0 Research

# Let’s Get Vysical: Perceptual Accuracy in Visual & Tactile Encodings

Zhongzheng Xu
Emory University  e-mail<EMAIL_ADDRESS>Kristin Williams
Emory University  e-mail<EMAIL_ADDRESS>Emily Wall
Emory University  e-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we explore the effectiveness of tactile data encodings using
swell paper in comparison to visual encodings displayed with SVGs for data
perception tasks. By replicating and adapting Cleveland and McGill’s graphical
perception study for the tactile modality, we establish a novel tactile
encoding hierarchy. In a study with 12 university students, we found that
participants perceived visual encodings more accurately when comparing values,
judging their ratios with lower cognitive load, and better self-evaluated
performance than tactile encodings. However, tactile encodings differed from
their visual counterparts in terms of how accurately values could be decoded
from them. This suggests that data physicalizations will require different
design guidance than that developed for visual encodings. By providing
empirical evidence for the perceptual accuracy of tactile encodings, our work
contributes to foundational research on forms of data representation that
prioritize tactile perception such as tactile graphics.

Human-centered computingVisualizationVisualization techniques; Human-centered
computingVisualizationVisualization design and evaluation methods

Introduction

Figure 1: Summary of experiment design.

Since Data Visualization emerged as a research field, scientists have
incorporated findings from visual perception and cognition to develop guidance
on effective and efficient ways of communicating data [5, 19]. For example,
experiments on preattentive features shed light on basic visual properties
that can be readily detected by low-level visual systems [19, 36, 37]. Such
results inform design guidelines[10], and have enabled both researchers and
designers to leverage empirical findings to develop effective visualizations
[39]. The efficacy of bar charts, for instance, is supported by empirical
findings that humans are best at judging positions and lengths, compared to
other encodings such as area or curvature[6, 5]. In fact, there is a wealth of
prior work on graphical perception, dating back several decades and continuing
today, grounding guidance on visual representations of data in empirical
studies [8, 26, 36, 5, 6].

While tangible representations of data pre-date the field of data
visualization, data physicalization has garnered recent interest, building
upon momentum in data visualization, tangible user interfaces, and data
accessibility [22, 29, 9]. Data physicalization enables the creation of
tangible representations of data that can be perceived and interpreted through
tactile senses. By leveraging haptic perception, data physicalization can
enhance user experience and engagement with data. For example, a tactile map
representing geographic data can provide a unique and interactive experience
by enabling touch-based navigation and exploration of the data [4, 21, 35].
Moreover, data physicalization provides a medium through which individuals
with visual or auditory impairments can access information [25].

For data physicalization to be effective, however, we need a better
understanding of the unique affordances and strengths each perceptual modality
allows through different encoding types. To date, visualization researchers
acknowledge that we have yet to move beyond creating 3D versions of familiar
visual graphics even when creating them for accessibility [22, 25]. This is
problematic, as designing to privilege tactile senses may require
fundamentally different considerations than designing to privilege visual
senses, research that currently lacks empirical support [22]. For instance, it
must account for differences in tactile acuity, known to decline with age yet
retaining high sensitivity in individuals who are literate in Braille [43].
This underscores the need for foundational research in tactile perception for
data physicalization. To address the current gap in empirical support for data
physicalization design, we conducted a within-subjects study comparing tactile
and visual perception of six encoding types (position aligned, position non-
aligned, length, area, shading, and curvature) with 12 university students. We
find that the same order of perceptual accuracy of encoding types found by
Cleveland and McGill [5, 6] does not hold for tactile graphics.

##  1 Related Work

Below, we first describe Cleveland and McGill’s graphical perception study
(1986) and its subsequent influence on the field of data visualization before
covering tactile perception and data physicalization. Cleveland and McGill’s
study laid the groundwork for data visualization by identifying elementary
perceptual tasks to inform graphical design, subsequently inspiring studies
that expanded on their findings. As the field of data physicalization,
particularly tactile graphics, gains momentum with innovative methods, there
is a need for comprehensive and systematic research on tactile data encodings.

###  1.1 Graphical Perception Studies

Cleveland and McGill identified elementary perceptual tasks for data
visualization to inform graphical design. In their 1984 study, Cleveland and
McGill investigated the efficiency and accuracy of decoding numerical values
from visual, graphical representations [5]. Inspired by other scientific
fields and Julesz’s concept of “pre-attentive particles”, they established
axioms of data visualization [23, 5]. To do so, they identified 10 elementary
perceptual tasks they thought were employed either individually or multiply
when analyzing visual data representations such as stacked bar charts and area
plots [5]. In a 1986 follow-up experiment, they presented participants with
isolated stimuli designed around these 10 tasks to eliminate context or
potential confounding factors to uncover the perceptual accuracy for each
encoding type [6]. They then proposed a hierarchy of visual encoding types in
order from most to least accurate: position judgments both 1) along a common
scale and 2) non-aligned identical scales, 3) length, 4) direction 5) angle,
6) slope, 7) area, 8) volume, 9) shading, and 10) saturation judgments.
Cleveland and McGill’s systematic taxonomy of graphical perception had clear
implications for designing data visualization.

Cleveland and McGill’s work sparked a multitude of follow-up studies in data
visualization. By distilling target visual elements as “visual variables”,
they could be used to investigate accuracy and response time in visual tasks,
show how to identify other fundamental visualization elements like size,
shape, and color, or to compare different chart types [30, 32, 38, 40, 18, 28,
42, 34, 26]. More recently, Heer and Bostock replicated Cleveland and McGill’s
findings [5] to demonstrate the viability of crowdsourcing platforms for
graphical perception experiments [20]. Researchers have extended Cleveland and
McGill’s elementary tasks to examine how visual variables interact and combine
to uncover capacity limits in attention or perception [16, 41, 7, 27].

