# Structural asymmetry along protein sequences
and co-translational folding

John M. McBride  Center for Soft and Living Matter, Institute for Basic
Science, Ulsan 44919, South Korea<EMAIL_ADDRESS><EMAIL_ADDRESS>Tsvi Tlusty  Center for Soft and Living Matter,
Institute for Basic Science, Ulsan 44919, South Korea  Departments of Physics
and Chemistry, Ulsan National Institute of Science and Technology, Ulsan
44919, South Korea<EMAIL_ADDRESS><EMAIL_ADDRESS>
## Abstract

Proteins are translated from the N- to the C-terminus, raising the basic
question of how this innate directionality affects their evolution. To explore
this question, we analyze $16,200$ structures from the protein data bank
(PDB). We find remarkable enrichment of $\alpha$-helices at the C terminus and
$\beta$-strands at the N terminus. Furthermore, this $\alpha\textrm{-}\beta$
asymmetry correlates with sequence length and contact order, both determinants
of folding rate, hinting at possible links to co-translational folding (CTF).
Hence, we propose the ‘slowest-first’ scheme, whereby protein sequences
evolved structural asymmetry to accelerate CTF: the slowest of the
cooperatively-folding segments are positioned near the N terminus so they have
more time to fold during translation. A phenomenological model predicts that
CTF can be accelerated by asymmetry, up to double the rate, when folding time
is commensurate with translation time; analysis of the PDB reveals that
structural asymmetry is indeed maximal in this regime. This correspondence is
greater in prokaryotes, which generally require faster protein production.
Altogether, this indicates that accelerating CTF is a substantial evolutionary
force whose interplay with stability and functionality is encoded in sequence
asymmetry.

## Introduction

All proteins are translated sequentially from the N- to the C-terminus, and
are thus inherently asymmetric [1]. One example of such N-to-C asymmetry is
signal peptides, which enable translocation across membranes, and are located
at the N terminus [2]. This raises the general question of whether and how
asymmetry in protein production is leveraged to gain evolutionary advantage.
Here we examine structural data from the protein data bank (PDB) in search of
traces of such adaptation. We analyzed the distribution of secondary structure
along the sequence for $16,200$ PDB proteins, finding two striking patterns of
asymmetry. First, disordered residues are principally located at the ends of
sequences, and depleted towards the middle. Second, $\beta$-strands are
enriched by $55\text{\,}\mathrm{\char 37\relax}\text{/}$ near the N terminus,
while $\alpha$-helices are enriched by $22\text{\,}\mathrm{\char
37\relax}\text{/}$ at the C terminus. These findings agree qualitatively with
previous reports [3, 4, 5, 6, 7, 8, 9, 10, 11]. This $\alpha\textrm{-}\beta$
asymmetry peaks at intermediate values of sequence length and contact order –
which both correlate negatively with folding rate – indicating a possible link
between secondary structure asymmetry and folding.

Hence, we further explore the possibility that $\alpha\textrm{-}\beta$
asymmetry may accelerate protein production, and is therefore a signature of
evolutionary adaptation. Production of functional proteins from mRNA comprises
two concerted processes: directional translation and cooperative folding. The
rate of translation is limited by trade-offs between speed, accuracy and
dissipation [12, 13, 14, 15]. Folding quickly has certain advantages: unfolded
proteins lead to aggregation, putting a significant burden on the cell [16,
17, 18]; faster folding allows quicker responses to environmental changes [19,
20]. Moreover, organisms whose fitness depends on fast self-reproduction would
benefit from accelerated protein production that can shorten division time
[21, 22]. Proteins may begin folding during translation, although the extent
to which this varies across the proteome is unclear [23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34], and depends on the collective dynamics of the folding
process. Thus, in principle, faster production times may be achieved if
proteins finish folding and translation at around the same time. This co-
translational folding (CTF) enables adaptations that increase yield and
kinetics of protein production [30, 31, 32, 35, 34]. For example, nascent
peptides interact with ribosomes and chaperones to reduce aggregation and
misfolding [36, 37, 38, 39, 40], while translation rates can be tuned to
facilitate correct folding [26, 41, 42, 43]. Specifically, we ask if
structural asymmetry may have evolved for fast and efficient production via
CTF.

