Protein escape at the ribosomal exit tunnel: Effects of native interactions, tunnel length and macromolecular crowding
Phuong Thuy Bui Trinh Xuan Hoang





Duy Tan University, 254 Nguyen Van Linh, Thanh Khe, Da Nang, Viet Nam


hoang@iop.vast.vn
Institute of Physics, Vietnam Academy of Science and Technology,
10 Dao Tan, Ba Dinh, Hanoi, Vietnam
Graduate University of Science and Technology, Vietnam Academy of
Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam




How fast a post-translational nascent protein escapes from the ribosomal
exit tunnel is relevant to its folding and protection against aggregation.
Here, by using Langevin molecular dynamics, we show that non-local native
	interactions help decreasing the escape time, and foldable proteins
	generally escape much
	faster than same-length self-repulsive homopolymers at low
	temperatures. The escape process, however, is slowed
	down by the local interactions that stabilize the α-helices.	
The escape time is found to increase with both the tunnel length and the
concentration of macromolecular crowders outside the tunnel. 
We show that a simple diffusion model described by the Smoluchowski equation
with an effective linear potential can be used to map out the escape time
distribution for various tunnel lengths and various crowder concentrations. The
consistency between the simulation data and the diffusion model however is
found only for the tunnel length smaller than a cross-over length of 
90 Å to 110 Å, above which the escape time increases much faster with
the tunnel length. It is suggested that the length of ribosomal exit
tunnel has been selected by evolution to facilitate both the efficient folding
and efficient escape of single domain proteins. We show that macromolecular
crowders lead to an increase of the escape time, and attractive crowders
are unfavorable for the folding of nascent polypeptide.


Protein escape at the ribosomal exit tunnel: Effects of native
interactions,  tunnel length and macromolecular crowding
    Trinh Xuan Hoang
    December 30, 2023
=======================================================================================================================





§ INTRODUCTION


Partially folded protein conformations can be found at the ribosome during
protein translation and after translation before the full release of a nascent
protein from the ribosomal exit tunnel. The folding during translation, namely
cotranslational folding, occurs when a nascent polypeptide chain undergoes
elongation due to biosynthesis (for reviews, see e.g.
Refs.<cit.>), while the
post-translational folding is associated with a full-length protein. In both
cases, the behavior of the nascent polypeptide is strongly influenced by the
ribosome, especially by the ribosomal exit tunnel through which the nascent
chain traverses to the cytosol or to another cellular compartment.
The length of the ribosomal exit tunnel spans from 80 Å to 100 Å,
depending on where the exit end is defined, whereas its width varies between
10 Å and 20 Å <cit.>. Such a geometry allows for the formation
of an α-helix or a β-hairpin inside the tunnel <cit.>,
and also promotes the α-helix formation <cit.>, but would
hardly accommodate even a small tertiary structure <cit.>.  The
tunnel was also suggested to have a recognitive
function leading to a translation arrest of certain amino acid sequences, such
as the SecM sequence <cit.>. Cotranslational folding has been
characterized with vectorial folding <cit.>,
i.e. the folding events proceed from the N- to the C-terminus; the
non-equilibrium effect of a growing chain <cit.>; and the impact
of the varying codon-dependent translation rate
<cit.>. Furthermore, folding of 
nascent proteins is assisted by the action of ribosome-associated
molecular chaperones <cit.>. All these effects are
indicative of a highly conditional and coordinated folding of
nascent protein at the ribosome, which is clearly different from refolding
<cit.> of a denatured protein in aqueous solvent. There have been
experiments <cit.> as well as simulations
<cit.> showing that the folding efficiency of
proteins is improved under biosynthesis conditions. It was also suggested that
the impact of cotranslational folding is evolutionarily imprinted on the
protein native states, as seen with an increased helix propensity
<cit.> and a decreased compactness <cit.> of the
chain near the C-terminus in the statistical analyses of protein 
structures from the protein data bank (PDB). 

While cotranslational folding is progressively understood, little is known
about post-translational folding at the ribosome. The latter is considered
to take place after the protein C-terminus is released from the peptidyl
transferate center (PTC), where the peptide bonds are formed. Certainly,
protein must escape from the ribosomal tunnel to fully acquire the native
conformation. A too slow escape would decrease the productivity of the
ribosome, while a too fast escape would make the nascent protein vulnerable to
aggregation <cit.>, as the partially folded protein may still have a
large exposure of hydrophobic segments.
In a recent study <cit.>, by using MD simulations, we have
shown that post-translational folding at the exit tunnel is concomitant with
the escape process and that the tunnel induces a vectorial folding of the
full-length protein. Such a folding has a greatly reduced number of pathways
and leads to an improved folding efficiency. Interestingly, it has been also
shown <cit.> that the escape time distribution of protein can be
captured by a simple one-dimensional diffusion model of a particle in a
linear potential field with an exact solution of the Smoluchowski equation.