###  1.2 Data Physicalization and Tactile Perception

Data physicalization—a term recently created by researchers—is the process of
representing data through physical objects [22]. A recent focus in this field
has been finding alternatives to overcome cost and reproducibility limitations
[3, 11, 13]. For example, Brittel compared the impact of fabrication materials
like 3D printed plastic and microcapsule paper on the discriminability of map
symbols [3]. In addition to fabrication techniques, novel approaches such as
actuated displays, microrobots integrated into tablets and multimodal methods
combining haptics and audio are being explored [14, 12, 31, 33].

A systematic review of data physicalization from 2010 to 2020 shows that
tactile graphics remain the most commonly used production technique in the
decade[4]. When designing chart types for data physicalization, we continue to
heavily borrow established chart types from the field of visualization, such
as bar plots or line charts [15, 4, 14]. There is a critical need for
empirical evidence supporting how physical variables—a tactile equivalent of
visual variables—affect the efficiency of perceiving tactile graphs, similar
to the way Cleveland and McGill’s contribution to data visualization [22].

The sense of touch, or tactile perception, has been studied in various fields
from haptics to psychology and neuroscience. The tactile modality supports
perception of textures, shapes, and objects [24]. Lederman and Klatzky
proposed a tactile encoding hierarchy, focusing on unique physical aspects
like vibration, temperature, and roughness [24]. It is important to note that
Lederman and Klatzky’s ranking explores different properties compared to our
study, as we focused on elementary visual encodings and their comparable
tactile representations.

##  2 Study Design

We used a within-subjects design to compare perceptual accuracy across visual
and tactile perception using support vector graphics (SVGs) rendered in a web
browser (visual) and printed on swell paper with a Thermoform printer
(tactile). We also incorporated a tabular condition as sanity check (analysis
of which is included in Supplemental Materials). A key objective was to
explore to what extent prior findings on the accuracy of encoding types in
graphical perception hold for tactile perception. The study design is
summarized in Figure 1.

Participants. We recruited 12 (5 female and 7 male) undergraduate college
students through student listservs at a US University. Studies took place in a
lab setting and were designed to last 1-1.5 hours.

Comparing Tactile and Visual Perception. We employed a within-subjects study
design with perceptual condition (visual, tactile) and encoding type (position
aligned, position non-aligned, length, area, shading, and curvature) as our
independent variables. Their order was randomized and counterbalanced across
participants. Participants completed 36 trials for each condition, during
which they sat next to the experimenter so that the experimenter could advance
to the next trial and manually record their responses. To enable equivalent
data collection in both the visual and tactile conditions, participants were
asked to provide their answers verbally.

Encoding Type. Within each perceptual condition, we investigated differences
when judging proportional differences (ratios) in stimuli across six encoding
types: position aligned, position non-aligned, length, area, shading, and
curvature. These were from the original Cleveland and McGill study (see the
“Encoding Types” column in Figure 1) [5, 6]. It identified 6 ranks for 10
elementary perceptual tasks, and so, we chose only one from each of the tasks
tied in rank as follows. We selected length from {length, direction, angle} as
it is used in one of the most common chart types in early education (bar
charts) [2]; curvature from {volume, curvature} and shading from {shading,
color saturation} they can be more easily compared across the conditions
without favoring one over the other (furthermore, volume and color saturation
cannot be encoded using our chosen tactile modality). To collect multiple
responses per encoding type we tested ratios ranging from 25-75% in intervals
of 10 so that each encoding type was tested in 6 trials. For each trial, we
asked participants 2 questions in turn:

  1. 1.

Identify the smaller of each stimuli pair (Forced Choice)

  2. 2.

Estimate the proportion (between $0-100$) of the smaller stimulus relative to
the larger (Estimation)

Trial Blocks Trials within the tactile and visual conditions were completed as
6 blocks of 6 trials each for a total of 36 unique trials in each condition.
Within each block, each encoding type appeared exactly once, as did each
ratio. The order of the trials within each block and the order of the blocks
was randomized. See Figure 1.

Apparatus. To compare across visual and tactile perception, we created a set
of 36 SVG files for each of the 6 encoding types and 6 ratios (in equal 10%
intervals from 25% to 75% using our arbitrarily chosen base standard value).
Participants used the Microsoft Edge web browser on a Lenovo ThinkPad T14 Gen
3 with a 14” display for the visual condition. We printed the SVG files on
Swell Touch paper—a microcapsule paper—and used the American Thermoform
machine to raise their graphics for the tactile condition [1]. Participants
were blindfolded for this condition. Example images of the visual and tactile
conditions are displayed in Figure LABEL:fig:_stimuli. For each trial,
participants were presented with a pair of stimuli of the given type. In the
visual condition, page adornments were limited to two buttons suitable for
navigating within the trial block. In the tactile condition, each pair was
printed on the Swell Form paper without any adornments. We have uploaded the
files and code used in our study to our Github repository, which can be
accessed via the link: https://github.com/CAV-
Lab/vis_tactile_vis23_supplemental.

###  2.1 Procedure

The study lasted up to 1.5 hours and was video recorded. After providing
informed consent, participants completed 4 main activities: a survey,
condition 1, condition 2, and a semi-structured interview. First, participants
completed a survey covering basic demographic questions and their background
and experience with visual and tactile data representations. Next, they
completed each of 2 conditions (tactile, visual) in a randomly assigned order.
Each condition began with 6 practice tasks. During the practice session,
participants were introduced to each encoding type in the given modality and
the set of questions they would be asked to answer. During this time, the
experimenter answered any questions participants had regarding the encoding
types or the task instructions. Once participants completed all of the
practice trials and indicated their comfort with the tasks, they proceeded to
complete the 6 blocks of trials for that condition. Following the completion
of each condition, participants rated their perceived accuracy on a 7-point
Likert scale (1 = least accurate, 7 = most accurate) and cognitive load on a
10-point scale using a subset of the NASA TLX questionnaire[17]. Then,
participants completed the second condition followed by their ratings. Lastly,
participants completed a semi-structured interview covering their experience
with the trials, task strategies, and preferences for encoding types.