Figure 1:  (A) Distribution of secondary structure along the sequence as a
function of distance from the N- and C-terminus, and (B) the structural
asymmetry – the ratio of the N and C distributions (in $\log_{2}$ scale; $\pm
1$ are 2:1 and 1:2 N/C ratios) – for all $16,200$ proteins (left), $4,702$
eukaryotic proteins (middle), and $10,966$ prokaryotic proteins (right).
Shading indicates bootstrapped $95$$\%$ confidence intervals in both A and B.
C-D: Mean $\textrm{asym}_{\alpha}$ (C) and $\textrm{asym}_{\beta}$ (D) as a
function of sequence length and contact order (Eq. 1). The data is split into
deciles and the bin edges are indicated on the axes. To reduce noise in the
figure due to undersampling, we only colour bins where there are 20 or more
proteins (full data is shown in SI Fig 1).

We show that the structural asymmetry observed in proteins is consistent with
a scheme for accelerating CTF based on the directional nature of translation
and the heterogeneity of folding rates along the sequence [44, 45, 46, 47, 48,
49] – e.g. cooperatively-folding protein segments containing $\beta$-sheets
may fold slower compared to those containing $\alpha$-helices [50]. In the
proposed slowest-first scheme, protein sequences take advantage of this
heterogeneity by evolving structural asymmetry: the slowest-folding segments
are enriched at the N-terminus [5, 6, 7, 8, 9, 10, 11], so that they are
translated first and have more time to fold. This scheme applies to proteins
composed of several independently-folding segments. Therefore, due to the
cooperative nature of the many-body folding process, the scheme is more likely
to apply to larger proteins.

A simple model predicts that, under the slowest-first scheme, production rate
can be almost doubled when folding time is equivalent to translation time. To
examine this hypothesis, we estimate the ratio of folding to translation time
of the PDB proteins and compare it with their $\alpha\textrm{-}\beta$
asymmetry, finding that asymmetry peaks when folding time is commensurate with
translation time. In this region, proteins are twice as likely to exhibit
$\alpha\textrm{-}\beta$ asymmetry that favours the slowest-first scheme. We
see more evidence for this scheme in prokaryotic proteins, which is consistent
with prokaryotes’ greater need for fast protein production due to more
frequent cell division. Taken together, these findings suggest that proteins
sequences have been adapted for accelerated CTF via structural asymmetry.

Figure 2:  Co-translational folding and the slowest-first mechanism. A: A
section of mRNA (red), encoding a protein segment capable of folding
cooperatively, is translated starting from the N terminal side (left). The
protein segment translocates through the ribosome channel (middle), and
undergoes folding once the full segment has been translated and is free from
steric constraints (right). The time from when the segment is about to undergo
translation, until the onset of folding, is labelled $\tau_{\textrm{onset}}$.
If this segment is a bottleneck for protein folding, the protein will fold
faster when this segment is located at the N terminus (top) instead of the C
terminus (bottom). B: Distribution of
$R=\tau_{\textrm{fold}}/\tau_{\textrm{trans}}$, the ratio of folding to
translation time (Eq. 3), for our entire sample, prokaryotic proteins, and
eukaryotic proteins. Solid lines are kernel density estimation fits to
histograms; dotted line indicates $R=1$; shading indicates bootstrapped
$95\text{\,}\mathrm{\char 37\relax}\text{/}$ confidence intervals. C:
Theoretical maximum speedup of production rate as a function of $R$ and
$\tau_{\textrm{onset}}$ (Eq. 4).

### Results

#### Protein secondary structure is asymmetric

Given the vectorial nature of protein translation, one may expect
corresponding asymmetries in protein structure. To probe this, we study a non-
redundant set of $16,200$ proteins from the Protein Data Bank (PDB) [51]. We
find that these PDB proteins exhibit significant asymmetry in secondary
structure (Fig. 1A-B), even when counting each SCOP family once (SI Fig 2).
For example, the first $20$ residues at the N terminus are on average
$55\text{\,}\mathrm{\char 37\relax}\text{/}$ more likely to form strands, and
the first $20$ residues at the C terminus are $22\text{\,}\mathrm{\char
37\relax}\text{/}$ more likely to form helices (in Fig. 1B this is reported in
$\log_{2}$ scale, but we report $\%$ here). This asymmetry is stronger for
prokaryotic proteins ($72\text{\,}\mathrm{\char 37\relax}\text{/}$;
$20\text{\,}\mathrm{\char 37\relax}\text{/}$) than for eukaryotic proteins
($20\text{\,}\mathrm{\char 37\relax}\text{/}$; $28\text{\,}\mathrm{\char
37\relax}\text{/}$). The substantial $\alpha\textrm{-}\beta$ asymmetry points
to an evolutionary driving force which we further investigate.

