Predictability of evolutionary trajectories in fitness landscapes

Alexander E. Lobkovsky1, Yuri I. Wolf1, Eugene V. Koonin1,∗

1 National Center for Biotechnology Information, National Library of Medicine,
National Institutes of Health, Bethesda, MD 20894

$\ast$ E-mail<EMAIL_ADDRESS>
## Abstract

Experimental studies on enzyme evolution show that only a small fraction of
all possible mutation trajectories are accessible to evolution. However, these
experiments deal with individual enzymes and explore a tiny part of the
fitness landscape. We report an exhaustive analysis of fitness landscapes
constructed with an off-lattice model of protein folding where fitness is
equated with robustness to misfolding. This model mimics the essential
features of the interactions between amino acids, is consistent with the key
paradigms of protein folding and reproduces the universal distribution of
evolutionary rates among orthologous proteins. We introduce mean path
divergence as a quantitative measure of the degree to which the starting and
ending points determine the path of evolution in fitness landscapes. Global
measures of landscape roughness are good predictors of path divergence in all
studied landscapes: the mean path divergence is greater in smooth landscapes
than in rough ones. The model-derived and experimental landscapes are
significantly smoother than random landscapes and resemble additive landscapes
perturbed with moderate amounts of noise; thus, these landscapes are
substantially robust to mutation. The model landscapes show a deficit of
suboptimal peaks even compared with noisy additive landscapes with similar
overall roughness. We suggest that smoothness and the substantial deficit of
peaks in the fitness landscapes of protein evolution are fundamental
consequences of the physics of protein folding.

## Author Summary

Is evolution deterministic, hence predictable, or stochastic, that is
unpredictable? What would happen if one could “replay the tape of evolution:”
will the outcomes of evolution be completely different or evolution is so
constrained that history will be repeated? Arguably, these questions are among
the most intriguing and most difficult in evolutionary biology. In other
words, the predictability of evolution depends on the fraction of the
trajectories on fitness landscapes that are accessible for evolutionary
exploration. Because direct experimental investigation of fitness landscapes
is technically challenging, the available studies only explore a minuscule
portion of the landscape for individual enzymes. We therefore sought to
investigate the topography of fitness landscapes within the framework of a
previously developed model of protein folding and evolution where fitness is
equated with robustness to misfolding. We show that model-derived and
experimental landscapes are significantly smoother than random landscapes and
resemble moderately perturbed additive landscapes; thus, these landscapes are
substantially robust to mutation. The model landscapes show a deficit of
suboptimal peaks even compared with noisy additive landscapes with similar
overall roughness. Thus, the smoothness and substantial deficit of peaks in
fitness landscapes of protein evolution could be fundamental consequences of
the physics of protein folding.

## Introduction

One of the most intriguing questions in evolutionary biology is: to what
extent evolution is deterministic and to what extent it is stochastic and
hence unpredictable? In other words, what happens if “the tape of evolution is
replayed:” are we going to see completely different outcomes or the
constraints are so strong that history will be repeated [1, 2, 3, 4]? If
evolution is envisaged as movement of a population across a fitness landscape,
the question can be reworded more specifically: among the numerous
trajectories connecting any two points on the landscape, what fraction is
accessible to evolution? Until recently, these remained purely theoretical
questions as experimental study of fitness landscapes in the actual sequence
space was impractical, due both to the technical difficulty of producing and
assaying numerous expressed sequence variants and to the more fundamental
problem of defining an adequate quantitative measure of fitness. However,
recent experimental studies of fitness landscapes could potentially shed light
on the problem of evolutionary path predictability.

The most thoroughly characterized feature of empirical fitness landscapes is
the structure near a peak. In experiments that examine the peak structure, a
high fitness sequence is typically subjected to either random mutations or an
exhaustive set of mutations at a small number of important sites. The
resulting library of mutants is then assayed to measure a proxy of fitness [5,
6, 7, 8, 9]. Significant sign epistasis (a situation in which the fitness
effect of a particular mutation can be either positive or negative depending
on the genetic context) near the peak has been typically observed whereas
local deviations from the additive model have been found to be uncorrelated
with the genetic context and derived from a nearly normal distribution [10,
11, 12, 13]. Because these studies characterize only a small region of the
landscape, they cannot be used to address the question of path predictability.

Another broad class of experiments probes the evolutionary trajectories from
low to high fitness. Usually, in such experiments, a random peptide is
subjected to repeated rounds of random mutagenesis and purifying selection [8,
14, 15, 16, 17]. During this process fitness grows with each generation and
eventually stagnates at a suboptimal plateau. The characteristics of the
fitness growth as well as the dependence of the plateau height on the library
size can be used to classify landscapes [18]. A quantitative comparison to the
$NK$ model of random epistatic landscapes ($N$ is the number of sites in an
evolving sequence and $K$ is the number of sites that affect the fitness
contribution of a particular site through epistatic interactions) can even
yield quantitative estimates of $N$ and $K$ [19, 20]. The directed evolution
studies explore the evolutionarily accessible portion of the landscape and
could in principle be used to shed light on the question of path
predictability. However, the inaccessible regions of the landscape remain
unexplored and the volume of data at this point is insufficient to obtain
quantitative conclusions regarding path predictability.

A different type of landscapes has been explored in various microarray
experiments where protein-DNA(RNA) binding affinity serves as the proxy for
fitness [21, 22]. These experiments produce vast, densely sampled landscapes.
A comparison with a sophisticated Landscape State Machine model of a
correlated fitness landscapes yields estimates of the model parameters [23,
24]. The DNA binding landscapes, in principle, contain the information
required for the analysis of path statistics, and could be a valuable resource
for advancing the understanding of evolutionary path predictability.

Empirical studies that exhaustively sample a region of the fitness landscape
allow one to actually assess the accessibility of the entire set of
theoretically possible evolutionary trajectories in a particular (small) area
of the fitness landscape. For example, all mutational paths between two states
of an enzyme, e.g., the transition from an antibiotic-sensitive to an
antibiotic resistant form of $\beta$-lactamase [25, 26, 27] or the transition
between different specificities of sesquiterpene synthase [28] have been
explored. The results of these experiments, which out of necessity explore
only short mutational paths of $<10$ amino acid replacements, suggest that
there is a substantial deterministic component to protein evolution: only a
small fraction of the possible paths are accessible for evolution [25, 29, 30,
31].

Theoretical analyses of available empirical fitness data reveal a tight link
between genetic and molecular interactions which are responsible for the
landscape ruggedness and ubiquitous sign epistasis [13, 32]. The emerging
quantitative analysis of fitness landscapes can shed light on some of the most
fundamental aspects of evolution but the interpretation of the currently
available experimental results requires utmost caution as only a minuscule
part of the sequence space can be explored, and that only for a few more or
less arbitrarily selected experimental systems.

Here we focus on the question of the predictability of mutational paths which
is intimately tied to the ruggedness/smoothness of the fitness landscapes. The
study of random landscapes of low dimensionality revealed an intuitively
plausible negative correlation between the roughness of a landscape and the
availability of pathways of monotonic fitness [33]. In the same study,
Carneiro and Hartl showed that experimentally characterized landscapes are
significantly smoother than their permuted counterparts and exhibit greater
peak accessibility [33].

To gain insights into the structure of the fitness landscapes of protein
evolution and in particular the accessibility of mutational paths we used a
previously developed simple model of protein folding and evolution [34]. The
key assumption of this model, which is based on the concept of misfolding-
driven evolution of proteins [35, 36, 37], is that the fitness of model
proteins is determined solely by the number of misfolded copies that are
produced before the required abundance of the correctly folded protein is
reached. We have previously shown that this model accurately reproduces the
shape of the universal distribution of the evolutionary rates among
orthologous protein-coding genes along with the dependencies of the
evolutionary rate on protein abundance and effective population size [34].
These results appear to suggest that our folding model (described in detail
the Methods section) is sufficiently rich to reproduce some of the salient
aspects of evolution. The model is also simple enough to allow exhaustive
exploration of the fitness landscapes, which prompted us to directly address
the problem of evolutionary path predictability.