The purpose of the present study is to explore the protein escape at the 
ribosomal exit tunnel with a focus on several effects, namely the role of
native interactions, the impact of tunnel length and the influence of
macromolecular crowders <cit.>. We use the same approach as
given in our previous work <cit.>, that is to consider simple
coarse-grained models for the protein, the exit tunnel and the crowders, which
enable multiple simulations of protein growth and escape by using the Langevin
equation. The diffusion model for protein escape previously
introduced <cit.> is improved in this study by considering an
absorbing boundary
condition. We find that escape time reflects well the changes in the system
properties, such as the native contact map, the tunnel length and the
crowder concentration with a remarkable consistency between simulations
and the theoretical diffusion model. Interestingly, the dependence of 
the escape time on the tunnel length suggests an explanation for the observed
length of the exit tunnel in real ribosomes. Our results obtained with
attractive crowders provide an insight into the effect of ribosome-associated
chaperones on the escape and folding of nascent proteins at the ribosome. 




§ METHODS




 §.§ Models of nascent protein, ribosomal exit tunnel and macromolecular
crowders


As a nascent protein we will focus on the B1 domain of protein G of length
N=56 amino acids with the PDB code of 1pga, denoted as GB1. The protein is 
considered in a Go-like model
<cit.>, in which each amino acid
is considered a single bead centered at the position of the
C_α atom. We adopt the same Go-like model as given in our previous
work <cit.> except that with a 10-12 Lennard-Jones (LJ) potential
for native contact interactions. In addition, we consider three types of native
contact maps, denoted as C1, C2 and C3, for the model.
The C1 map is defined by a cut-off distance of 7.5 Å
between the C_α atoms in the native conformation.  The C2 and C3 maps
are obtained based on an all-atom consideration <cit.> of the
protein PDB structure: contact between two amino acids is identified if there
are least two non-hydrogen atoms belonging to the two amino acids, found at a
distance shorter than λ times the sum of their atomic van der Waals
radii. C2 map has λ=1.27 whereas C3 has λ=1.5.  The choice of
λ=1.27 is such that the C2 map has the same number of native contacts
and the C1 for the GB1 protein.  The interaction
between a pair of amino acids forming a native contact takes the form of a
10-12 LJ potential <cit.>

    V(r_ij) = ϵ[5(r_ij^*/r_ij)^12 - 
    6(r_ij^*/r_ij)^10]  ,

where ϵ is an energy unit in the system corresponding to the strength
of the LJ potential, r_ij is the distance between 
residues i and j, and r_ij^* is the corresponding distance in the
native state. The use of 10-12 LJ potential makes the folding transition
more cooperative <cit.> than the 6–12 LJ potential (used in 
previous work <cit.>) as indicated by the height of the specific heat
peak (see Fig. S1 of the supplementary material).
Additionally, we will consider also a number of small single-domain
proteins to study the effects of native interactions and the tunnel length
on the escape process.




The ribosomal tunnel is modeled as a hollow cylinder of repulsive walls with
diameter d=15 Å and length L (Fig.  <ref>). It has been shown
<cit.> that this diameter allows for the formation of an
α-helix and a β-hairpin inside the tunnel but not tertiary
structures. In the present study, L is allowed to changed between 0 and
140 Å. The cylinder
has one of its circular bases open and attached to a repulsive flat wall
mimicking the ribosome's outer surface.  Macromolecular crowders are modeled as
soft spheres of radius R=10 Å (R is chosen
approximately equal to the radius of gyration of GB1, R_g=10.2 Å).
Assume that the x axis is the tunnel axis, the crowders are confined between
the ribosome's wall and another wall parallel to it at a distance l=100 Å
along the x direction. Periodic boundary conditions are applied for the y
and z directions with a box size equal to l. The crowders' volume fraction
is given by ϕ=M (4π/3) R^3/l^3, with M the number of crowders.




The interactions between an amino acid and a wall, between a crowder and a
wall, and between two crowders are all repulsive and given in the form of a
shifted and truncated LJ potential

    V_rep(r) = {[ 4 ϵ[(σ/r)^12 - (σ/r)^6 ]
     + ϵ               ,    r ≤ 2^1/6σ;                             0               ,    r > 2^1/6σ ].  ,

where σ=5 Å is a characteristic length equal to the typical diameter
of an amino acid; r is the distance between an amino acid and the nearest
virtual residue <cit.> of diameter σ, embedded under the
surface a wall or a crowder, or between two such virtual residues (see
Fig. <ref> for the definition of virtual residue).

Additionally, we will consider also a case in which the crowders are weakly
attractive to amino acids with the interaction given by the 6–12 LJ potential

    V_att(r) = 4 ϵ_1 [(σ/r)^12 - (σ/r)^6 ]  
     ,

with energy parameter ϵ_1 < ϵ.