In both N- and C-termini, the $\alpha$-helix and $\beta$-strand distributions
have a well-defined shape: they exhibit periodicity in the positioning of
these elements along the sequence. This periodicity is matched by several
$\alpha\beta$-type protein folds where $\alpha$-helices and $\beta$-strands
are arranged in alternating order (SI Fig. 3). These folds tend to be more
abundant in prokaryotic proteins (SI Table 1); for example, ferredoxin-like
folds exhibit high $\alpha\textrm{-}\beta$ asymmetry, significant periodicity
at the N-terminus, and are $\sim 3$ times more common in prokaryotes.

The distribution of disordered regions exhibits a different pattern of
asymmetry: disordered residues are enriched at both ends of proteins compared
to the middle [3, 4]. Eukaryotic proteins are significantly more disordered,
where the probability of disorder is well approximated by $\sim D^{-0.5}$,
where $D$ is the distance from the end, while in prokaryotic proteins the
probability of disorder decays as $\sim D^{-1}$. Proteins also tend to be more
disordered at the N termini [3]: eukaryotic proteins are
$30\text{\,}\mathrm{\char 37\relax}\text{/}$ more likely to be disordered
within the first $100$ residues of the N terminus compared to the C terminus
(prokaryotes: $17\text{\,}\mathrm{\char 37\relax}\text{/}$). Although
prokaryotic proteins are less disordered than eukaryotic ones, the ratio of
the numbers of residues in $\beta$ strands and $\alpha$ helices is the same.

#### Structural asymmetry correlates with sequence length and contact order

To better understand the $\alpha\textrm{-}\beta$ asymmetry, we examined
correlations with sequence length, $L$, and contact order, CO. CO is the
average sequence distance between intra-protein contacts,

$\textrm{CO}=\big{\langle}\lvert j-i\rvert\big{\rangle}~{},$ (1)

where $i$ and $j$ are pairs of residue indices for each contact [52]. High CO
is likely to result in greater entropy loss between the unfolded and
transition state, thus increasing folding time.

To quantify secondary structure asymmetry, we calculate the magnitude of
asymmetry normalized by length,

$\displaystyle\textrm{asym}_{\alpha}=$
$\displaystyle\left(N_{\alpha}-C_{\alpha}\right)/L~{},$ (2)
$\displaystyle\textrm{asym}_{\beta}=$
$\displaystyle\left(N_{\beta}-C_{\beta}\right)/L~{},$

where $N_{\alpha}$ ($N_{\beta}$) and $C_{\alpha}$ ($C_{\beta}$) are the number
of residues in a $\alpha$-helices ($\beta$-strands) in the N and C halves of a
protein sequence.

Figure 3:  A: $\alpha\textrm{-}\beta$ asymmetry distributions as a function of
$R$, the folding/translation time ratio (Eq. 3). Proteins are divided into
deciles according to $R$; bin edges are shown on the y-axis. B: N terminal
enrichment – the degree to which strands/helices are enriched in the N over
the C terminus (Eq. 6) – is shown for the deciles given in B. C: N terminal
enrichment as a function of $R$ for $4,702$ eukaryotic proteins and $10,966$
prokaryotic proteins. Proteins are divided into bins according to $R$; bin
edges, shown on the y-axis, are the same as in A-B. Whiskers indicate
bootstrapped $95\%$ confidence intervals.

We find that $\alpha\textrm{-}\beta$ asymmetry is a non-monotonic function of
both $L$ and CO (Fig. 1C-D; this is to some extent expected since $L$ and CO
are correlated, $r=0.65$). In particular, there is a region of intermediate
length ($179-416$) and intermediate CO ($21-35$) where structural asymmetry is
most apparent (SI Fig. 4). The fact that both quantities correlate negatively
with folding rate ($L$, $r=-0.68$; CO, $r=-0.64$; SI Fig. 5) [53, 52, 54, 55],
taken together with proteins’ inherent asymmetry due to vectorial translation,
leads us to suspect that the origins of this $\alpha\textrm{-}\beta$ asymmetry
may be related to co-translational folding.