We build on the efforts of Carneiro and Hartl [33] who examined the statistics
of evolutionary trajectories. Although counting monotonic fitness paths
reveals important features of the landscapes, we argue that reliable
retrodiction of the evolutionary past is possible (i.e., evolution is quasi-
deterministic) only when the available monotonic paths are similar to each
other in a quantifiable way. We therefore propose a measure of path divergence
to quantify the difference between the available monotonic paths. Our aims are
to investigate the structure of the fitness landscapes of protein evolution
and to elucidate the connection between the roughness of landscapes and the
predictability of mutational trajectories. We analyze three classes of fitness
landscapes: landscapes in which fitness is derived from the folding robustness
of model polymers; additive random landscapes perturbed by noise; and
experimental landscapes derived from the combinatorial mutation analysis of
drug resistance and enzymatic activity. We show that all three classes of
landscapes are markedly smoother than their randomly permuted counterparts and
all exhibit a similar qualitative connection between roughness and path
predictability. However, at the same level of path predictability, the folding
landscapes have substantially fewer fitness peaks. Given that the statistical
properties of the model landscapes can be directly traced to the constraints
imposed by the energetics and kinetics of a folding heterpolymer, we
hypothesize that the relative smoothness and the suppression of suboptimal
peaks in fitness landscapes of protein evolution are fundamental consequences
of protein folding physics.

## Results

### Quantitative characterization of fitness landscapes

Carneiro and Hartl compared small random landscapes to several empirical
fitness landscapes using deviation from additivity as a measure of roughness
[33]. They found that empirical landscapes were significantly smoother than
their random counterparts and that the degree of smoothness was correlated
with the number of monotonic paths to the main summit. Deviation from
additivity of a landscape is computed by fitting an additive model in which
the fitness of each sequence is different from the peak fitness by the sum of
contributions of the substitutions that differentiate it from the peak
sequence. The negative fitness contributions of the substitutions to the peak
fitness are adjusted to minimize the sum $S$ of squares of the differences
between the actual fitnesses in the landscape and the fitnesses predicted by
the additive model. Deviation from additivity is defined as $\sqrt{S/L}$,
where $L$ is the number of points in the landscape.

Because roughness of a multidimensional landscape with variable degree
connectivity is not an intuitive concept, we introduce three additional
quantitative measures to probe alternative facets of the concept of roughness.
First, local roughness is the root mean squared difference between the fitness
of a point and its neighbors, averaged over the entire landscape. As defined,
local roughness conflates the measures of roughness and “steepness.” For
example, a globally smooth landscape, in which fitness depends only on the
distance from the peak, will have a non-zero local roughness. However, because
there is a large number of directions that change the distance from the peak
by one, the local roughness of a globally smooth landscape will be vanishingly
small. In addition, our landscapes tend to be globally flat–so that the
average decrease in fitness due to a single mutation step away from the main
peak is much smaller than the local fitness variability–everywhere except a
small region around the main peak (see Fig. 7). Therefore, the landscape-
average local roughness in our case is a true measure of the local fitness
variability.

Second, the fraction of peaks is the number of points with no fitter neighbors
divided by the total number of points in the landscape. A strictly additive
landscape has a single peak [30] whereas the peak fraction in landscapes
derived from the folding model as well as the corresponding randomized
landscapes depends on the method of landscape construction, alphabet size and
sequence length.

Third, the roughness of a landscape can be assessed by identifying its tree
component. The tree component is the set of all nodes with no more than one
neighbor of higher fitness. Thus, the tree component includes peaks and
plateaus. Monotonic fitness paths along the tree component form a single or
several disjoint tree structures without loops. In the limit of high selection
pressure, a mutational trajectory that finds itself on the tree component has
a single path to the nearest peak or plateau, i.e. evolution on the tree
component is completely deterministic. We use the mean distance to the tree
component, i.e. the distance to the tree component averaged over the
landscape, as a measure of roughness. In a fully additive landscape, only the
peak sequence and its immediate neighbors belong to the tree component and
therefore the mean distance to the tree component is a measure of the diameter
of an additive landscape (which, for example, could be defined as the maximum
pairwise distance between points on the landscape). Kauffman and Levin have
shown that in a large class of correlated random landscapes, the mean distance
to the tree component grows only logarithmically with the number of points in
the landscape [19].

We utilize two quantitative measures of the predictability of evolutionary
trajectories. First is fraction of monotonic paths to the main peak $F_{m}$
which is computed by counting the number $n_{i}$ of simple (without reverse
substitutions or multiple substitutions at the same site) monotonic paths to
the main peak from each point $i$ on the landscape, dividing it by the total
number of simple paths $h_{i}!$ (where $h_{i}$ is the Hamming distance from
point $i$ to the peak), and averaging over the landscape via

$F_{m}=\frac{1}{L}\sum_{i}\frac{n_{i}}{h_{i}!},$ (1)

where $L$ is the number of points in the landscape and the sum excludes the
main peak. The monotonic path fraction $F_{m}$ measures the scarcity of
accessible evolutionary paths when selection is strong. When the monotonic
path fraction is small, evolution is more constrained.

Second, the mean path divergence, is a fine-grained measure of evolutionary
(un)predictability. We first define the divergence $d(p_{1},p_{2})$ of a pair
of paths $p_{1}$ and $p_{2}$, as the average of the shortest Hamming distances
from each point on one path to the other path. Suppose that we have a way of
generating stochastic evolutionary paths. The outcome of a large number of
evolutionary dynamics simulations is a collection of paths with their
associated probabilities of occurrence. In general, the probability of
occurrence of an evolutionary path is proportional to the product of fixation
probabilities of its constituent mutation steps. Given a bundle of paths with
the same starting and ending points, we define its mean path divergence to be

$\bar{d}=\sum_{p_{1}\neq p_{2}}d(p_{1},p_{2})\,O(p_{1})\,O(p_{2}),$ (2)

where $O(p)$ is the probability of occurrence of path $p$ in the ensemble. In
other words, if two paths were drawn from the bundle at random with
probabilities proportional to $O(p)$, their expected divergence would be
$\bar{d}$. Alternatively, if we were to fix one path to be the most likely
path in the bundle and to select the second path at random with probability
proportional to $O(p)$, the divergence would be proportional to $\bar{d}$ as
well.

In an additive landscape, the mutational trajectory is maximally ambiguous. As
every substitution that brings the sequence closer to the peak increases
fitness, substitutions can occur in any order and all shortest mutational
trajectories to the peak–without reverse substitutions or multiple
substitutions at the same site–are monotonic in fitness. In the strong
selection limit of our model defined below, all monotonic trajectories have
roughly the same probability of occurrence, so the mutational path cannot be
predicted.

The mean path divergence is a better measure of the predictability of
evolutionary trajectories than the number or fraction of accessible paths.
Even when only a small fraction of paths are monotonic in fitness, these paths
could potentially be quite different, perhaps randomly scattered over the
landscape. In such a case, prediction of the evolutionary trajectory would be
inaccurate despite the scarcity of accessible paths which will be reflected in
a high value of path divergence.

Equation (2) introduces the mean path divergence of a bundle of paths with the
same starting and ending points. The landscape-wide mean path divergence is
measured by constructing representative path bundles with all possible [start,
peak] pairs including suboptimal peaks as trajectory termination points. Path
divergence is averaged over all bundles with the starting and ending points
separated by the same Hamming distance. To construct the path bundles, we
employed a low mutation rate model in which the attempted substitutions are
either eliminated or fixed in the population before the next mutation attempt
occurs.

We invoke the misfolding-cost hypothesis to assign a fitness to a sequence
that folds with probability $P$ to a particular structure. To produce an
abundance $A$ of correctly folded copies, an average of $A(1-P)/P$ of
misfolded copies are produced. The “fitness” of a sequence should be a
monotonically decreasing function of the cost incurred by the misfolded
proteins. Previously we showed that qualitative conclusions drawn from the
average population dynamics on the fitness landscape did not depend on the
precise functional relationship between the number of misfolded copies and
fitness [34]. We use simply the negative of the number of misfolded copies and
assign a fitness $w=-A/P$, to a sequence whose probability of folding to the
reference structure is $P$. Because the exact population dynamics model is not
important, we use diploid population dynamics in the low mutation rate limit.
Therefore, the probability of fixation of a mutant $j$ in the background of
$i$ is given by

$\pi(i\rightarrow
j)=\frac{1-e^{-2(w_{j}-w_{i})}}{1-e^{-4N_{e}(w_{j}-w_{i})}},$ (3)

where $N_{e}$ is the effective population size [38] which in all simulations
was fixed at $N_{e}=10,000$. The required abundance $A$ is a measure of the
strength of selection. In the limit of large $A$, the probability of fixation
of a beneficial mutation is unity whereas neutral and deleterious mutations
are never fixed. In this limit, all uphill steps in the fitness landscape are
equally likely and all monotonic uphill trajectories have equal evolutionary
significance.