###  2.2 Hypotheses

Based on prior research, we hypothesize that we will be able to replicate
prior results from Cleveland and McGill in the visual condition, obtain novel
perceptual accuracy results for the tactile condition, and identify trade-offs
between visual and tactile stimuli types.

  * H1: Comparison Accuracy. Participants will compare data values more accurately using visual encodings than tactile encodings.

  * H2: Proportional Accuracy. Participants will perceive proportion differences more accurately when presented with visual encodings than tactile encodings.

  * H3: Encoding Type Order of Accuracy. Participants will perceive the proportional difference between visual encodings in the order of accuracy found in Cleveland and McGill’s experiments: position aligned, position non-aligned, length, area, curvature, shading [5]. However, this order will differ for comparable tactile encodings.

  * H4: Cognitive Load. Participants will experience higher cognitive load for the tactile condition than the visual condition.

  * H5: Perceived Accuracy. Participants will perceive their performance to be better for visual encodings than tactile encodings.

###  2.3 Measures

  * H1: Comparison Accuracy. We counted the number of correct answers (to the question identify the smaller of each stimuli pair) for the tactile and visual condition respectively. The count for each condition was then compared.

  * H2: Proportional Accuracy. We adopted the same methodology in Cleveland and McGill’s study [6]. For each response, we computed the absolute error (AE) within each conditions using: $AE=|RespondedProportion-TrueProportion|$, where RespondedProportion represents the participant’s response for the proportion judgment, and TrueProportion refers to the actual encoded ratio for the corresponding trial.

  * H3: Encoding Type Order of Accuracy. We evaluated and compared the Mean Absolute Error (MAE) of each encoding type type within conditions.

  * H4: Cognitive Load. We adapted the NASA TLX questions to measure participants’ cognitive load during the tasks. Participants were asked to rate each modality according to the level of mental demand (“simple vs. demanding”) and frustration (“relaxed vs. stressed”) on a scale from 1 to 10 (1 = low, 10 = high). Questions on physical demand, temporal demand, and effort were excluded since our study did not involve any physically demanding tasks, and participants were free to choose their own pace and take breaks.

  * H5: Perceived Accuracy. We asked participants to rate how accurate they thought they were when completing the tasks in each condition using a 7-point Likert scale (1 = least accurate, 7 = most accurate).

##  3 Results

We found that the accuracy of perceiving proportional differences from
elementary visual encodings, when presented in their tactile forms deviates
from the established visual hierarchy. Before running any statistical test to
assess H2 and H3, we cleaned the data by removing outliers which we defined as
>3 SD.

H1: Comparison Accuracy. Participants were slightly more accurate when
completing tasks using visual encodings (3/432 incorrect) than when using
tactile encodings (10/432 incorrect). However, a Matched Pairs Signed test
revealed the encoding type did not have a significant effect on task accuracy
(M = -3.500, p = 0.092). Our results show that there is no significant
difference in accuracy when participants identify the smaller stimulus in the
pair.

H2: Proportional Accuracy. The MAE for the tactile condition was 12.60%
($SD=10.21\%$, $range=0-50\%$), and the MAE for the visual condition was
10.25% ($SD=8.65\%$, $range=0-50\%$). Because the data were not normally
distributed, we applied Aligned Rank Transform (ART) to the cleaned data, then
performed Repeated Measures Analysis of Variance (RMANOVA) for the visual and
tactile conditions, revealing a significant difference between them
($F(1,836)=4.651$, $p=0.031$). In our contrast test, we used Holm-adjusted
$p$-values, and the result between the tactile and visual conditions was
$t(836)=2.146$, $p=0.032$. Our results show that there is a significant
difference in participants’ perceptual accuracy between the visual and tactile
condition.

H3: Encoding Type Order of Accuracy. Analyses performed on the cleaned data
using ART and RMANOVA revealed a significant main effect for encoding type
($F(5,836)=24.504$, $p<.001$). Figure 2 presents the MAE for each encoding
type and modality, with the legend showing Cleveland and McGill’s ranking for
comparison [5, 6]. We observe a clear difference in the order for shading
(light blue) and curvature (purple); whereas Cleveland and McGill posit
curvature to be more accurate than shading [5], we observe the opposite.
Nonetheless, these two encoding types remain amongst the lower half of
encoding types by perceptual accuracy. At the top of the ranking, Cleveland
and McGill’s findings indicate that the three highest accuracy encoding types
are, from most to least accurate, position aligned, position non-aligned, and
length [5]; our results, while all relatively close in terms of MAE, flip the
second and third most accurate encodings.

Figure 2 further demonstrates that the perceptual accuracy hierarchy for
tactile encodings (left) varies compared to the visual hierarchy (right). In
the tactile condition, “length” emerged as the encoding with the lowest error
in participants’ judgments. Interestingly, “position along a common scale”
demonstrated the lowest error in the visual condition but ranked in the lower
half for the tactile condition. “Area,” classified in the lower half of the
visual hierarchy, ranks as the third-best encoding in the tactile hierarchy,
although the MAE is only marginally different between tactile and visual.
However, in our subsequent contrast tests using the Holm-adjustment, we
compared each encoding type with all others within both visual and tactile
conditions. “Curvature” was the only encoding that showed a significant
difference from five other encoding types in the visual condition and from
four types in the tactile condition. Only “position-aligned” showcased a
significant difference between the visual and tactile conditions. In summary,
we observe notable differences in our visual hierarchy compared to that
obtained by Cleveland and McGill, and, although we observed differences
between tactile and visual rankings qualitatively, our findings didn’t
indicate any significant distinction between the tactile and visual
hierarchies.

Figure 2: MAE of the six encoding types for tactile (left) and visual (right).
Lower MAE corresponds to higher accuracy. In the legend, we present the visual
encoding hierarchy hypothesized by Cleveland and McGill [5, 6].