#### Co-translational folding appears to be widespread

During protein production, the ribosome advances along the mRNA from the N to
the C terminus (Fig. 2A). Proteins may fold in stages during translation [24,
56, 28, 27, 57, 25, 58, 59, 60, 61, 26], but the extent to which this happens
in still unclear [33, 62, 57, 63, 64, 65, 37, 66]. In principle, one way to
maximise the rate of production and to minimise aggregation is by making
proteins fold faster than they are translated, or at a similar rate. We can
obtain a rough approximation of how often this occurs by estimating folding
rates and translation rates of proteins. We estimate the folding rate
$k_{\mathrm{fold}}$ using a power law scaling with length (not CO) fitted to
data from the protein folding kinetics database (PFDB) [54] (Methods). We
assume an average translation rate $k_{\mathrm{trans}}$ that depends on the
organism. Thus we can estimate the ratio $R$ of folding time
$\tau_{\textrm{fold}}$ to translation time $\tau_{\textrm{trans}}$,

$R=\frac{\tau_{\textrm{fold}}}{\tau_{\textrm{trans}}}=\frac{1/k_{\mathrm{fold}}}{L/k_{\mathrm{trans}}}~{}.$
(3)

The estimated $R$ distribution exhibits a peak in the region of commensurate
times $R\approx 1$ (Fig. 2B). For the $68\text{\,}\mathrm{\char
37\relax}\text{/}$ of proteins (CI $5388\text{\,}\mathrm{\char
37\relax}\text{/}$, SI Fig. 6) that lie in the region $R\leq 1$, folding may
be quicker than translation, suggesting that co-translational folding (CTF) is
common. In comparison, a more rigorous method estimated that in
$37\text{\,}\mathrm{\char 37\relax}\text{/}$ of proteins in E. coli, at least
one domain will fully fold before translation finishes [67]. Examining
prokaryotic proteins and eukaryotic proteins separately reveals a sharper peak
in the $R$ distribution for prokaryotic proteins in the region of commensurate
folding and translation times, $1/10<R<10$. Notably, a greater fraction of
prokaryotic proteins ($56\text{\,}\mathrm{\char 37\relax}\text{/}$) are in
this regime compared to eukaryotic proteins ($41\text{\,}\mathrm{\char
37\relax}\text{/}$).

#### Folding rate asymmetry can speed up co-translational folding

Fig. 2A shows two scenarios where the folding rate $\tau_{\textrm{fold}}$ is
determined by a rate-limiting fold (red segment) [68, 49, 69, 66]. This
bottleneck represents an independent, cooperatively-folding segment (i.e. a
‘foldon’ [47]) that folds slowly. If the bottleneck is located at the N
terminus (top in Fig. 2A), then the production time is minimal,
$\tau_{\textrm{min}}=\max(\tau_{\textrm{fold}}+\tau_{\textrm{onset}},\tau_{\textrm{trans}}),$
where $\tau_{\textrm{onset}}$ is the time it takes for the segment to be
translated, pass through the ${\sim}10$ nm long ribosome tunnel and be free of
steric constraints [30, 55, 56, 70]. In the other extreme, if the rate-
limiting fold includes the C terminus (bottom in Fig. 2A), production time is
maximized [71],
$\tau_{\textrm{max}}=\tau_{\textrm{fold}}+\tau_{\textrm{trans}}.$ In this
case, the last element can escape the ribosome quickly after being translated
since it is not delayed by downstream translation [72]. Thus, production rate
can be accelerated by a factor,

$\textrm{speedup}=\frac{\tau_{\textrm{max}}}{\tau_{\textrm{min}}}~{}.$ (4)

In the limit $\tau_{\textrm{onset}}\ll\tau_{\textrm{trans}}$, one finds from
Eqs. 3-4 that the speedup as a function of
$R=\tau_{\textrm{fold}}/\tau_{\textrm{trans}}$ (Fig. 2D) is

$\textrm{speedup}=1+e^{-\lvert\ln{R}\rvert}~{}.$ (5)

A maximal, twofold speedup is achieved when translation time equals folding
time, $R=1$, and taking $\tau_{\textrm{onset}}>0$ shifts this maximum towards
$R<1$.

In practice, $\tau_{\textrm{onset}}$ depends on the size of the cooperatively-
folding protein segment. It has been suggested that such a ‘foldon’ can have
as few as 20 residues [47, 73], but in general, cooperatively-folding units
appear to be larger than this – proteins that are observed to fold via two-
state kinetics can vary greatly in size [74, 75, 76]. Thus, this speedup may
only be applicable to larger proteins that fold via multi-state kinetics; this
is compatible with the prediction that a speedup is substantial only when
$\tau_{\textrm{trans}}\approx\tau_{\textrm{fold}}$.

#### Structural asymmetry is maximum for commensurate folding and translation
times

The speedup curve (Fig. 2D) implies that proteins can benefit the most from
structural asymmetry when $R=\tau_{\textrm{fold}}/\tau_{\textrm{trans}}\approx
1$. Hence, we estimate the magnitude of $\alpha\textrm{-}\beta$ asymmetry as a
function of $R$, and plot the distributions in Fig. 3A. We note that $R$ is
not estimated very precisely, and thus it is useful to average out the errors
by examining the proteins in large bins (deciles). At intermediate $R$, the
means of the distributions shift away from zero, indicative of strong bias.