In the analysis that follows, we study the association between landscape
roughness and path predictability for the folding landscapes and their
randomized (also referred to as permuted or scrambled) versions. In the
scrambled landscapes, the topology (i.e. connectivity) of the landscape is
preserved but the fitness values are randomly shuffled. We also compare the
roughness and path predictability characteristics of the model and the
experimental landscapes for $\beta$-lactamase [25] and sesquiterpene synthase
[28] to those for noisy additive landscapes with a continuously tunable amount
of roughness.

### Evolutionary path predictability in fitness landscapes

#### Deviation from additivity, local roughness, peak fraction, and monotonic
paths

We first establish that the folding and the experimental landscapes are
significantly different from their randomly permuted counterparts. The
deviation from additivity of the folding landscapes is typically several
standard deviations below the mean of their scrambled counterparts. Although
the additivity hypothesis accounts for less than 40% of the fitness
variability (computed by comparing the sum of the squares of the fitnesses in
the landscape to the sum of the squares of the residuals of the additive
fitness model fit) in all but one of the folding landscapes, the deviation
from additivity of the permuted landscapes is substantially greater (Fig. 1A).
The experimental landscapes follow the same pattern, in agreement with the
earlier findings of Carneiro and Hartl [33]. Furthermore, both in the folding
and in the experimental landscapes, the fraction of monotonic paths to the
main peak is several standard deviations greater than in the respective
scrambled landscapes (Fig. 1B). An even more striking disparity exists between
the fraction of peaks in the folding landscapes and their permuted versions:
the folding landscapes contain at least an order of magnitude fewer peaks than
their scrambled counterparts; the experimental landscapes resemble the folding
landscapes more closely than their own randomized versions (Fig. 1C).

To further characterize the deviation of the folding and experimental
landscapes from their permuted counterparts, each landscape metric was
measured and the mean and standard deviation were computed among 100 randomly
permuted landscapes. We then compute the Z-score (deviation from the mean
measured in the units of the standard deviation) of the original non-permuted
landscape compared to the ensemble of the permuted landscapes. This Z-score
shows how much more correlated the original landscape is, as measured by the
chosen characteristic, compared to its scrambled counterparts (Figure 2).
Notably, despite the considerable scatter of the Z-score values for the
folding landscapes, they all showed extremely large difference (mean Z-score
greater than 20 standard deviations) from the scrambled landscapes for all
measures, with the sole exception of the monotonic path fraction (Figure 2).
The two experimental landscapes also significantly differed from the scrambled
landscapes albeit less so than the folding landscapes, again with the
exception of the monotonic path fraction in which case the two classes of
landscapes had similar Z-scores (Figure 2).

Aside from the significant correlation (Pearson $\rho=-0.68$) between peak
fraction and mean distance to the tree component, there was little or no
correlation between the four measures of landscape roughness (Fig. 3).
Roughness of landscapes of high and variable dimensionality is impossible to
capture by a single quantity. Therefore, the different measures seam to reveal
distinct aspects of landscape architecture. The strong negative correlation
between the peak fraction and mean distance to the tree component is due to
the fact that each peak spawns a distinct subset of the tree component. The
higher the density of peaks on the landscape, the larger fraction of the
landscape that is covered by the tree component. Therefore the average
distance to the tree component declines with the increasing density of peaks.

#### Path divergence

Starting from a random non-peak sequence in the landscape, we introduced
random mutations and accepted or rejected them according to equation (3) until
the trajectory arrived at a fitness peak. This procedure was repeated a large
number of times, and path bundles were constructed for all pairs of starting
and ending sequences. Then the mean path divergence was computed for each path
bundle using equation (2) and averaged over all bundles for which starting and
ending points were separated by the same Hamming distance. When selection is
weak, all mutations which do not result in a sequence with zero folding
probability are accepted. Thus, evolution is a random walk on the landscape
and the statistical properties of evolutionary trajectories are fully
determined by the topology of the landscape (i.e. the connectivity of each
node). Conversely, in the strong selection limit, only mutations that increase
fitness are fixed. The mean path divergence varies smoothly between the two
limits (Fig. 4) and saturates at high selection pressure. In our analysis, we
focus on the strong selection limit plateau. In the weak selection limit, the
diversity of trajectories stems solely from the number of neighbors of each
point; by contrast, in the strong selection limit, the statistics of the
monotonic trajectories depend on the roughness of the landscape. Thus, the
weak selection limit probes only the topology of the landscape whereas the
strong selection limit also exposes its topography which appears to be
critical for assessing predictability of evolution.

#### Predictors and correlates of path divergence and monotonic path fraction

All four measures of landscape roughness can serve as predictors of path
divergence and monotonic path fraction to some degree (Fig. 5), in agreement
with the notion that each of these measures reflects salient properties of
fitness landscapes. The properties of the folding and empirical landscapes are
consistent with those of additive landscapes that were perturbed by a moderate
amount of noise (see Methods for details). A striking exception is the dearth
of peaks and monotonic paths in folding landscapes all other characteristics
being similar. Deviation from additivity and fraction of peaks are negatively
correlated with path divergence. This relationship captures the intuitive
notion that in rough landscapes there are fewer accessible evolutionary paths
than in smooth landscapes, and furthermore, in rough landscapes, even those
paths that are accessible show the tendency to aggregate within small areas on
the landscape. Indeed, in both the folding model-derived landscapes and the
experimental landscapes, the mean path divergence for all Hamming distances
between the starting and ending points was dramatically greater than in
scrambled landscapes (Fig. 6). Interpreting these findings in terms closer to
biology, the fitness landscapes derived from the model as well as experimental
landscapes show greater robustness to mutations than random landscapes: a
random mutation in a model-derived or experimental fitness landscape is more
likely than expected for random landscapes to have no deleterious effect,
leading to another monotonic path to the peak. Consequently, evolution on the
model-derived and experimental landscapes is less predictable (deterministic)
than it would be on uncorrelated random landscapes.

In contrast to deviation from additivity, the mean distance to the tree
component is positively correlated with path divergence. When the tree
component comprises a large fraction of the landscape, the mean distance to
the nearest tree branch is small. Consequently, the path divergence is reduced
as the paths that reach the tree component do not deviate from each other from
that point onward. By the same token, when the tree component is large, there
are fewer monotonic paths.

The origin of the positive correlation between the local roughness and path
divergence (Fig. 5) is less obvious. Paradoxically, greater noise results in
lower mean local roughness of noisy additive landscapes. The lowering of the
overall mean fitness with noise and, more importantly, the flattening of the
mean fitness dependence on the distance from the peak (Fig. 7) appear to
provide an explanation for this counter-intuitive result. Indeed we found that
in noisy additive landscapes there is a characteristic fitness value of
approximately 0.2 above which roughness increases with increasing noise and
below which roughness declines with increasing noise. Given that roughly 75%
of the points on the landscape have fitnesses below 0.2, the landscape-
averaged local roughness declines with increasing noise amplitude.

## Discussion

Here we examined the fraction of monotonic paths and introduced mean path
divergence as quantitative measures of the degree to which the starting and
ending points determine the path of evolution on fitness landscapes. The lower
the mean path divergence value, the more deterministic (and predictable)
evolution is. Global measures of landscape roughness correlate with path
divergence in the three analyzed classes of fitness landscapes: additive
landscapes perturbed by noise, landscapes derived from our protein folding
model and two small empirical landscapes. The folding landscapes are
substantially smoother than their permuted counterparts. As a result, although
in all analyzed landscapes only a small fraction of the theoretically possible
evolutionary trajectories is accessible, this fraction is much greater in the
folding and experimental landscapes than it is in randomized landscapes. In
addition, the mean path divergence in the randomized landscapes is
significantly smaller than in the original landscapes. Thus, the model and
empirical landscapes possess similar global architectures with many more
diverged monotonic paths to the high peaks than uncorrelated landscapes with
the same distribution of fitness values. Consequently, evolution in fitness
landscapes is substantially more robust to random mutations and less
deterministic (less predictable) than expected by chance. These findings are
compatible with the concept that might appear counter-intuitive but is
buttressed by results of population genetic modeling, namely, that robustness
of evolving biological systems promotes their evolvability [39, 40, 41].
Additionally, the folding landscapes exhibit a substantial deficit of peaks
compared to perturbed additive landscapes and experimental landscapes, a
property that translates into a substantially greater fraction of paths
leading to the main peak.