H4: Cognitive Load. Using a scale ranging from $1-10$ (1 = low, 10 = high
cognitive load), we asked participants to rate each modality according to the
level of mental demand (“simple vs. demanding”) and frustration (“relaxed vs.
stressed”). Participants rated tactile as requiring higher mental demand than
visual: tactile received an average score of 6.50 / 10 ($SD=2.39$,
$range=2-8$) and visual received an average 5.50 / 10 ($SD=2.32$,
$range=2-9$), although the difference was not statistically significant (a
Wilcoxon test indicated $V=35$ and $p=0.1522$). Participants rated tactile as
more frustrating than visual: tactile received an average score of 4.33 / 10
($SD=2.19$, $range=2-9$) and visual received an average 3.75 / 10 ($SD=1.60$,
$range=1-7$); however, the difference was likewise not statistically
significant (a Wilcoxon test indicated $V=31$ and $p=0.3346$). While
qualitative data suggests that the tactile condition leads to greater mental
demand and frustration, we do not observe a significant difference in
cognitive load between the two modalities.

Figure 3: A side-by-side comparison of the visual and tactile modalities,
where each encoding type is distinguished by color. The x-axis represents the
true proportion, and the y-axis displays the absolute error (%). The dots
represent MAE associated with each true proportion. Curves are fitted using a
polynomial of degree 2.

H5: Perceived Accuracy. On a scale of 1 = least accurate to 7= most accurate,
participants perceived the tactile condition as less accurate than the visual
condition: tactile received an average rating of 3.25 / 7 ($SD=0.75$,
$range=2-4$) and visual received an average of 4.5 / 7 ($SD=1.09$,
$range=2-6$). A Wilcoxon signed-rank test was then performed
($V=7.5,p=0.02297$), suggesting that there is a significant difference in how
participants rated their performance between the two modalities, with higher
perceived accuracy in the visual condition.

##  4 Discussion & Conclusion

In Figure 3, we observe most encoding types had the highest error in the
50-60% range of true percentage difference with lowest error at the extreme
ends of true percentage, consistent with Cleveland and McGill [5, 6]. Encoding
types that were most accurate for visual (position aligned, position non-
aligned, and length) tended to have relatively flat fitted curves, suggesting
minimal changes in error as a function of true percentage. These same
encodings for tactile (left) resulted in much higher error rates, however.
Some encoding types can interestingly be observed to increase in error as a
function of true percentage: position aligned in tactile (left) and shading in
visual (right). When we consider raw error compared to absolute error, we
further observe a systematic error of overestimation for some encoding types
such as curvature and area in the visual condition, while we see
underestimation for others such as shading in the visual condition and
position aligned, length, and shading in the tactile condition (see
Supplemental Materials). These observations, in conjunction with different
observed rankings of perceptual accuracy, provide empirical support to prior
claims that a one-to-one translation of visual encodings into a tactile form
would be insufficient [25] and suggest the need for new guidance for tactile
graphics.

Limitations and Future Work. A primary limitation of our study stems from our
participant group, which consisted of a small sample of 12 college students
for the in-lab experiment. This demographic and sample size likely does not
encompass the full range of tactile acuity [43]. As the field holds promise
for the accessibility community, we aim to include participants with blindness
or low vision in future studies to understand if differences may arise in the
rankings of tactile encodings when individuals have different levels of acuity
with tactile encodings, including experience with Braille.

Conclusion.  In this paper, we compared perceptual accuracy for visual and
tactile modalities. Our findings indicate that participants have better
overall accuracy for visual modality than tactile. While we observed marginal
differences in the order of accuracy of perception of visual encoding types
(position aligned, position non-aligned, length, area, shading, curvature)
found by Cleveland and McGill, we found that this ranking deviated for tactile
encodings (length, position non-aligned, area, position aligned, shading,
curvature) [5, 6]. This result highlights the current empirical gap in guiding
the design of charts and graphs, and underscores the need for further
foundational research in tactile data visualization.

###### Acknowledgements.

The authors wish to thank Matt Kay for his generous supply of clever paper
titles.

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# Spontaneous Mutations from Terahertz Proton Tunneling

Noah Bray-Ali  Email address<EMAIL_ADDRESS>Science Synergy
Los Angeles, CA 90045

###### Abstract

Protons in the gap between base pairs of the double helix store the code of
life by breaking the chiral symmetry that swaps the sense strand with its
complementary partner. When these hydrogen bonds break during replication and
transcription, pairs of protons switch sides restoring chiral symmetry and
destroying genetic information. The observed rate of such spontaneous
mutations follows in the sudden approximation for bond breaking provided
protons in bonds between bases tunnel across the gap with terahertz
frequencies.

##  I Introduction

Looking down the helical axis of deoxyribonucleic acid (DNA), we find a
winding stack of pairs of nucleotide bases linking the two strands of its
double helix structure [1]. Remarkably, the molecule has a chiral symmetry
axis passing through the gap between base pairs and running roughly from the
minor groove of the double helix to its major groove: Half a turn about the
chiral axis swaps the two helices. Nevertheless, the code of life breaks this
chiral symmetry and stores genes in the sequence of bases along just one
helix: the sense strand.

Figure 1: Pairing arrangements of adenine before (above) and after (below) it
has undergone tautomeric shift. The major groove of the double helix lies
above the base pairs while the minor groove lies below them. We are looking
down the axis of the helix. The chiral symmetry axis runs from the minor to
the major groove along a line roughly half way between the bases in each pair.
After the shift, the bases pair using the same atom (N for nitrogen) on either
side of the bond closest to the major groove, restoring the chiral symmetry
which interchanges the sense strand and its complement. (Adapted with
permission from Ref. [3]).

Spontaneous mutations restore the chiral symmetry of the double helix and let
genes change by shifting the tautomeric form of bases (Fig. 1). Following such
a shift, hydrogen bonds form between identical atoms. In wild-type bonds, by
contrast, oxygen provides the lone pair of electrons and nitrogen accepts the
proton in the hydrogen bond nearest the major groove of the double helix. In
mutant bonds, the pair of protons tunnels back and forth coherently across the
gap like the pair of electrons in the covalent chemical bond [2]. And just as
the covalent bond is spin symmetric, so also the mutant hydrogen bonds are
chiral symmetric: When we split the mutant pair, there is an equal chance of
finding either tautomeric form on a given base provided only that the other
base has the complementary form. The genetic information is lost [3].