To capture the magnitude of these shifts we calculate the N terminal
enrichment, $E$, defined as the fraction of proteins with positive asymmetry
(i.e. enriched at the N terminus) minus the fraction of proteins with negative
asymmetry (enriched at the C terminus), for both helices and strands:

$\displaystyle E_{\alpha}=$ $\displaystyle
P(\textrm{asym}_{\alpha}>0)-P(\textrm{asym}_{\alpha}<0),$ (6) $\displaystyle
E_{\beta}=$ $\displaystyle
P(\textrm{asym}_{\beta}>0)-P(\textrm{asym}_{\beta}<0).$

Fig. 3B shows that in the $R$ deciles with maximum asymmetry, proteins in the
PDB are $2.0$ times as likely to be enriched in $\beta$-strands in the N
terminus, while $\alpha$-helices are $1.9$ times more likely to be enriched in
the C-terminal half. This maximum is found when $-0.6\leq\log_{10}R\leq 0.1$
(the $95\text{\,}\mathrm{\char 37\relax}\text{/}$ confidence intervals for the
$k_{\mathrm{fold}}$ estimate give $-1.1\leq\log_{10}R\leq 0.7$; SI Fig. 7).
This region of maximal asymmetry overlaps with the region of maximal speedup
(Fig. 2D, Eq. 5), suggesting that asymmetry evolves because it enhances CTF.
To help put these figures in context, in the pentile where proteins have
maximum asymmetry ($-0.6\leq\log_{10}R\leq 0.1$), $11\%$ of proteins have
$\textrm{asym}_{\beta}\geq 0.05$ (which, for the mean length, $L=303$, and an
average $\beta$-strand length of $5$ residues, corresponds to about three
extra $\beta$-strands near the N terminus) while half as many have
$\textrm{asym}_{\beta}\leq-0.05$ (the opposite case) (SI Fig 8).

#### Prokaryotes exhibit greater asymmetry than Eukaryotes

We looked at $\alpha\textrm{-}\beta$ asymmetry for prokaryotic and eukaryotic
proteins separately, finding that when asymmetry is maximum, prokaryotes
exhibit more asymmetry than eukaryotes – strands are $36\text{\,}\mathrm{\char
37\relax}\text{/}$ more likely to be enriched at the N terminus in prokaryotes
compared to eukaryotes (Fig. 3C). Typically, prokaryotic cells divide more
frequently than eukaryotic cells [22], and thus have a greater need for fast
production of functional proteins. The analysis is therefore consistent with
the slowest-first scheme that implies that the stronger pressure on
prokaryotes should lead to greater asymmetry.

Figure 4:  A: Multi-domain proteins can be optimized via asymmetry between
domains, and/or within domains. B-C: N terminal enrichment within domains as a
function of $R$ for single-domain proteins (B: $14,442$ domains) and multi-
domain proteins (C: $23,832$ domains). Domains are split into deciles based on
$R$, and the bin edges are shown on the y-axis; whiskers indicate bootstrapped
$95\%$ confidence intervals. D-E: Domain sequence length and contact order
distributions for proteins with different numbers of domains.

#### Multi-domain proteins are optimized for CTF via distinct mechanisms

Multi-domain proteins can be potentially adapted at two levels: within
domains, and between domains (Fig. 4A). To test this, we isolated individual
domains in the PDB (using Pfam) [77], and calculated CO and
$\alpha\textrm{-}\beta$ asymmetry for each domain as in Fig. 3. While intra-
domain optimization of secondary structure clearly occurs within single-domain
proteins, it is much weaker within multi-domain proteins (Fig. 4B-C). Inter-
domain optimization entails ordering the slowest-folding domains at the N
terminus, for which we find no significant bias (SI Fig. 9). Instead, we find
that as the number of domains increases, the size and CO of individual domains
decreases (Fig. 4D-E). Thus CTF is maintained in multi-domain proteins mostly
by using faster-folding domains throughout.

### Discussion

We examined the hypothesis that proteins are selected for CTF to hasten
protein production and reduce aggregation/misfolding, but this may not be
equally true for all proteins. As an example, we showed that in prokaryotes,
which have a greater burden of cell growth, proteins tend to have more
asymmetry than in eukaryotes. Topological constraints may preclude early
folding of the N terminus if the C-terminus is also part of the folding
nucleus [66]. More generally, CTF may be hindered in some proteins by
interactions with the ribosome [57]. Long-lived proteins [78] may derive
little benefit from an increase in production speed. On the other hand,
proteins produced in large quantities need to fold quickly as aggregation can
increase non-linearly with concentration [79]. Some of these predictions can
be tested when sufficient data for protein lifespan [80], expression levels
[81], and structure become available. While we showed that
$\alpha\textrm{-}\beta$ asymmetry is apparent in a broad set of proteins,
further analysis of an extended data set may be able to detect the sub-classes
of proteins that will benefit most from $\alpha\textrm{-}\beta$ asymmetry.