When it comes to the interpretation of the properties of fitness landscapes
described here, an inevitable and important question is whether the folding
model employed here is sufficiently complex and realistic to yield
biologically relevant information. In selecting the complexity of our folding
model, we attempted to construct the simplest model which exhibits 1) a rich
spectrum of low energy conformations across the sequence space, and 2) a non-
trivial distribution of substitutions effects on the low energy conformations.
An important choice is whether the location of monomers is confined to a
lattice or can be varied continuously. When the configuration space is
continuous, the distribution of energy barriers between energetically optimal
conformations can extend to zero. Therefore, the subtlety of distinctions
between conformations can lead to a richer structure of the fitness landscape.
We chose not increase the complexity of the model further and treated monomers
as point-like particles in a chain where the distance between nearest
neighbors is fixed but the angle between successive links in the chain in
unrestricted. Our level of abstraction is therefore somewhere between lattice
models and all-atom descriptions of proteins [42, 43, 44, 45, 46, 47, 48, 49,
50, 51].

Another important choice is the number of the model monomer types. Again, we
opted for an intermediate level of abstraction and chose four types of
monomers: hydrophobic, hydrophilic, and charged. This choice drastically
reduces the size of the sequence space while retaining some of the
substitution complexity whereby hydrophilic and charged monomers can be
swapped under some conditions without radically altering the native state. The
intermediate level of abstraction in our approach has its pros and cons.
Although the model reproduces key features of protein folding such as the
existence of the hydrophobic folding nucleus and two-stage folding kinetics
[52, 53], compact conformations certainly do not represent proteins. Rather,
we might think of our monomers as representing structurally grouped regions
several (perhaps up to a dozen) amino-acids in length. Compact conformations
in the model might therefore be analogous to tertiary structures of proteins.
Representing sequence space with only four monomer types and treating
mutations without reference to the underlying DNA or genetic code does not
accurately reflect the natural mutation process. However, our goal was to
isolate the features of fitness landscapes which could be traced directly to
the constraints imposed by the heteropolymer folding kinetics and energetics.
We therefore used a simple sequence space and a homogeneous mutation model to
avoid compounding the fitness landscape structure by the complexity derived
from the mutation process.

Most importantly, our folding model has been shown to reproduce the observed
universal distribution of the evolutionary rates of protein-coding genes as
well as the dependencies of the evolutionary rate on protein abundance and
effective population sizes [34]. Therefore, despite its simplicity, the
behavior of this model might reflect important aspects of protein evolution.
In particular, the conclusions drawn from the analysis of the model landscapes
exhaustively explored here could also apply to the fitness landscapes of
protein evolution. In the previous work, we concluded that the universal
distribution of evolutionary rates and other features of protein evolution
follow from the fundamental physics of protein folding [34]. The results
presented here suggest that the (relative) smoothness and a substantial
deficit of peaks in the fitness landscapes of protein evolution that lead to
mutational robustness and the ensuing evolvability could similarly follow from
the fact that proteins are heteropolymers that have to fold in three
dimensions to perform their functions.

The experimental landscapes considered here are decidedly incomplete. Due to
experimental limitations, only the analysis of binary substitutions at a
handful of sites is feasible at this time. The incompleteness of the empirical
landscapes analyzed in this work could be the cause of the observed lack of
peak suppression. This proposition will be put to test by the study of larger
parts of experimental landscapes that are becoming increasingly available.

## Materials and Methods

### Folding model

The goal of this study is to explore the relationship between roughness and
path divergence in realistic fitness landscapes. Our polymer folding model
provides a simple way of constructing such landscapes. The model has been
described in detail previously [34].

In brief, the model polymer is a flexible chain of monomers in which the the
nearest neighbors interact via a stiff harmonic spring potential with rest
length $a=1$. The angles between the successive links in the chain are
unrestricted. There are four types of monomers: hydrophobic H, hydrophilic P,
and charged + and –. Next nearest neighbors $i$ and $j$ in the chain and
beyond interact via a pairwise potential

$U_{ij}(r_{ij})=\frac{A_{ij}}{r_{ij}^{12}}-\frac{C_{ij}}{r_{ij}^{6}}+\frac{q_{i}q_{j}e^{-D\,r_{ij}}}{r_{ij}},$
(4)

where $r_{ij}$ is the distance between monomers $i$ and $j$, $q_{i}$ is the
monomer’s charge, $D$ is the Debye-Hückel screening length, and $A_{ij}$ and
$C_{ij}$ depend on the pair in question. The interaction parameters are chosen
to mimic the essential features of the amino-acid interactions. To emulate the
effects of solvent, we assign a stronger attraction to the HH pair than to the
PP, ++, and – – pairs. There is also a long range repulsion between H and P
and even stronger repulsion between H and the charged monomers. The values of
the parameters are $q_{\pm}=\pm 2$, Debye-Hückel screening length $D=3$. The
Lennard-Jones coefficients $A_{ij}$ and $C_{ij}$ are

$\displaystyle A_{HH}=4,\ A_{HP}=A_{H+}=2,\ A_{PP}=A_{P+}=A_{++}=1,$
$\displaystyle C_{HH}=8,\ C_{HP}=-1,\ C_{H+}=-3,\ C_{PP}=C_{P+}=C_{++}=2.$ (5)

Note that a $+$ can be substituted by a $-$ in the subscripts and the
coefficients are symmetric with respect to the interchange of the indices.

The energy of the chain is

$E=\sum_{|i-j|>1}U_{ij}+\frac{b\,T}{2}\sum_{i=1}^{N-1}(r_{i,i+1}-a)^{2},$ (6)

where the first term is the sum of the pairwise energies given by Eq. (4) over
non-nearest neighbor pairs, and the second term reflects the springs
connecting nearest neighbors. The spring constant is proportional to
temperature $T$. The parameters are fixed for all simulation runs at $b=300$,
and the quench temperature $T=1$. To mimic the observed tendency of the $N$
and $C$ termini to be in close proximity, we fixed the endpoint monomers of
the model sequences to be of $+$ and $-$ types.

Dynamics of folding are simulated via overdamped Brownian kinetics which are
appropriate when inertial and hydrodynamic effects are not important. Units
are chosen so that each component $\alpha$ of the $i$’th monomer’s coordinates
$x_{\alpha i}$ is updated according to

$x_{\alpha i}(t+\Delta t)=x_{\alpha i}(t)-\frac{\Delta t}{T}\,\frac{\partial
E}{\partial x_{\alpha i}}(t)+W_{\alpha i}(t),$ (7)

where $\Delta t$ is the time step and $W_{\alpha i}(t)$ is a random variable
with zero mean, variance $2\Delta t$, uncorrelated with $W$ for other times,
monomers and spatial directions.

### Native structure ensemble and correct folding probability

The “native structure” of a particular sequence is represented by an
equilibrium ensemble of conformations. The ensemble is constructed by
identifying the typical folded conformation and measuring the characteristic
RMSD $D$ due to thermal fluctuations in the folded state. Three thousand
quenches are then performed and the resulting folded conformations are
accumulated. The equilibrium ensemble that represents the native structure is
defined as the largest cluster of quenched conformations within RMSD distance
$D$ from each other. Thus, each conformation in the ensemble differs from any
other by an amount comparable to the differences introduced by thermal
fluctuations alone.

The concept of the native structure ensemble allows us to compute the
probability that a sequence folds to a particular structure in a natural,
physically plausible fashion. Given a native structure ensemble we assess its
conformation space density by computing the distance $d_{i}$ between each
member $i$ of the ensemble and its closest neighbor. Given the set
$\\{d_{i}\\}$ of these shortest distances we compute the median $Q$ and the
median absolute deviation (MAD) $V$. A new conformation is deemed to belong to
the ensemble if the shortest distance from this conformation to the members of
the ensemble is smaller than $R=Q+3V$.

