary base of $\rm X(Y)$
and $\rm{XY/\overline{X}\,\overline{Y}}$ is the NNBP resulting from
hybridizing dinucleotides $\rm 5^{\prime}-XY-3^{\prime}$ and $\rm
5^{\prime}-\overline{Y}\,\overline{X}-3^{\prime}$. The energies of
$\rm{XY/\overline{X}\,\overline{Y}}$ and $\rm\overline{Y}\,\overline{X}/YX$
are equal due to complementary strand symmetry. For example, the RNA sequence
$\rm 5^{\prime}-CUUAGC-3^{\prime}$ forms a duplex with its complementary
strand, $\rm 5^{\prime}-GCUAAG-3^{\prime}$. According to the NN model the
energy of hybridization of such a sequence equals $\rm\Delta g_{CU/GA}+\Delta
g_{UU/AA}+\Delta g_{UA/AU}+\Delta g_{AG/UC}+\Delta g_{GC/CG}$ with $\rm\Delta
g_{CU/GA}=\Delta g_{AG/UC}$ due to complementary strand symmetry. The
derivation of the 10 (8 if circular symmetry is applied) NNBPs free-energies
has been carried out by unzipping a 2.2kbp and 6.8kbp DNA and a 2kbp RNA
hairpins at different salt conditions (Huguet et al. 2010; 2017; Rissone et
al. 2022).

In thermodynamics, the free-energy difference equals the mechanical work in a
reversible process that requires the system to evolve along a sequence of
equilibrium states. This is only applicable to the DNA case where the
unzipping and rezipping FDCs do not show hysteresis (Fig. 5A). On the
contrary, RNA unzipping is an out-of-equilibrium process in our experimental
timescales: a large hysteresis is observed between unzipping and rezipping
FDCs due to the formation of multiple long-lived (off-pathway) states (Fig.
5B). An equilibrium FDC (black line) had to be computed from the RNA
unzipping/rezipping experimental data. This has been achieved by developing a
statistical method based on the (extended) fluctuation relations (Bennett
1976; Jarzynski 1997; Shirts et al. 2003; Junier et al. 2009) that allowed for
the computation of the equilibrium free energy during the isothermal unzipping
process (see Supp. Info in (Rissone et al. 2022)). Finally, a Monte Carlo
optimization algorithm has been tailored to relate the experimental data with
the numerical FDC prediction, ultimately permitting measuring the NNBP
energies in DNA and RNA (Huguet et al. 2010; 2017; Rissone and Ritort 2022).
The experimentally derived values for RNA in sodium and magnesium are shown in
Fig. 6A along with values reported in the literature (the Mfold set) (Freier
et al. 1986; Mathews et al. 1999; Walter et al. 1994; Xia et al. 1998; Zuker
2003). As the latter are only available at 1M NaCl, the comparison required
applying a correction to the measured NNBP energies.

###  4.2 Salt Dependency of the Hybridization Free Energy

The effect of a monovalent salt concentration, $[\rm Mon+]$ in molar units, on
the hairpin free energy of formation is described by the relation

$\Delta g_{0,i}[{\rm Mon+}]=\Delta g_{0,i}[{\rm 1M}]-m\log{[{\rm Mon+}]}\,,$
(1)

where $\Delta g_{0,i}[\rm 1M]$ is the free energy of formation of motif $i$ at
1M of monovalent salt at zero force and $m=0.10\pm 0.01$ kcal/mol is the RNA
NNBP-homogeneous monovalent salt correction (Bizarro et al. 2012). However,
Eq.(1) only holds for monovalent ions and its extension to divalent ions
requires to account for the effect of the divalent salt on the stabilization
of the double-helix.