The chance of spontaneous mutation $P=2.0\times 10^{-10}$ gives roughly one
tautomeric shift for every few billion base pairs [4]. In the sudden
approximation, the wild-type proton pair wavefunction for a base pair between
adenine $A$ and thymine $T$ takes the form [5, 6]

$\left|W\right\rangle=\left|AT\right\rangle+\epsilon\left|A^{*}T^{*}\right\rangle,$
(1)

where, $\left|AT\right\rangle$ is the dominant tautomeric form,
$\left|A^{*}T^{*}\right\rangle$ has tautomeric shifts in both bases, and
$\epsilon=\sqrt{2P}=2.0\times 10^{-5}$ is the probability amplitude for this
to take place.

During replication and transcription, the base pair splits quickly compared to
the time-scale for tautomeric shifts. From the wild-type wavefunction in Eq.
(1), the chance of finding the shifted form is then simply $\epsilon^{2}=2P$
the square of the probability amplitude for the shift. This is twice the
chance of spontaneous mutation $P$, since the shifted adenine $A^{*}$ has
equal chance of being found in either tautomeric form once it binds to wild-
type cytosine $C$. Chiral symmetry then gives the mutant proton pair
wavefunction

$\left|M\right\rangle=\frac{1}{\sqrt{2}}\left|A^{*}C\right\rangle+\frac{1}{\sqrt{2}}\left|AC^{*}\right\rangle.$
(2)

The equal probability amplitude for the two tautomeric forms in the mutant
wavefunction follows from the fact that both bases use nitrogen atoms to make
hydrogen bonds.

In this Article, we show that terahertz frequency proton tunneling between
base pairs creates the wild-type and mutant proton pair wavefunctions
responsible for spontaneous mutation. The Methods section shows how to express
the wild-type tautomeric shift probability amplitude
$\epsilon=2t^{2}/(U\Delta)$ in terms of the energy-scale $t=h\times 1.7~{}{\rm
THz}=7.0\times 10^{-3}~{}{\rm eV}$ for this proton tunneling, the energy
splitting between tautomers $\Delta=h\times 24~{}{\rm THz}=0.10~{}{\rm eV}$,
and the charge transfer energy $U=h\times 1200~{}{\rm THz}=4.9~{}{\rm eV}$ for
putting both protons on the same base [7]. The Results section argues that the
stability of the gene follows from the emergence of the radio-frequency scale
$J=2t^{2}/U=h\times 480~{}{\rm MHz}$ for the protons to swap sides. The
slowness of the corresponding proton swap time $h/J=2.1\times 10^{6}~{}{\rm
fs}$ on the femto-second time-scale $h/\Delta=40~{}{\rm fs}$ set by the
tautomeric energy splitting ensures the wild-type tautomeric shift has small
probability amplitude $\epsilon=J/\Delta=2.0\times 10^{-5}$. Conversely, the
bonds between bases break during replication and transcription on the femto-
second time-scales corresponding to changes in electronic structure. Charge
transfer $h/U=0.83~{}{\rm fs}$ gives a roughly similar time-scale. This leaves
little time for the protons in the wild-type wavefunction to adjust to the
changes. Instead, we get the chance for spontaneous mutation
$P=\epsilon^{2}/2=2.0\times 10^{-10}$ in the sudden approximation to bond
breaking. We end with Discussion of the meaning of these results for evolution
and for aging [8].

##  II Methods

The wild-type proton pair wavefunction $\left|W\right\rangle$ is the lowest
energy state of the effective Hamiltonian [5, 9]

$\displaystyle H_{\rm eff}=$ $\displaystyle-$
$\displaystyle\frac{\Delta}{2}\left(\left|AT\right\rangle\left\langle
AT\right|-\left|A^{*}T^{*}\right\rangle\left\langle A^{*}T^{*}\right|\right)$
(3) $\displaystyle-$ $\displaystyle J\left(\left|AT\right\rangle\left\langle
A^{*}T^{*}\right|+\left|A^{*}T^{*}\right\rangle\left\langle AT\right|\right),$

where, the first term splits the tautomers and the second term swaps the
protons. The swap energy $J=2\times 10^{-6}~{}{\rm eV}$ is small compared to
the tautomeric splitting $\Delta=0.10~{}{\rm eV}$. This means we can find the
wild-type wavefunction
$\left|W\right\rangle=\left|AT\right\rangle+\epsilon\left|A^{*}T^{*}\right\rangle$
by time-independent first-order perturbation theory.

Dropping terms beyond first-order in the small quantities $J/\Delta$ and
$\epsilon$, we simply act on $\left|W\right\rangle$ with the effective
Hamiltonian $H_{\rm
eff}\left|W\right\rangle=-(\Delta/2)\left|AT\right\rangle+(\epsilon\Delta/2-J)\left|A^{*}T^{*}\right\rangle$.
We then match this with the product
$-(\Delta/2)\left|W\right\rangle=-(\Delta/2)\left|AT\right\rangle-(\epsilon\Delta/2)\left|A^{*}T^{*}\right\rangle$.
Physically, the matching means the energy of the proton pairs does not change
at first-order in $J/\Delta$ from the value it would have without proton
swaps. Carrying out the matching, we find that the probability amplitude for
proton swap $\epsilon=J/\Delta=2\times 10^{-5}$ is small enough that we can
simply stop at first-order in perturbation theory for the wild-type
wavefunction.