Given a native structure ensemble of some sequence $s_{1}$ we compute the
probability $P$ that sequence $s_{2}$ (which could be $s_{1}$ itself) folds to
the this structure by accumulating $M=100$ equilibrated quenched conformations
of $s_{2}$ and using the above criterion to determine the fraction $P$ that
belong to the native structure ensemble of $s_{1}$. Because $M=100$ sample
conformations are computed, the smallest measurable $P$ is $1/M=0.01$. The
sample size used to measure $P,$ dictated by the computational demands of the
model, introduces a random component to the model fitness landscapes. As we
report below, model landscapes turn out to be substantially smoother than
random. Therefore the underlying global structure of the model landscapes
appears to survive the modest amount of randomness introduced by the
relatively small sample size used for measuring $P$.

### Search for compact robust folders

Robust folders (sequences with a high probability of correct folding) tend to
have large linear regions stretched by repulsive Coulomb interactions. Because
the linear regions have no contacts with other monomers, we focused our
attention on compact conformations with a high monomer contact density.
Substitutions in these higher complexity conformations were more likely to
exhibit non-trivial effects. To find compact robust folders in the vast
available sequence space of $23$-mers (the sequences are of length $N=25$ but
the endpoint monomer types are fixed) with $4$ monomer types, we implemented a
simulated annealing search which optimized the correct folding probability $P$
divided by the cube of the native conformation’s radius of gyration. The
search produced over 800 sequences with $P>0.5$ and at least two distinct
regions of the polymer in mutual contact.

### Assembly of the folding fitness landscapes

We examined each single substitution mutant of a robustly folding sequence and
computed the folding probability $P$ to the structure of the original
sequence. All mutants with $P>0$ were added to the landscape and if $P\geq
0.1$ their mutants were also examined. This process is repeated until all
mutants of the last sequence under consideration have $P<0.1$.

From our study of complete landscapes we estimate that on average for each
sequence with $P>0$ which is included into the landscape, roughly 6 others
with $P=0$ need to be examined. Since each quench and equilibration takes
about 2–4 seconds, landscape construction takes roughly 30 minutes to an hour
per included sequence. Thus landscapes larger than 10,000 sequences take
months to compile.

At the time of submission, 39 complete landscapes have been constructed, the
largest comprising 12969 sequences.

### Additive landscapes perturbed by noise

The organization of the folding fitness landscapes and experimental landscapes
were compared with perfectly additive landscapes perturbed by noise
constructed as follows. Each substitution to the peak fitness sequence was
assigned a negative fitness differential drawn at random from an exponential
distribution with parameter $\lambda=3$. The sum over the fitness
differentials of a particular set of substitution was modified by either
additive of multiplicative noise [54]. Additive noise is drawn from a Gaussian
distribution with zero mean and standard deviation $\nu$ which was varied
between $0.05$ and $0.5$. The multiplicative perturbation is achieved by
multiplying the fitness by a number drawn from a uniform distribution $[0,1)$
raised to a positive power $\mu$ varied between $0.1$ and $10.$ When $\mu$ is
small, multiplicative factors are close to unity and the perturbation is small
as well. If the perturbed fitness was positive, the mutant was included into
the landscape. The noise amplitude was varied to obtain a family of landscapes
of continuously varying roughness. Only the data for the additive landscapes
with multiplicative noise were included in this manuscript. Landscapes
perturbed by other types of noise exhibited essentially the same qualitative
behavior.

### Experimental landscapes

The studies on experimental fitness landscapes typically involve constructing
a library of all possible combinations of binary mutations at a small number
of sites. The first study included in the present analysis measured the
minimum inhibitory concentrations (MIC) of an antibiotic for a complete
spectrum of mutants with modified TEM $\beta$-lactamases; the transition from
the antibiotic-sensitive to the antibiotic-resistant form requires five
mutation, so the landscape encompassed 120 mutational trajectories between the
most distant points on the landscape (or 32 sequences) [25]. The logarithm of
MIC was used as the proxy for fitness. In the second study, catalytic activity
of 419 sesquiterpene synthase mutants that differed by at most 9 substitutions
was measured [28]. We used the catalytic specificity (propensity for producing
a particular reaction product rather than a broad spectrum of products) of the
mutant enzymes as the proxy for fitness. Before performing the analysis, the
fitnesses in the experimental landscapes are mapped onto the $[0.01,1)$
interval to enable meaningful quantitative comparisons of the roughness
measures.

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## Figure Legends

Figure 1: Roughness, monotonic paths and suboptimal peak suppression in
folding and experimental landscapes. (A) Deviation from additivity for the
folding landscapes (larger symbols), their scrambled versions (smaller
symbols) and the two experimental landscapes. Error bars show one standard
deviation within the ensemble of permuted landscapes. (B) Fraction of
monotonic paths to the main peak in folding, scrambled and experimental
landscapes. (C) The number of peaks is vastly greater in scrambled landscapes
than in folding or experimental landscapes (with the exception of the
sesquiterpene synthase landscape). Figure 2: The Z-scores of different
characteristics of the original folding and experimental landscapes measured
with respect to the ensembles of their randomly permuted counterparts. Figure
3: Correlations between different quantitative characteristics of the folding
landscapes. Each panel quotes the Spearman rank correlation coefficient
between the particular pair of characteristics. Figure 4: Mean path divergence
as a function of selection pressure for a folding landscape with 5936 nodes
and 65 peaks. Solid lines are labeled by the Hamming distance between the
pairs of starting and ending points of the trajectory bundles over which the
path divergence is averaged. Figure 5: The dependence of the path divergence
(top row) and the monotonic path fraction (bottom row) on the measures of
landscape roughness. The dots of different color correspond to noisy additive
landscapes with differing amounts of multiplicative noise: low (red), two
intermediate levels (green smaller than blue), and high (magenta). Yellow
circles represent the folding landscapes, the cyan squares–the
$\beta$-lactamase landscape, and the red triangles–the sesquiterpene synthase
landscape. Figure 6: Mean path divergence in folding and experimental
landscapes (larger symbols) landscapes, as well as their scrambled versions
(smaller symbols) as a function of Hamming distance from the main peak. Figure
7: Fitness averaged over all points at a particular distance $H$ from the peak
for folding landscapes, additive landscapes with the same three levels of
multiplicative noise used in Fig. 5 and the sesquiterpene synthase landscape.