RNAs are highly charged polyanions whose stability strongly depends on solvent
ionic conditions. The ability of divalent ions, such as $\rm Mg^{2+}$, to
stabilize RNA structures at much lower concentrations than monovalent ions is
known since the 70 _s_ (Cole et al. 1972). This effect is usually quantified
by the so-called 100:1 rule of thumb which states that the concentration of
divalent salt equals 100-fold that of monovalent salt. This phenomenological
rule has been experimentally tested by unzipping the 2kbp RNA hairpin at 500
mM NaCl and 10 mM MgCl2 (Rissone et al. 2022). By plotting the RNA NNBP energy
values in $\rm Na^{+}$ versus those in $\rm Mg^{2+}$ (Fig. 6B), we demonstrate
that the divalent salt concentration is equal to $77\pm 49$ the one of
monovalent salt, which is compatible with the phenomenological rule.

This result has also been validated by measuring the free energy of formation
of short RNA duplexes unzipped in a broad range of monovalent and divalent
salt concentrations (Bizarro et al. 2012). By plotting these results versus
the salt concentration, the data can be collapsed into a single master curve
by multiplying the $[{\rm Div}\,2+]$ by a factor $\sim 80$ (Fig. 6C). The
agreement between the results proves that the RNA hybridization free energy of
the hairpin, i.e. the energy of the hairpin native conformation, satisfies
(within errors) the 100:1 rule of thumb. Finally, a fit to data (dashed grey
line) proved the validity of the logarithmic salt correction to the formation
energy in Eq.(1).

Figure 6:  RNA energetics. ($\mathbf{A}$) NNBPs free-energies measured by
unzipping a 2kbp RNA hairpin at 500mM NaCl (triangles) and 10mM MgCl2
(circles) (Rissone et al. 2022). Notice that the values are scaled to 1M of
equivalent sodium concentration in order to compare with the literature (grey
squares). ($\mathbf{B}$) Experimental validation of the 100:1 rule of thumb. A
fit to data (solid black line) gives $77(\pm 49)$:1 (Rissone et al. 2022).
($\mathbf{C}$) Hairpin (total) free-energy of formation measured in RNA
unzipping experiments of a 20bp hairpin (Bizarro et al. 2012) in sodium (blue
squares) at 50mM, 150mM, 550mM, 1050mM and magnesium (orange circles) at
0.01mM, 0.10mM, 0.50mM, 1mM, 4mM, 10mM. A fit to data (dashed line) shows the
logarithmic dependence of the salt correction. All results in magnesium are
reported in sodium equivalents according to the 100:1 rule.

##  5 RNA Folding Kinetics

The typical unzipping patterns of DNA and RNA hairpins are very different, as
shown in Fig. 5A and Fig. 5B. To characterize the strong irreversibility
observed in RNA, it has been hypothesized that stem-loop structures form along
the two unpaired RNA strands during the unzipping (rezipping) process. As the
unzipping reaction progresses, forming such structures close to the junction
(that separates the native stem from the unpaired ssRNA) slows down hairpin
hybridization. Consequently, the system gets trapped into off-pathway
metastable conformations generating the observed hysteresis (Fig. 7A, top-
left).

Figure 7:  RNA unzipping and rezipping kinetics. ($\mathbf{A}$) Stem-loops BEL
of a 2kbp RNA hairpin. The stem-loops formation along the ssRNA drives the
folding process by stabilizing off-pathway states that slow down the stem
hybridization (top-left). The BEL correlates with the hysteresis measured
along the FDC (bottom and top-right inset). Panel adapted from (Rissone and
Ritort 2022). ($\mathbf{B}$) Correlation of the forming stem-loops with the
UR, ER, and UE hysteresis (see text) as a function of stem-loops length, $L$.
($\mathbf{C}$) Unzipping of a short (52 bases) RNA hairpin at 10 mM MgCl2. The
presence of divalent ions causes the molecule to misfold (inset).