Terahertz proton tunneling generates the effective Hamiltonian $H_{\rm eff}$
at second-order in time-independent perturbation theory from the full
Hamiltonian [10]

$H=\begin{pmatrix}-\Delta/2&0&-t&-t\\\ 0&\Delta/2&-t&-t\\\ -t&-t&U&0\\\
-t&-t&0&U\end{pmatrix}$ (4)

where, the matrix acts on the column vector $(a,b,c,d)^{T}$ for the proton
pair wave function
$a|AT\rangle+b|A^{*}T^{*}\rangle+c|A^{+}T^{-}\rangle+d|A^{-}T^{+}\rangle$. The
zwitterion states $|A^{+}T^{-}\rangle$ and $|A^{-}T^{+}\rangle$ have charge
transfer energy $U=4.9~{}{\rm eV}$ compared to the tautomers $|AT\rangle$ and
$|A^{*}T^{*}\rangle$. They arise from the tautomers when protons hop. The base
on which the proton lands has positive charge from the extra proton. At the
same time, the base from which the proton came is left with negative charge
from the extra lone pair of electrons. The small size of the tunneling energy
$t=U/700=7\times 10^{-3}~{}{\rm eV}$ compared to the charge transfer energy
means we can effectively eliminate the high-energy zwitterion states from the
low-energy tautomeric physics through a simple change of basis.

In the language of quantum optics, the zwitterions have a “bright” state
$|+\rangle$ that mixes with the tautomers and a “dark” state $|-\rangle$ that
does not mix with them [11]. The dark state
$|-\rangle=(|A^{+}T^{-}\rangle-|A^{-}T^{+}\rangle)/\sqrt{2}$ has equal
probability to land in either zwitterion but opposite probability amplitude.
It retains the energy $U$ that it would have without proton tunneling.
Destructive interference from the two zwitterions decouples the dark state
from the tautomers.

Meanwhile, the zwitterion bright state
$|+\rangle=-(\sqrt{2}t/U)(|AT\rangle+|A^{*}T^{*}\rangle)+(|A^{+}T^{-}\rangle+|A^{-}T^{+}\rangle)/\sqrt{2}$
has equal probability amplitude for both zwitterions. It lowers its energy by
$-2J$ through constructive interference, where, $J=2t^{2}/U=h\times 480~{}{\rm
MHz}$ is the radio-frequency energy scale for proton swaps. This new scale
emerges from the terahertz-frequency $t=h\times 1.7~{}{\rm THz}$ for proton
tunneling when we change basis to the zwitterion bright and dark states.

Coming back to the tautomers, the dominant and rare forms mix with the
zwitterion states to make a new pair of tautomers whose dynamics are governed
by the effective Hamiltonian in Eq. (3). The role of the dominant tautomer in
the effective Hamiltonian is played by the state
$|AT\rangle+(t/U)\left(|A^{+}T^{-}\rangle+|A^{-}T^{+}\rangle\right)$. It is
easy to check that this state has no overlap with the bright and dark
zwitterion states [12]. Similarly, for the rare tautomer, we must take
$|A^{*}T^{*}\rangle+(t/U)\left(|A^{+}T^{-}\rangle+|A^{-}T^{+}\rangle\right)$
to ensure it too has no overlap with the states of the new zwitterion basis.

With this simple change of basis, the zwitterion bright and dark states split
off from the low frequency dynamics of the tautomers relevant for spontaneous
mutation. The full Hamiltonian simplifies to the new matrix [13]

$V^{T}HV=\begin{pmatrix}-\Delta/2&J&0&0\\\ J&\Delta/2&0&0\\\ 0&0&U-2J&0\\\
0&0&0&U\end{pmatrix}$ (5)

where, $V$ (and $V^{T})$ are the matrices whose columns (rows) are the vectors
that give the wavefunctions for the new basis states in terms of the original
basis. The matrix now acts on column vectors that give the proton pair
wavefunction in the new basis of states. The new basis is simply the new
dominant tautomer, the new rare tautomer, the bright zwitterion, and the dark
zwitterion.

The effective Hamiltonian from Eq. (3) appears in the upper left corner and
creates proton swaps that exchange the new tautomer basis states. Meanwhile,
the zwitterion bright and dark states are stationary states of the full
Hamiltonian with energies $U-2J$ and $U$, respectively. The change of basis
and the new form for the full Hamiltonian in that basis are exact up to and
including terms of order $(t/U)^{2}=2.0\times 10^{-6}$. The energy $t=U/700$
gained by terahertz proton tunneling between base pairs is small compared to
the ultraviolet-scale energy $U=4.9~{}{\rm eV}$ needed to excite electrons in
the resonating rings of the nucleotide bases and accommodate the charge
transfer. This means we can simply stop at second-order in the time-
independent perturbation theory for the effective Hamiltonian of the
tautomeric shifts that generate spontaneous mutation [14].

##  III Results

The chance of spontaneous mutation $P=(J/\Delta)^{2}/2=2.0\times 10^{-10}$
emerges from the wild-type proton pair wavefunction $|W\rangle$ of nucleotide
base pairs in DNA given in Eq. (1). The wavefunction has amplitude $J/\Delta$
to swap protons between the bases, where, $J=2t^{2}/U=h\times 480~{}{\rm MHz}$
is the microwave energy scale that emerges from terahertz frequency $t=h\times
1.7~{}{\rm THz}$ proton tunneling and the ultraviolet energy $U=4.9~{}{\rm
eV}$ needed to accomodate the charge transfer. When bonds between bases break
during replication and transcription, the femto-second time scale for bond-
breaking is fast compared to the pico-seconds needed for protons to tunnel
between bases. In the sudden approximation, the protons land in the rare
tautomer with probability $(J/\Delta)^{2}\approx 3\times 10^{-10}$ given by
the square of the probability amplitude for the proton swap within the wild-
type proton pair wavefunction.

The tautomeric shift induced by bond-breaking forces the new base pairs that
form to have the mutant wavefunction $|M\rangle$ given in Eq. (2). It has
equal amplitude for a given base to be found in either tautomeric form. This
restores the chiral symmetry between the two strands of the double helix and
destroys the genetic information that was stored within the old base pair.

The physical picture for the proton swap proceeds in two coherent steps.
First, the proton closer to the major groove, say, tunnels across the gap
between bases. This costs charge transfer energy but is not forbidden by any
symmetry principle. Next, the other proton, located closer to the minor
groove, tunnels across the gap to gain back the charge transfer energy. At the
end of the coherent second-order proccess, the protons have swapped.