36 | 20 | 50 | -999 | -999 | 355 | 91 | 24 | -96 | 26 | 3
37 | 12 | 2 | -999 | -999 | 95 | 85 | 18 | 10 | 20 | 3
39 | 65 | 4 | -999 | -999 | 260 | 244 | 2 | 16 | 8 | 3
45 | 28 | 3 | -999 | -999 | 245 | 179 | 12 | 66 | 14 | 3
51 | 3 | 1 | -999 | -999 | 140 | 8 | 40 | 132 | 41 | 3
54 | 17 | 2 | 17 | 8 | 260 | 294 | 9 | -34 | 12 | 1
63 | 5 | 1 | -999 | -999 | 300 | 35 | 47 | -95 | 47 | 3
64 | 55 | 2 | -999 | -999 | 120 | 115 | 3 | 5 | 9 | 3
65 | 3 | 1 | -999 | -999 | 330 | 11 | 47 | -41 | 48 | 3
66 | 23 | 1 | -999 | -999 | 320 | 309 | 6 | 11 | 10 | 3
67 | 4 | 1 | -999 | -999 | 300 | 15 | 32 | -75 | 33 | 3
83 | 10 | 3 | 11 | 20 | 69 | 256 | 16 | 173 | 18 | 2
89 | 2 | 2 | -999 | -999 | 110 | 330 | 25 | 140 | 26 | 3
101 | 9 | 1 | -999 | -999 | 305 | 80 | 24 | -134 | 25 | 3
129 | 11 | 2 | -999 | -999 | 100 | 30 | 4 | 70 | 9 | 3
157 | 45 | 15 | -999 | -999 | 315 | 29 | 2 | -74 | 8 | 3
163 | 6 | 5 | -999 | -999 | 260 | 267 | 42 | -7 | 43 | 3
165 | 21 | 17 | -999 | -999 | 10 | 21 | 5 | -11 | 9 | 3
173 | 13 | 15 | -999 | -999 | 40 | 268 | 39 | 132 | 39 | 3
195 | 10 | 23 | -999 | -999 | 220 | 24 | 117 | -164 | 118 | 3
197 | 15 | 26 | -999 | -999 | 345 | 11 | 21 | -26 | 23 | 3
200 | 34 | 13 | 44 | 14 | 260 | 30 | 2 | -130 | 8 | 2
203 | 13 | 2 | -999 | -999 | 25 | 132 | 21 | -107 | 22 | 3
212 | 15 | 1 | 25 | 12 | 115 | 114 | 14 | 1 | 16 | 1
240 | 8 | 2 | 24 | 3 | 238 | 333 | 9 | -95 | 12 | 2
249 | 14 | 4 | 21 | 7 | 235 | 298 | 18 | -63 | 20 | 2
252 | 15 | 2 | -999 | -999 | 237 | 144 | 32 | 93 | 33 | 3
255 | 19 | 2 | 28 | 3 | 10 | 24 | 3 | -14 | 8 | 2
263 | 44 | 12 | -999 | -999 | 80 | 75 | 6 | 5 | 10 | 3
265 | 17 | 5 | 44 | 7 | 5 | 32 | 4 | -27 | 9 | 2
268 | 13 | 1 | -999 | -999 | 310 | 161 | 11 | 149 | 14 | 3
271 | 16 | 5 | 45 | 14 | 29 | 48 | 6 | -19 | 10 | 2
272 | 60 | 18 | 63 | 27 | 260 | 277 | 10 | -17 | 13 | 1
282 | 42 | 8 | 55 | 8 | 45 | 39 | 2 | 6 | 8 | 2
284 | 32 | 12 | 35 | 16 | 140 | 316 | 20 | -176 | 22 | 2
285 | 35 | 7 | 47 | 11 | 340 | 8 | 4 | -28 | 9 | 1
287 | 10 | 4 | -999 | -999 | 117 | 213 | 40 | -96 | 41 | 3
288 | 11 | 2 | 38 | 18 | 320 | 229 | 6 | 91 | 10 | 2
289 | 12 | 1 | -999 | -999 | 290 | 284 | 8 | 6 | 11 | 3
291 | 13 | 20 | -999 | -999 | 25 | 197 | 9 | -172 | 12 | 3
295 | 3 | 2 | -999 | -999 | 352 | 287 | 28 | 65 | 29 | 3
296 | 7 | 23 | -999 | -999 | 355 | 234 | 18 | 121 | 20 | 3
298 | 25 | 2 | 35 | 2 | 170 | 144 | 6 | 26 | 10 | 2
300 | 4 | 1 | 16 | 2 | 189 | 206 | 101 | -17 | 101 | 2
303 | 52 | 90 | 55 | 90 | 10 | 209 | 11 | 161 | 14 | 1
306 | 7 | 1 | 14 | 5 | 270 | 70 | 42 | -160 | 43 | 2
309 | 15 | 83 | -999 | -999 | 165 | 202 | 44 | -37 | 44 | 3
313 | 6 | 2 | -999 | -999 | 169 | 261 | 10 | -92 | 13 | 3
314 | 11 | 2 | -999 | -999 | 285 | 222 | 5 | 62 | 9 | 3
315 | 37 | 42 | -999 | -999 | 215 | 206 | 10 | 9 | 13 | 3
318 | 9 | 2 | -999 | -999 | 38 | 241 | 5 | 157 | 9 | 3
319 | 11 | 12 | 37 | 45 | 91 | 164 | 45 | -73 | 46 | 1
320 | 7 | 2 | 9 | 2 | 290 | 53 | 56 | -123 | 56 | 2
321 | 1 | 1 | 18 | 2 | 35 | 258 | 16 | 137 | 18 | 2
322 | 3 | 1 | 14 | 19 | 35 | 247 | 28 | 148 | 29 | 1
324 | 30 | 2 | 36 | 4 | 225 | 286 | 4 | -61 | 9 | 2
326 | 17 | 2 | 22 | 3 | 100 | 70 | 6 | 30 | 10 | 2
328 | 18 | 1 | 20 | 15 | 130 | 111 | 8 | 19 | 11 | 2
329 | 38 | 2 | 46 | 4 | 225 | 202 | 2 | 23 | 8 | 2
330 | 9 | 1 | 39 | 2 | 40 | 172 | 12 | -132 | 14 | 2
331 | 4 | 2 | 25 | 30 | 20 | 248 | 7 | 132 | 11 | 1
332 | 5 | 1 | 43 | 20 | 140 | 297 | 12 | -157 | 14 | 1
333 | 2 | 1 | 10 | 2 | 210 | 285 | 16 | -75 | 17 | 3
335 | 12 | 2 | 15 | 2 | 205 | 32 | 7 | 174 | 10 | 2
336 | 26 | 2 | -999 | -999 | 90 | 147 | 4 | -57 | 9 | 3
338 | 4 | 2 | 4 | 8 | 290 | 313 | 22 | -23 | 23 | 2
339 | 77 | 3 | 88 | 10 | 70 | 44 | 1 | 26 | 8 | 2
340 | 19 | 2 | 25 | 3 | 100 | 48 | 4 | 52 | 9 | 2
341 | 11 | 2 | 36 | 10 | 135 | 4 | 10 | 131 | 13 | 1
342 | 14 | 2 | 14 | 5 | 350 | 330 | 6 | 20 | 10 | 2
343 | 49 | 2 | 123 | 12 | 340 | 358 | 2 | -18 | 8 | 2
344 | 45 | 2 | -999 | -999 | 10 | 356 | 2 | 14 | 8 | 3
345 | 6 | 2 | 32 | 2 | 70 | 327 | 25 | 103 | 26 | 2
346 | 15 | 2 | 216 | 9 | 0 | 34 | 6 | -34 | 10 | 1
349 | 20 | 2 | 31 | 6 | 245 | 227 | 2 | 18 | 8 | 2
351 | 6 | 20 | 10 | 20 | 330 | 230 | 17 | 100 | 18 | 2
352 | 5 | 2 | 7 | 5 | 270 | 309 | 23 | -39 | 25 | 2
353 | 67 | 4 | 70 | 5 | 265 | 286 | 1 | -21 | 8 | 2
354 | 8 | 4 | 11 | 5 | 115 | 221 | 30 | -106 | 31 | 3
356 | 10 | 2 | 38 | 18 | 45 | 10 | 17 | 35 | 19 | 1
358 | 2 | 1 | 14 | 2 | 300 | 264 | 10 | 36 | 13 | 2
359 | 12 | 2 | 59 | 3 | 170 | 78 | 26 | 92 | 28 | 2
360 | 74 | 4 | 103 | 8 | 220 | 226 | 1 | -6 | 8 | 1
361 | 29 | 2 | 45 | 9 | 50 | 54 | 3 | -4 | 8 | 1
362 | 27 | 2 | 27 | 11 | 95 | 74 | 3 | 20 | 9 | 2
365 | 11 | 2 | 13 | 2 | 145 | 178 | 34 | -33 | 35 | 2
366 | 11 | 2 | -999 | -999 | 5 | 359 | 12 | 6 | 14 | 3
367 | 8 | 2 | 35 | 2 | 45 | 7 | 28 | 38 | 29 | 2
368 | 8 | 1 | -999 | -999 | 269 | 271 | 14 | -2 | 16 | 3