Our model proposes an absolute maximum of a twofold acceleration in protein
production, which at first glance, may seem insignificant compared to the
range of protein folding times ($\sim 9$ orders of magnitude). However, we
show that this acceleration is only possible when folding time is similar to
translation time (i.e. on the order of seconds to minutes). We expect that,
even for a single protein, decreasing production time by seconds to minutes
can increase fitness, and when this is applied over a large group of proteins
this should be quite substantial.

To experimentally test the slowest-first mechanism, we suggest studying CTF of
multiple proteins with $R\approx 1$, which differ in $\textrm{asym}_{\alpha}$
and $\textrm{asym}_{\beta}$. In particular, we propose to use proteins whose
sequences are related by circular permutation, while having identical
structures [82, 83, 84, 65]. Circular permutants with opposite structural
asymmetry, as the example in Fig. 5, should fold at significantly different
rates. Additional experimental control of $R$ is possible via synonymous codon
mutations [85] or in vitro expression systems [29]. Thus, one can test whether
asymmetry in secondary structure can lead to acceleration of CTF, and how this
depends on $R$.

Figure 5:  Secondary structure for nuclear transport factor 2 H66A mutant
(PDB: 1ASK [86]) and a circular permutant, 1ASK-CP67, which may fold faster
during translation.

#### CTF for multi-domain proteins is more complex

Multi-domain proteins exhibit less asymmetry than single-domain proteins. Due
to interactions between domains [87, 56, 40, 70, 42], optimization via
asymmetry may not be feasible — instead, a safe strategy is to fold each
domain before translating subsequent domains. To explain the lack of intra-
domain $\alpha\textrm{-}\beta$ asymmetry (Fig. 4C), we propose a simple
mechanical argument. When a $\beta$-sheet forms, the protein chain contracts.
This results in a pulling force on both the ribosome [88, 89], and on any
upstream domains. This extra resistance to $\beta$-sheet formation may
preclude the early formation of $\beta$-strands at the N terminal side of a
domain. If this is true, then the domain in position 1 should still exhibit
$\alpha\textrm{-}\beta$ asymmetry; we currently lack sufficient statistical
power to conclusively test this (SI Fig. 10). Further tests could look at CTF
of a $\beta$-rich domain in the the presence or absence of an upstream domain
[90, 91].

#### Disorder is enriched at both sequence ends

The N and C termini principally share a notable tendency for disorder near the
end, which suggests that they are affected by the same physical end effect.
The amino acid at the end is linked to the chain by only one peptide bond,
leaving it more configurational freedom than an amino acid in the centre of
the protein, which is constrained by two bonds. This entropic contribution to
the free energy of the loose ends, of order $k_{B}T$, can induce disorder in
marginally stable structures.

Since disordered regions do not need time to fold, placing them towards the
C-terminus gives the other residues more time to fold. Yet, we find a similar,
slightly stronger, tendency for disorder near the N-terminus (green curves in
Fig. 1B), particularly in eukaryotes. This may result from other determinants
of protein evolution; e.g., disordered regions tend to interact with some
ribosome-associating chaperones [92, 93]. If disorder at the N terminus is
related to chaperones, we expect that asymmetry will be higher for slow-
folding proteins as they are more prone to aggregation. We find that bias for
disorder at the N-terminus is strongest for slow-folding proteins (high $R$,
$L$ and CO; SI Fig.11), but only for prokaryotes, not eukaryotes. Given the
absence of a correlation between $R$, $L$ and CO and disorder asymmetry in
eukaryotic proteins, the question of why eukaryotic proteins are more
disordered at the N terminus remains open.

#### Considering tertiary structure

We used secondary structure as a proxy for folding rate, but there are also
contributions from tertiary structure. To evaluate this assumption, we ran
coarse-grained simulations of CTF of three structurally-asymmetric proteins
while varying $R$, for both the original sequence and of the reverse sequence.
We find that these proteins fold faster when $\beta$-strands are translated
first, in the relevant region of $R\sim 1$ (SI Fig. 12).