This physical picture for spontaneous mutation coincides with that of “super-
exchange” in quantum magnetism [15, 16]. In place of proton pairs, consider a
pair of electrons in the partially filled $d$-shell ($f$-shell) of two
transition-metal (rare-earth) ions separated by a closed shell ligand such as
fluorine or oxygen in an electrically insulating, transparent salt such as
manganese fluoride (${\rm MnF_{2}}$). Instead of the location relative to the
groove (major or minor) used by the proton pair, the electrons use their spin
state, up or down. The difference in spin permits a pair of electrons to sit
in the same orbital on one or the other of the two neighboring transition
metal ions. This costs a large charge transfer energy but is not forbidden by
any symmetry principle. Meanwhile, the spin of the electron that hops back
need not match the spin of the one that hops first. In this way, the spin of
the electron on neighboring sites can exchange on an energy scale that exceeds
the thermal energy available at room temperature [17].

##  IV Discussion

Terahertz proton tunneling in the hydrogen bonds between nucleotide base pairs
within DNA generates radio-frequency proton pair swaps that drive spontaneous
mutation. Yet both processes are slow compared to the femto-second scale bond-
breaking during replication and transcription. The sudden splitting makes the
pair choose at random which tautomer to take: the low-energy form found in
free nucleotides within the cytoplasm or the high-energy form that generates
mutations. Remarkably, the chance they land in the rare tautomer gives roughly
twice the observed spontaneous mutation rate. The factor of two reflects the
restoration of the chiral symmetry between the two strands of the double helix
structure when the rare tautomer binds to a free nucleotide.

The spontaneous mutation rate acts as a molecular clock driving evolution.
Each generation must replicate its genome to pass along genetic information to
its descendants. However, the molecular process of replication itself
generates changes in the sequence of bases within DNA. These spontaneous
mutations restore the symmetry between the sense strand and its complementary
partner. They destroy genetic information at random within the genome at a
regular rate. The spontaneous mutation rate can be expressed in terms of the
energy splitting between tautomers and the emergent energy scale for proton
pair swap between nucleotides.

Spontaneous mutation also sets the life span of organisms by making cells age
[8]. Each time the cell uses the code of life to live its life, it must break
bonds between base pairs to read the base sequence on the sense strand. Like
replication, the transcription process itself generates spontaneous mutations.
The sudden splitting of the strands makes base pairs choose at random which
tautomer to take.

During transcription, when a base pair in DNA lands in the rare form, the base
on the sense strand then must form a mutant bond with the “wrong” base in the
complementary messenger ribonucleic (mRNA) molecule. This mutant bond then
breaks to release the single-stranded mRNA. By the chiral symmetry created
within the mutant bond, though, the base on the sense strand of the DNA has an
even chance of coming back as either the dominant or the rare tautomer.

To complete transcription, the sense strand base returns to bind its partner.
If it returns in the dominant form, however, the bases do not bind: Both bases
now have their protons close to the same groove (major or minor) of the double
helix and the electrostatic repulsion between protons keeps the bases apart.
To bind the strands, the transcription machinery must remove one of the bases
and replace it with a free nucleotide in the dominant form.

Since the sense strand makes and breaks twice as many bonds as its partner
does, the machinery prefers to use the complementary base as its template for
repair. This locks in the spontaneous mutation as a mutant bond with chiral
symmetry between the two strands and complete loss of genetic information. In
this way, the act of using the code of life to live in fact destroys the code
itself. The rate of loss of genetic information due to transcription is set by
the same balance of energy scales that sets the spontaneous mutation rate. The
balance sets the pace of aging and fixes the life span of a given generation.

By way of conclusion, it is worth recalling the key steps. The analysis began
by recognizing that protons tunnel with terahertz frequency between the
nucleotides in the base pairs that bind the double helix structure of DNA and
that store the code of life. The proton that hops back across the gap,
however, need not be the same one that hopped first. The resulting proton swap
between base pairs leaves both bases in the pair in a rare, high-energy
tautomer form. The emergent radio-frequency proton pair swap energy-scale
competes with the tautomer energy splitting to set the spontaneous mutation
rate. During replication and transcription, the hydrogen bonds between base
pairs break fast and force the base pair to choose which form to take. This
random choice drives the genetic drift between generations and leads to
evolution. It also leads to the aging of organisms and sets the length of each
generation.

## Acknowledgements

This research was supported in part by the National Science Foundation under
Grant No. NSF PHY-1748958, by the Department of Energy under grant No. DE-
FG02- 00ER41132, and by the Mainz Institute of Theoretical Physics within the
Cluster of Excellence PRISMA+ (Project ID 39083149).