371 | 24 | 3 | 11 | 37 | 220 | 201 | 10 | 19 | 13 | 1
372 | 16 | 2 | 86 | 111 | 325 | 256 | 9 | 69 | 12 | 1
373 | 23 | 3 | -999 | -999 | 300 | 308 | 2 | -8 | 8 | 3
374 | 18 | 2 | -999 | -999 | 315 | 298 | 7 | 18 | 11 | 3
375 | 88 | 3 | -999 | -999 | 350 | 354 | 1 | -4 | 8 | 3
376 | 21 | 2 | -999 | -999 | 50 | 28 | 5 | 22 | 9 | 3
377 | 42 | 2 | -999 | -999 | 20 | 286 | 3 | 94 | 9 | 3
378 | 9 | 2 | -999 | -999 | 45 | 207 | 24 | -162 | 25 | 3
381 | 11 | 3 | -999 | -999 | 340 | 346 | 6 | -6 | 10 | 3
383 | 88 | 13 | -999 | -999 | 310 | 319 | 1 | -9 | 8 | 3
384 | 4 | 2 | -999 | -999 | 45 | 128 | 9 | -83 | 12 | 3
385 | 30 | 2 | -999 | -999 | 348 | 359 | 3 | -11 | 8 | 3
386 | 48 | 2 | -999 | -999 | 345 | 5 | 2 | -20 | 8 | 3
389 | 15 | 1 | -999 | -999 | 285 | 234 | 11 | 51 | 14 | 3
390 | 20 | 2 | -999 | -999 | 315 | 307 | 4 | 8 | 9 | 3
391 | 10 | 1 | -999 | -999 | 10 | 174 | 7 | -164 | 11 | 3
393 | 9 | 1 | -999 | -999 | 300 | 167 | 7 | 133 | 10 | 3
394 | 57 | 2 | -999 | -999 | 135 | 138 | 1 | -3 | 8 | 3
395 | 11 | 1 | -999 | -999 | 210 | 168 | 6 | 42 | 10 | 3
396 | 27 | 2 | -999 | -999 | 90 | 98 | 3 | -8 | 9 | 3
397 | 31 | 2 | -999 | -999 | 315 | 286 | 2 | 29 | 8 | 3
404 | 17 | 2 | -999 | -999 | 100 | 124 | 2 | -24 | 8 | 3
405 | 11 | 2 | -999 | -999 | 355 | 340 | 15 | 15 | 17 | 3
406 | 12 | 2 | -999 | -999 | 110 | 197 | 18 | -87 | 20 | 3
407 | 10 | 2 | -999 | -999 | 205 | 129 | 4 | 76 | 9 | 3
408 | 4 | 1 | -999 | -999 | 325 | 118 | 8 | -153 | 11 | 3
409 | 19 | 2 | -999 | -999 | 20 | 200 | 8 | -180 | 11 | 3
410 | 14 | 1 | -999 | -999 | 5 | 180 | 10 | -175 | 13 | 3
411 | 5 | 1 | -999 | -999 | 220 | 111 | 5 | 109 | 9 | 3
412 | 4 | 1 | -999 | -999 | 340 | 116 | 10 | -136 | 12 | 3
414 | 7 | 1 | -999 | -999 | 310 | 132 | 28 | 178 | 29 | 3
422 | 9 | 2 | -999 | -999 | 100 | 109 | 4 | -9 | 9 | 3
425 | 17 | 2 | -999 | -999 | 100 | 117 | 3 | -17 | 9 | 3
426 | 15 | 2 | -999 | -999 | 200 | 109 | 3 | 91 | 9 | 3
430 | 10 | 1 | -999 | -999 | 260 | 125 | 9 | 135 | 12 | 3
431 | 9 | 1 | -999 | -999 | 80 | 50 | 5 | 30 | 9 | 3
434 | 76 | 4 | -999 | -999 | 95 | 97 | 1 | -2 | 8 | 3
440 | 11 | 31 | -999 | -999 | 85 | 140 | 71 | -55 | 72 | 3
445 | 6 | 7 | -999 | -999 | 270 | 88 | 36 | -178 | 37 | 3
448 | 6 | 3 | -999 | -999 | 35 | 68 | 5 | -33 | 10 | 3
449 | 11 | 3 | -999 | -999 | 205 | 100 | 4 | 104 | 9 | 3
450 | 9 | 5 | -999 | -999 | 280 | 121 | 98 | 159 | 99 | 3
453 | 11 | 2 | -999 | -999 | 350 | 97 | 4 | -107 | 9 | 3
458 | 14 | 4 | -999 | -999 | 195 | 81 | 4 | 114 | 9 | 3
462 | 14 | 1 | -999 | -999 | 155 | 128 | 8 | 27 | 12 | 3
463 | 8 | 2 | -999 | -999 | 300 | 107 | 34 | -167 | 35 | 3
464 | 9 | 1 | -999 | -999 | 170 | 110 | 11 | 60 | 13 | 3
467 | 3 | 1 | -999 | -999 | 345 | 42 | 10 | -57 | 12 | 3
468 | 11 | 7 | -999 | -999 | 345 | 82 | 21 | -97 | 23 | 3
469 | 10 | 2 | -999 | -999 | 110 | 260 | 99 | -150 | 99 | 3
470 | 13 | 2 | -999 | -999 | 85 | 186 | 41 | -101 | 41 | 3
471 | 10 | 2 | -999 | -999 | 175 | 322 | 58 | -147 | 59 | 3
473 | 7 | 2 | -999 | -999 | 325 | 359 | 16 | -34 | 18 | 3
474 | 18 | 2 | -999 | -999 | 280 | 280 | 8 | -0 | 11 | 3
484 | 26 | 3 | -999 | -999 | 325 | 263 | 7 | 62 | 11 | 3
485 | 7 | 2 | -999 | -999 | 5 | 74 | 4 | -69 | 9 | 3
489 | 38 | 7 | -999 | -999 | 35 | 80 | 1 | -45 | 8 | 3
497 | 8 | 1 | -999 | -999 | 265 | 7 | 10 | -102 | 13 | 3
506 | 11 | 2 | -999 | -999 | 245 | 212 | 95 | 33 | 95 | 3
509 | 18 | 10 | -999 | -999 | 25 | 31 | 9 | -6 | 12 | 3
512 | 20 | 6 | -999 | -999 | 220 | 141 | 22 | 79 | 24 | 3
513 | 11 | 2 | -999 | -999 | 140 | 270 | 62 | -130 | 63 | 3
514 | 21 | 2 | -999 | -999 | 215 | 188 | 11 | 27 | 14 | 3
515 | 38 | 2 | -999 | -999 | 235 | 232 | 2 | 3 | 8 | 3
517 | 20 | 2 | -999 | -999 | 160 | 207 | 15 | -47 | 17 | 3
518 | 7 | 2 | -999 | -999 | 65 | 41 | 68 | 24 | 68 | 3
520 | 17 | 14 | -999 | -999 | 180 | 57 | 5 | 123 | 10 | 3
534 | 5 | 3 | -999 | -999 | 120 | 3 | 21 | 117 | 22 | 3
539 | 33 | 10 | -999 | -999 | 0 | 89 | 9 | -89 | 12 | 3
544 | 38 | 3 | -999 | -999 | 225 | 218 | 2 | 7 | 8 | 3
545 | 2 | 14 | -999 | -999 | 155 | 216 | 115 | -61 | 115 | 3
548 | 10 | 12 | -999 | -999 | 280 | 33 | 38 | -113 | 39 | 3
555 | 14 | 2 | -999 | -999 | 290 | 321 | 9 | -31 | 12 | 3
558 | 16 | 3 | -999 | -999 | 225 | 144 | 36 | 81 | 37 | 3
561 | 17 | 5 | -999 | -999 | 145 | 46 | 4 | 98 | 9 | 3
575 | 95 | 9 | -999 | -999 | 195 | 194 | 1 | 0 | 8 | 3
579 | 13 | 3 | -999 | -999 | 250 | 290 | 15 | -40 | 17 | 3
581 | 13 | 3 | -999 | -999 | 250 | 282 | 18 | -32 | 19 | 3
583 | 24 | 3 | -999 | -999 | 290 | 233 | 4 | 57 | 9 | 3
584 | 21 | 3 | -999 | -999 | 90 | 194 | 11 | -104 | 14 | 3
587 | 4 | 12 | -999 | -999 | 235 | 46 | 112 | -172 | 113 | 3
589 | 10 | 2 | -999 | -999 | 350 | 312 | 18 | 38 | 20 | 3
607 | 22 | 6 | -999 | -999 | 175 | 211 | 39 | -36 | 40 | 3
614 | 7 | 3 | -999 | -999 | 45 | 55 | 50 | -10 | 51 | 3
616 | 45 | 4 | -999 | -999 | 280 | 218 | 4 | 62 | 9 | 3