This scenario can be modeled by introducing the barrier energy landscape (BEL)
(Rissone et al. 2022; Rissone and Ritort 2022) that includes all combinations
of a number $k$ of stem-loops ($k=1,2,\dots K$, with $K$ a maximum total
number) that can form in the two unpaired RNA strands. Let $n$ be the number
of hybridized base pairs in a hairpin with a total number of N base pairs
(Fig. 7A, top-left). Each unpaired strand contains N-n bases, and the two
unpaired strands taken together contain 2(N-n) bases. Therefore, we can
consider that stem loops can form in a single unpaired strand of length
2(N-n). For a given number $k$ of stem-loops of size $L$, we will consider
them as randomly distributed along the 2(N-n) bases strand (Fig.A). For a
given applied force $f$, the BEL is defined as

$\Delta G_{L}(f)=-k_{\rm B}T\log\sum_{k=0}^{K}\exp\left(-\frac{\Delta
g_{L}(k,f)}{k_{\rm B}T}\right)\,,$ (2)

where $\Delta g_{L}(k,f)$ is the total free-energy contribution of $k\geq 0$
stem-loops along the unpaired strand and $K=\lfloor 2(N-n)/L\rfloor$ is the
maximum number of stem-loops (Rissone and Ritort 2022). Notice that for $k=0$
no stem loops are formed. In principle, stem loops cannot overlap because only
one structure can be formed with the same bases. For sake of simplicity, we
will consider the approximated case where stem loops can overlap.

The loop-BEL computed in Eq.(2) correlates with the amount of observed
hysteresis (Fig. 7A, bottom and top-right inset), measured as the area between
unzipping-rezipping (UR), unzipping-equilibrium (UE) and equilibrium-rezipping
(ER) curves (Rissone et al. 2022). In particular, the study of this
correlation as a function of the stem-loops size, $L$, shows that the
stability of stem-loops increases with $L$ reaching a maximum for $L\sim 20$
bases and gently decaying for larger sizes (Fig. 7B).

Stem-loops formation is a kinetic process that competes with native
hybridization explaining the different behaviors observed in DNA and RNA
unzipping. A main source of irreversibility in RNA is the higher stability of
RNA stacking. In fact, the average NNBP RNA free-energy is $\langle\Delta
g_{0}\rangle^{{\rm RNA}}\approx-2.2$ kcal/mol while for DNA $\langle\Delta
g_{0}\rangle^{{\rm DNA}}\approx-1.7$ kcal/mol. This $\approx 0.5$ kcal/mol
difference could be sufficient for the RNA hairpin to slow down hybridization
without the need for stem loops forming along the unpaired strands making the
loop-BEL unnecessary. However, there is mounting evidence that RNAs can form a
multiplicity of structures as compared to DNA. In particular, in (Rissone et
al. 2022) it was found that a 52 bases native hairpin can also form
alternative misfolded structures not predicted by secondary-structure RNA
numerical models. By competing with the native pathway of the 2.2kb RNA
hairpin, the stem loops slow down the hybridization reaction stabilizing the
on-pathway intermediates in the FDC leading to the observed hysteresis.

Moreover, the observed irreversibility is enhanced in presence of magnesium
(Rissone et al. 2022). The metal ions induce higher flexibility to the RNA
chain and higher stability to the RNA helices due to the coordination effect
of the two positive charges and a reduction in the backbone’s charge
repulsion. An essential consequence of the reduced charge repulsion is that it
allows more frequent close encounters between the different RNA segments,
facilitating the formation of tertiary contacts (Tan and Chen 2009). Several
studies pointed out that $\rm Mg^{2+}$ ions are more efficient than $\rm
Na^{+}$ in stabilizing RNA tertiary folds (Chu et al. 2007; Chen 2008; Draper
2008; Walter et al. 2008). The same phenomenon is not observed for DNA, where
salt concentrations as high as 10 mM $\rm Mg^{2+}$ and 1 M $\rm Na^{+}$
(equivalent concentrations as per the salt rule) induce similar foldings (Tan
and Chen 2007). On the contrary, unzipping experiments at 10 mM $\rm Mg^{2+}$
on the previously mentioned 52 bases RNA hairpin showed that magnesium induces
misfolding (Fig. 7C), whereas only the native conformation is present at the
equivalent concentration of 1 M $\rm Na^{+}$ (Rissone et al. 2022). In
general, it has been observed that in mixed conditions of monovalent and
divalent salt, low concentrations of the latter are sufficient to stabilize
RNA tertiary structures in presence of specific binding sites (Heilman-Miller
et al. 2001).