We also studied the effect of tertiary structure by looking at asymmetry in
surface accessibility. $\beta$ strands at the N terminus are less likely to be
exposed to solvent than $\beta$ strands at the C terminus; this bias is
stronger for prokaryotic proteins (first 20 residues:
$41\text{\,}\mathrm{\char 37\relax}\text{/}$) compared to eukaryotic proteins
($13\text{\,}\mathrm{\char 37\relax}\text{/}$) (SI Fig 12). Since solvent-
exposed $\beta$-strands are less likely to form part of a folding nucleus
[94], this suggests that $\beta$-strands at the N terminus are more likely to
nucleate folding compared to those at the C terminus.

#### Alternatives explanations for asymmetry

We can think of only few examples of inherent asymmetry in proteins. The ends
have a slight chemical difference (carbon vs nitrogen), but one atom seems
insignificant. Amino acids are chiral, but we cannot say how this could lead
to asymmetry in secondary structure. The only remaining driving force for
asymmetry, in our view, is the unidrectional translation process.

One theory exists that predicts asymmetry in $\alpha$-helices: Marenduzzo et
al. show that in a model of a growing, self-interacting string, the string
forms helices at the growing (C) end [7], and suggest that this bias affected
protein evolution. However, this fails to address why there are more beta
sheets at the N terminal, nor does it explain why asymmetry depends on protein
length.

We could also consider that $\beta$-strands are typically less open to
solvent, and hence we also see asymmetry in solvent accessibility (SI Fig 12).
Thus, it is possible that $\beta$-strand asymmetry is a by-product of a
driving force for having the N terminal side less exposed to solvent. The only
explanation we can think of for such a driving force is to have the folding
nucleus located at the N terminus, which is effectively covered by the
‘slowest-first’ hypothesis since a folding nucleus can be considered a rate-
limiting fold.

#### We need more data on folding rates

The data used to fit Eq. 7 are sparse ($122$ proteins), biased towards small,
single-domain proteins, and typically obtained from in vitro refolding
experiments [54]. To test whether our conclusions are robust to sampling, we
estimate confidence intervals using bootstrapping with sample sizes equal to
the original sample size, and half that amount; we perform this test on both
the reduced version of the PFDB data set used in the main figures, and on a
second version of the PFDB data set (Methods; SI Fig. 7). In addition, we
calculate the main results using using a different protein folding data set,
ACPro [95], which partially overlaps with PFDB, but includes larger proteins
(SI Fig. 14). In all of the above analyses, the point of maximum asymmetry is
found to be $1/100<R<100$, which corresponds to the region where CTF speed-up
is possible. However, to fully overcome the aforementioned limitations,
further experiments are needed.

#### Analysis is consistent with hypothesis that proteins are selected for CTF
via secondary structure

To sum, in the proposed the slowest-first mechanism, CTF can be accelerated by
positioning the slowest-folding parts of a protein near the N terminus so that
they have more time to fold. A survey of the PDB shows that the estimated
acceleration correlates with asymmetry in secondary structure. In particular,
the rate of production can be almost doubled when translation time is similar
to folding time, and indeed these proteins exhibit the maximal asymmetry in
secondary structure distribution. Altogether, there appears to be substantial
evolutionary selection, manifested in sequence asymmetry, for proteins that
can fold during translation.

## Methods

#### Data

We extracted a set of $16,200$ proteins from the Protein Data Bank (PDB) [51].
We include proteins where the SEQRES records exactly match the corresponding
Uniprot sequence (not mutated, spliced, or truncated) [96]. We include
proteins that have been fused or have purification tags, but for these we
again only include the part that exactly matches the Uniprot sequence. For
each unique protein sequence, we only include the most recent structure. We
used SIFTS to map PDB and Uniprot entries [97]. We exclude proteins with
predicted signal peptides as little is known about whether such proteins
undergo CTF; we used Signal-P5.0 to identify signal peptides [98]. Using the
above criteria we extracted a set of $38,274$ domains by matching PDB entries
to Pfam domains [77]. We assume that domains identified in Pfam via homology
are independent folding units. $\alpha$ helix and $\beta$-strands are
identified through annotations in the PDB; disorder is inferred from residues
with missing coordinates. To calculate contact order, we only consider
contacts between residues where $\alpha$ carbons are within
$10\text{\,}\mathrm{\SIUnitSymbolAngstrom}\text{/}$; we confirm that the
correlation in Fig. 1D is robust to choice of this cutoff (SI Fig. 15).