618 | 17 | 4 | -999 | -999 | 230 | 158 | 20 | 72 | 22 | 3
619 | 44 | 7 | -999 | -999 | 290 | 230 | 25 | 60 | 26 | 3
624 | 28 | 3 | -999 | -999 | 25 | 215 | 4 | 170 | 9 | 3
626 | 3 | 2 | -999 | -999 | 5 | 12 | 7 | -7 | 11 | 3
634 | 9 | 2 | -999 | -999 | 350 | 75 | 44 | -85 | 45 | 3
635 | 6 | 3 | -999 | -999 | 55 | 18 | 6 | 38 | 10 | 3
636 | 21 | 3 | -999 | -999 | 80 | 214 | 6 | -134 | 10 | 3
641 | 25 | 4 | -999 | -999 | 100 | 218 | 13 | -118 | 15 | 3
648 | 48 | 20 | -999 | -999 | 180 | 206 | 40 | -26 | 41 | 3
653 | 15 | 2 | -999 | -999 | 225 | 171 | 23 | 54 | 25 | 3
655 | 25 | 3 | -999 | -999 | 305 | 217 | 4 | 88 | 9 | 3
656 | 34 | 3 | -999 | -999 | 280 | 218 | 2 | 62 | 8 | 3
657 | 24 | 3 | -999 | -999 | 255 | 210 | 10 | 45 | 13 | 3
662 | 13 | 3 | -999 | -999 | 70 | 25 | 3 | 45 | 9 | 3
667 | 29 | 2 | -999 | -999 | 120 | 120 | 7 | -0 | 10 | 3
670 | 33 | 3 | -999 | -999 | 140 | 232 | 3 | -92 | 8 | 3
673 | 21 | 3 | -999 | -999 | 130 | 227 | 5 | -97 | 10 | 3
681 | 9 | 2 | -999 | -999 | 120 | 91 | 80 | 29 | 80 | 3
687 | 20 | 3 | -999 | -999 | 105 | 191 | 9 | -86 | 12 | 3
691 | 15 | 2 | -999 | -999 | 25 | 286 | 11 | 99 | 14 | 3
692 | 17 | 1 | -999 | -999 | 55 | 96 | 11 | -41 | 14 | 3
700 | 16 | 3 | -999 | -999 | 105 | 206 | 18 | -101 | 19 | 3
701 | 15 | 3 | -999 | -999 | 180 | 232 | 15 | -52 | 17 | 3
705 | 24 | 3 | -999 | -999 | 160 | 226 | 4 | -66 | 9 | 3
709 | 22 | 2 | -999 | -999 | 115 | 139 | 11 | -24 | 13 | 3
712 | 17 | 4 | -999 | -999 | 63 | 19 | 3 | 44 | 9 | 3
713 | 4 | 2 | -999 | -999 | 51 | 21 | 36 | 30 | 37 | 3
714 | 8 | 2 | -999 | -999 | 351 | 64 | 36 | -73 | 36 | 3
715 | 5 | 3 | -999 | -999 | 29 | 4 | 7 | 25 | 11 | 3
717 | 7 | 1 | -999 | -999 | 60 | 54 | 24 | 6 | 25 | 3
718 | 34 | 3 | -999 | -999 | 170 | 192 | 3 | -22 | 9 | 3
720 | 17 | 2 | -999 | -999 | 38 | 176 | 15 | -138 | 17 | 3
722 | 5 | 2 | -999 | -999 | 45 | 23 | 74 | 22 | 74 | 3
723 | 39 | 61 | -999 | -999 | 170 | 156 | 63 | 14 | 64 | 3
727 | 1 | 2 | -999 | -999 | 148 | 345 | 20 | 163 | 21 | 3
728 | 10 | 12 | -999 | -999 | 156 | 241 | 24 | -85 | 25 | 3
743 | 9 | 23 | -999 | -999 | 340 | 36 | 20 | -56 | 22 | 3
746 | 10 | 2 | -999 | -999 | 144 | 200 | 30 | -56 | 32 | 3
748 | 92 | 34 | -999 | -999 | 195 | 184 | 1 | 11 | 8 | 3
750 | 8 | 3 | -999 | -999 | 343 | 16 | 8 | -33 | 11 | 3
762 | 29 | 18 | -999 | -999 | 104 | 31 | 3 | 73 | 9 | 3
764 | 5 | 2 | -999 | -999 | 144 | 14 | 30 | 130 | 31 | 3
773 | 17 | 3 | -999 | -999 | 208 | 199 | 7 | 9 | 11 | 3
779 | 8 | 1 | -999 | -999 | 137 | 157 | 28 | -20 | 29 | 3
780 | 16 | 2 | -999 | -999 | 210 | 183 | 6 | 27 | 10 | 3
782 | 21 | 3 | -999 | -999 | 95 | 12 | 4 | 83 | 9 | 3
784 | 18 | 7 | -999 | -999 | 358 | 35 | 24 | -37 | 25 | 3
789 | 3 | 2 | -999 | -999 | 340 | 291 | 16 | 49 | 18 | 3
790 | 12 | 4 | -999 | -999 | 169 | 81 | 21 | 88 | 23 | 3
792 | 4 | 2 | -999 | -999 | 354 | 270 | 6 | 84 | 10 | 3
793 | 23 | 38 | -999 | -999 | 1 | 266 | 23 | 95 | 25 | 3
795 | 50 | 25 | -999 | -999 | 356 | 331 | 15 | 25 | 17 | 3
797 | 21 | 2 | -999 | -999 | 278 | 308 | 5 | -30 | 9 | 3
807 | 11 | 4 | -999 | -999 | 11 | 347 | 18 | 24 | 19 | 3
809 | 6 | 4 | -999 | -999 | 81 | 78 | 6 | 3 | 10 | 3
826 | 12 | 1 | -999 | -999 | 226 | 88 | 7 | 138 | 11 | 3
828 | 12 | 2 | -999 | -999 | 99 | 74 | 4 | 25 | 9 | 3
834 | 15 | 2 | -999 | -999 | 174 | 225 | 42 | -51 | 43 | 3
838 | 13 | 2 | -999 | -999 | 61 | 133 | 22 | -72 | 24 | 3
840 | 10 | 2 | -999 | -999 | 65 | 64 | 4 | 2 | 9 | 3
851 | 91 | 5 | -999 | -999 | 231 | 214 | 1 | 17 | 8 | 3
852 | 47 | 9 | -999 | -999 | 325 | 190 | 3 | 134 | 8 | 3
856 | 8 | 11 | -999 | -999 | 39 | 67 | 27 | -28 | 29 | 3
857 | 27 | 9 | -999 | -999 | 271 | 266 | 27 | 5 | 28 | 3
860 | 50 | 10 | -999 | -999 | 241 | 195 | 3 | 46 | 8 | 3
863 | 24 | 3 | -999 | -999 | 102 | 245 | 5 | -143 | 9 | 3
864 | 29 | 3 | -999 | -999 | 67 | 197 | 5 | -130 | 9 | 3
865 | 106 | 6 | -999 | -999 | 116 | 279 | 9 | -163 | 12 | 3
869 | 16 | 4 | -999 | -999 | 170 | 48 | 5 | 122 | 9 | 3
873 | 40 | 8 | -999 | -999 | 216 | 4 | 3 | -148 | 8 | 3
876 | 26 | 4 | -999 | -999 | 259 | 195 | 10 | 64 | 12 | 3
880 | 34 | 6 | -999 | -999 | 40 | 196 | 25 | -156 | 26 | 3
884 | 16 | 3 | -999 | -999 | 218 | 225 | 11 | -7 | 14 | 3
885 | 26 | 3 | -999 | -999 | 78 | 182 | 6 | -104 | 10 | 3
887 | 24 | 1 | -999 | -999 | 123 | 121 | 9 | 2 | 12 | 3
892 | 10 | 2 | -999 | -999 | 57 | 51 | 7 | 6 | 10 | 3
895 | 18 | 3 | -999 | -999 | 229 | 214 | 6 | 15 | 10 | 3

Table 4: Summary of Principal Conclusions  Property | Value
---|---
Galactic O Star Catalog Findings
Overall Runaway Fraction | 22${}^{+3}_{-3}$
Multiplicity fraction among 25 runaways |  $>$28%
16th/50th/84th percentiles of $v_{\rm 2D}$ | 5.6/11.4/32.6 km s-1
Bowshock Nebula Central Star Findings
Runaway Fraction | 24${}^{+9}_{-7}$%
Multiplicity Fraction |  $>$36%$\pm$6%
16th/50th/84th percentiles of $v_{\rm 2D}$ | 6.5/14.6/32.6 km s-1

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