##  6 Future Perspectives

Force spectroscopy is an exquisite tool to probe chemical interactions in
biomolecules. The finely tuned balance between hydrogen bonding and stacking
energies in nucleic acids makes them extremely sensitive to environmental
changes, sequence mutations, and chemical modifications (Song and Yi 2017).
Single-molecule fluorescence and nanopore microscopy are complementary tools
that permit the detection and monitor conformational transitions at zero
force. In contrast, force spectroscopy cannot probe molecular states at zero
force because the end-to-end molecular extension cannot be detected. Therefore
the combination of force with fluorescence (optical fleezers) (Whitley et al.
2017) and nanopores (optical trap nanopore) (Keyser et al. 2006; Trepagnier et
al. 2007; Yuan et al. 2020) offers exciting prospects for the future.

Force spectroscopy permits the direct estimation of free energy differences at
room temperature by direct work measurements, $W=\Delta G$. A new direction of
expansion is now possible using temperature-controlled optical tweezers (Mao
et al. 2005; Mahamdeh and Schäffer 2009; De Lorenzo et al. 2015). Controlling
temperature is often tricky, especially when the heating region is large
compared to the typical micrometer-sized dimensions of the experimental
trapping area. In this case, thermal expansion and convection effects lead to
uncontrolled drift effects. In a recent setup (De Lorenzo et al. 2015), a
heating laser can be switched on and off to controllably heat up the
experimental trapping region without convection and drift effects. The
instrument also operates at low temperatures by using an icebox kept at water-
freezing temperatures (1-4∘C). This temperature-jump optical trap has been
recently used to derive folding enthalpies, and entropies of DNA (De Lorenzo
et al. 2015) and proteins (Rico-Pasto et al. 2022). The instrument permits
measuring heat capacity changes, cold denaturation, and other previously
inaccessible phenomena. Fascinating is the study of RNA at very low
temperatures where thermal fluctuations are reduced, folding kinetics slowed
down, and details of the molecular interactions intensified. By lowering the
temperature, monitoring kinetics provides a natural microscope to amplify the
finest energetic features driving RNA folding.

While DNA is often referred to as life’s molecule (Frank-Kamenetskii 1993),
RNA is the dark matter of the genome (Darnell 2011; Mattick and Amaral 2022),
underlining how much we still do not know about this fascinating molecule. RNA
presents such remarkable features that biophysical models must be refined to
understand its many behaviors. We need physical models based on molecular
energy landscapes to unravel the folding kinetics for RNA folding. Concepts
borrowed from physics such as rugged free energy landscapes (Kirkpatrick and
Thirumalai 2015), and molecular replica symmetry breaking (Ritort 2022). Ideas
from soft and condensed matter physics may come into play shortly to explain
why RNA is so unique, in stark difference from the stable DNA counterpart. The
critical question is understanding under which conditions DNA might behave as
RNA. DNA lacks the reactive $2^{\prime}$–OH from ribose in RNA, impairing its
catalytic activity. Yet, DNA might fold in some conditions as RNA does.
Foreseeable experiments in the future are the study of folding kinetics of
ssRNAs and differences with ssDNA (Viader-Godoy et al. 2021; Rissone and
Ritort 2022). Other unexplored areas are awaiting discovery, such as RNAs with
chemical modifications, RNAs at low temperatures, RNAs in crowded
environments, etc. From custom-designed RNAs to biological RNAs (coding and
non-coding), our knowledge of this fantastic molecule has never stopped
growing and will be so for decades.

###### Acknowledgements.

P.R. was supported by the Angelo Della Riccia foundation. I. P. and F.R. were
supported by Spanish Research Council Grant PID2019-111148GB-I00 and the
Institució Catalana de Recerca i Estudis Avançats (F. R., Academia Prize
2018).

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