We use the protein folding kinetics database (PFDB) for estimating folding
rates [54]. For our main results we only used entries with realistic physical
conditions ( $5<\textrm{pH}<8$, and
$$20\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$<T<$40\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$$)
and ignored folding rates which had been extrapolated to
$T=$25\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$$; in total, $122$
proteins. We test a second version of the PFDB data set without excluding
proteins, and using folding rates which were extrapolated to
$T=$25\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$$; $141$ proteins. We
also use the ACPro data set [95] to test the robustness of our conclusions;
$125$ proteins.

#### Predicting folding and translation rate

The folding rate, $k_{\mathrm{fold}}$ (in units of 1/sec), is estimated by a
power-law fit as a function of the protein’s length:

$\log_{10}k_{\mathrm{fold}}=A+B\,\log_{10}L~{},$ (7)

where $L$ is sequence length in residues; $A$ and $B$ are free parameters. We
fit these parameters using data from the PFDB [54] to get
$95\text{\,}\mathrm{\char 37\relax}\text{/}$ confidence intervals of
$A=13.8\pm 2.1$ and $B=-6.1\pm 1.2$ (with correlation coefficient $r=-0.68$,
and p-value $p<0.005$). We consider that proteins that are observed to fold
via multi-state kinetics may be badly approximated by a single folding rate;
we thus repeat the analyses with the 89 two-state proteins in the PFDB,
finding the area of maximum asymmetry within $-1.1<\log_{10}R<1.5$. The
estimate from Eq. 7 is limited for the following reasons: (i) It is extracted
from a small set of 122 proteins. (ii) It disregards the effects of secondary
structure, contact order order, and other important determinants. (iii) The
data is from in vitro measurements. (vi) The data is biased towards small,
single-domain proteins. Thus, it is only a rough predictor for the folding
rates of individual proteins in the set, as the standard deviation between
estimated and empirical folding rates is $1.22$. For all these reasons, we use
Eq. 7 as an estimator of the average folding rate of sets of proteins of
similar length $L$ where the large sampling size of each bin is expected to
reduce the errors as ${\sim}N^{-1/2}$.

We tested whether the predicted folding rates of proteins in the PDB are
within certain approximate bounds on realistic folding rates. A lower bound to
folding time has been estimated at ${\sim}L/100$
$\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ [99], while we take the
doubling time of E. coli, roughly $20$ minutes, as an approximate upper bound.
Of course, many proteins rely on chaperones, so their bare estimated folding
time may be longer than the upper bound, while others come from organisms with
much longer doubling times. Even so, according to Eq. 7 only
$8\text{\,}\mathrm{\char 37\relax}\text{/}$ of proteins are estimated to have
a folding time greater than $20$ minutes, while only $7\text{\,}\mathrm{\char
37\relax}\text{/}$ of proteins are estimated to fold faster than the lower
bound. Given the magnitude of the error in estimating the folding time of
individual proteins, Eq. 7 appears to yield estimates that are mostly within
the biologically reasonable regime. Furthermore, in estimating the folding
rate of large proteins, a common assumption is that they consist of multiple
independently-folding domains [100] – which considerably reduces the estimated
folding time of the slowest proteins – but we neglect to make this assumption.

In principle, we could have used structural/topological measures (such as
contact order, long-range order, etc. [101]) to slightly improve the fit to
Eq. 7. However these typically involve numerous methodological choices and
additional parameters [102], and the scaling relations are entirely empirical.
In contrast, scaling of folding time with length has a robust theoretical
background [103, 104, 105, 106, 107, 53, 108, 109]; the exact form of of the
scaling is debated, but a power law is favoured slightly [104, 110].

We assume the translation rate, $k_{\mathrm{trans}}$, depends on the organism
(host organism for viral proteins), such that $k_{\mathrm{trans}}$ is $5$
amino acids per second for eukaryotes and $10$ for prokaryotes.

### Data Availability

The non-redundant sets of proteins and domains, along with the data used in
the figures and Supplementary Information, will be made available on Zenodo.

### Code Availability

Simulation and analysis code, along with code used to make all figures, are
accessible at https://github.com/jomimc/FoldAsymCode.

### Acknowledgements

We acknowledge Albert J. Libchaber for stimulating discussions and comments on
the manuscript. This work was supported by the Institute for Basic Science,
Project Code IBS-R020-D1.

### Competing Interests

The authors declare that they have no competing financial interests.

### Correspondence

Correspondence and requests for materials should be addressed to J.M. (email:
<EMAIL_ADDRESS>and T.T. (tsvitlusty@gmail.com).

### Author Contributions

J.M. and T.T. designed research; J.M. performed research; J.M. analyzed data;
J.M. and T.T. wrote the paper.

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