## References

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  * [2] For the covalent chemical bond, see L. Pauling, “The Shared-Electron Chemical Bond,” Proc. Natl. Acad. Sci, 14, 359 (1928). In the analogy, the tautomeric form of the nucleotide base plays the role of the spin of the electron. Just as the two spin states (up and down) distinguish the two electrons within the covalent bond, so also the size of the nearest groove (major and minor) distinguish the two protons binding the base pair.
  * [3] J. D. Watson and F. H. C. Crick, “The Structure of DNA,” Cold Spring Harb. Symp. Quant. Biol. 18, 123 (1953).
  * [4] J. W. Drake, “Comparative rate of spontaneous mutations,” Nature 221, 1132 (1960). We take the rate $P=2.0\times 10^{-10}$ of spontaneous mutations per base pair per replication reported for both Escerchia coli and Salmonella typhimurium.
  * [5] For the source of the notation, see P. A. M. Dirac “A new notation for quantum mechanics,” Proc. Camb. Phil. Soc. 35, 416 (1939).
  * [6] For a statement of the sudden approximation, see the Editor’s remark on the second to last page of M. Born “Zur Quantenmechanik der Stossvorgange,” Z. Physik 37, 863 (1926) [English translation “On the quantum mechanics of collision processes” by D. H. Delphenich].
  * [7] The charge transfer energy $U=hc/(254~{}{\rm nm})=4.9~{}{\rm eV}$ gives the peak in the absorbance spectrum of DNA in ultraviolet light with wavelength $254~{}{\rm nm}$, where, $c$ is the speed of light. The splitting between tautomers is $\Delta=0.10~{}{\rm eV}$. The constant $h=4.14~{}{\rm eV/THz}$ expresses the equivalence between energy $E=hf$ and frequency $f$. For a discussion of this equivalence in the context of the energy of light with different frequencies, see A. Einstein, “Uber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristichen Gesichtspunkt,” Ann. der Phys. 322, 132 (1905) [English Translation: “On a Heuristic Viewpoint Concerning the Production and Transformation of Light” by D. ter Haar]. For a derivation of the equivalence based on the analogy between the phase of the wavefunction and the action of particle motion, see section 3 of the paper by E. Schrodinger in [9].
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  * [10] The $N\times N$ matrix of numbers $(M_{ij})$, with $i=1,\ldots,N$ running over the rows and $j=1,\dots,N$ indexing the columns ($N=4$ in our case), acts “linearly” on the column vector $v=(v_{1},v_{2},\ldots,v_{N})^{T}$ describing the wavefunction. This generates a new column vector $Mv=(M_{11}v_{1}+M_{12}v_{2}+\ldots+M_{1N}v_{N},M_{21}v_{1}+M_{22}v_{2}+\ldots+M_{2N}v_{N},\ldots,M_{N1}v_{1}+M_{N2}v_{2}+\ldots+M_{NN}v_{N})^{T}$. The linearity of the Hamiltonian means that the “superposition” $av+bw=(av_{1}+bw_{1},av_{2}+bw_{2},\ldots,av_{N}+bw_{N})$ of two wavefunctions described by column vectors $v$ and $w$ with the same energy $E$ will also be a stationary state with energy $E$ for any numbers $a$ and $b$. In fact, the energy is simply the number one gets $Hv=Ev$ when acting with the Hamiltonian on a stationary state.
  * [11] E. Arimondo and G. Orriols, “Non-absorbing atomic coherences by coherent 2-photon transitions in a 3-level optical pumping,” Lett. Nuovo Cimento 17, 333–338 (1976).
  * [12] The overlap $\langle\psi|\phi\rangle=a_{1}^{*}b_{1}+a_{2}^{*}b_{2}+\ldots a_{N}^{*}b_{N}$ between state $|\psi\rangle$ with column vector $(a_{1},a_{2},\ldots,a_{N})^{T}$ and state $|\phi\rangle$ with column vector $(b_{1},b_{2},\ldots,b_{N})^{T}$ expresses the probability amplitude for finding state $|\phi\rangle$ in the state $|\psi\rangle$. Here, the complex conjugation $(x+iy)^{*}=x-iy$ exchanges the unit imaginary numbers $\pm i$ while leaving fixed the real numbers $x$ and $y$.
  * [13] The matrix product $MV=(Mv_{1},Mv_{2},\ldots,Mv_{N})$ of matrix $M$ and matrix $V=(v_{1},v_{2},\ldots,v_{N})$ is simply the matrix whose column vectors $Mv_{1},Mv_{2},\ldots,Mv_{N}$ come from acting with matrix $M$ on the column vectors $v_{1},v_{2},\ldots v_{N}$ of matrix $V$. The change of basis matrix $V$ in Eq. (5) takes the form  $V=\begin{pmatrix}1&0&-\sqrt{2}t/U&0\\\ 0&1&-\sqrt{2}t/U&0\\\ t/U&t/U&1/\sqrt{2}&1/\sqrt{2}\\\ t/U&t/U&1/\sqrt{2}&-1/\sqrt{2}\end{pmatrix}$ (6) where, the columns of $V$ are the column vectors that give the states of the new basis (dominant tautomer, rare tautomer, bright zwitterion, dark zwitterion) in terms of the old basis ($|AT\rangle,|A^{*}T^{*}\rangle,|A^{+}T^{-}\rangle,|A^{-}T^{+}\rangle)$.
  * [14] The mixture of the zwitterion bright state into both the rare and dominant tautomer states means that the dominant tautomer is far more likely to land in a zwitterion than in the rare tautomer when the strands of the double helix split during replication and transcription: The zwitterion probability is $2\times(t/U)=2.9\times 10^{-3}$ in the sudden approximation while the tautomeric shift occurs with probability $(J/\Delta)^{2}=2\times(t/U)\times(t/\Delta)=4.0\times 10^{-10}$ that is much smaller in the same approximation. However, DNA is an acid in aqueous solution within the living cell and there is no shortage of protons and hydroxyl (${\rm OH^{-}}$) ions at normal pH to restore the zwitterion bases in the split strands to their dominant tautomeric forms. Furthermore, this relaxation is fast compared to the time it takes for new bases to come into the “fork” where replication and transcription take place. By the time the new bases arrive on the scene, the bases in the split pair present the dominant tautomeric form. By contrast, the rare tautomer on the split strand is as stable as the dominant tautomer: Both are protected by the large charge transfer energy $U=4.9~{}{\rm eV}$ needed to add or remove a proton from the neutral tautomeric state.
  * [15] H. A. Kramers, “L’interaction Entre les Atoms Magnétogènes dans un Cristal Paramagnétique,” Physica 1, 182 (1934) [English translation: “The Interaction between Magnetic Atoms in a Paramagnetic Crystal”].
  * [16] P. W. Anderson, “Antiferromagnetism. Theory of Superexchange Interaction,” Phys. Rev. 79, 350 (1950).
  * [17] The insulating parent compounds of the high-transition temperature cuprate superconductors have typical super-exchange energy $J=0.13~{}{\rm eV}=1500~{}{\rm K}/k$, where, $k=0.861~{}{\rm eV/K}$ is the Boltzmann constant. For comparison, room temperature is equivalent to energy $k\times 300~{}{\rm K}=0.026~{}{\rm eV}$ nearly fifty times smaller.