The motions of the protein and the crowders are simulated by using the Langevin
equations <cit.>. One assumes that all amino acids have
the same mass, m, while the crowders has a molecular mass m_c equal to the
mass of the protein, i.e. m_c = N m.  Similarly, the friction coefficient of
amino acid is ζ_a, whereas that of crowder is ζ_c = N ζ_a. The
Langevin equations are integrated by using a Verlet algorithm introduced in
Ref. <cit.> with time step Δ t = 0.002 τ, where τ =
√(m σ^2/ϵ) is the time unit in the system. In the simulations
we use ζ_a = 5   m τ^-1, for which the dynamics of the system are
in the overdamped limit <cit.>.  Fig. <ref> shows that
the employed dynamics lead to the same diffusion characteristics for a folded
protein and for crowders in the solution. These characteristics are also close
to that of a Brownian motion for times larger than τ. 

Following previous work <cit.>, the nascent protein is grown inside
the tunnel at the PTC from the N terminus to the C terminus with the growth
time per amino acid t_g = 100 τ. This growth speed is sufficiently slow
to produce fully translated conformations of similar structural characteristics
to those obtained by a much slower growth speed <cit.>. The escape
time is measured from the moment the protein has grown to its full length until
all of its amino acids are escaped from the tunnel.



 §.§ Diffusion model of protein escape


Our previous study <cit.> has shown that the protein escape at
the exit tunnel in absence of crowders is a downhill process corresponding
to a free energy which monotonically decreases along a reaction coordinate
associated with the escape degree, such as the number of residues outside the
tunnel or the position of the C-terminus. Such a process is consistent with the
diffusion of a particle in an one-dimensional external potential field U(x)
where U is a decreasing function of x. This diffusion process is described
by the Smoluchowski equation
<cit.>

    ∂/∂ t  p(x,t|x_0,t_0) =  
    ∂/∂ x D
     (β∂ U(x)/∂ x + ∂/∂ x)  p(x,t|x_0,t_0) ,

where p(x,t|x_0,t_0) is a conditional probability density of finding the
particle at position x and at time t, given that it was found previously at
position x_0 at time t_0; D is diffusion constant, assumed to be position
independent; and β=(k_B T)^-1 is the inverse temperature with k_B the
Boltzmann constant. Assume that the external potential field has a linear form,
U(x) = - k x, with k a constant. 
For a nascent protein at the tunnel, the constant k presents an average
slope of the dependence of the free energy of the protein on the escape
coordinate.
In such a case, a solution of Eq.
(<ref>) for an unconstrained particle is given by

    p(x,t) ≡ p(x,t|0,0)=1/√(4π D t)exp[
    -(x - Dβ k t)^2/4Dt] .

given that the initial condition is p(x,0)=δ(x).
This solution gives the mean displacement of the particle

    ⟨ x ⟩ = (Dβ k) t  ,

with a diffusion speed equal to Dβ k.
For a Brownian particle, D depends on the temperature T and on the
friction coefficient ζ according to the Einstein's relation

    D = k_B T/ζ .
	


The escape time of nascent protein at the tunnel corresponds to the first
passage time (FPT) of a diffused particle subject to the initial condition at
x=0 and an absorbing boundary condition at x=L. The latter condition is
given as

    p(L,t) = 0 .

The FPT distribution for this absorbing boundary condition can be found
in Ref. <cit.> and is given by

    g (t)=L/√(4π D t^3)exp[
    -(L - Dβ k t)^2/4Dt] .

Using the distribution in Eq. (<ref>) one obtains the mean escape
time 

    μ_t ≡⟨ t ⟩ = ∫_0^∞ t   g(t)  dt 
    = L/D β k ,

and the standard deviation 

    σ_t ≡ (⟨ t^2 ⟩ - ⟨ t ⟩^2)^1/2
    = √(2 β k L)/D (β k)^2 .

It follows that the ratio σ_t/μ_t is independent of D.  Note that
both μ_t and σ_t diverges when k=0, for which g(t) becomes the
heavy-tailed Lévy distribution. 
It can be expected that D and β k may depend on L and on the
crowders' volume fraction ϕ. These dependences will be investigated in the
present study. 



§ RESULTS AND DISCUSSION




 §.§ Effect of native interactions


Our previous study <cit.> has shown that the folding of nascent
proteins speeds up their escape process at the ribosomal tunnel. Here, we
investigate how this enhancement is sensitive to the details of native
interactions and how the escape of a protein is different from that of a
homopolymer. For this investigation, we fix the length of the tunnel to be
L=80 Å and consider the protein without crowders.

We consider the protein GB1 with three different native contact maps, C1, C2
and C3, as described in the Methods section.  Both the C1 and C2 maps have 102
native contacts, but of which only 72 contacts are common. The C2 map has more
long-range contacts than the C1 one. The relative contact order (CO)
<cit.> of the C2 map (≈ 0.3444), is higher than that of the
C1 map (≈ 0.3283). The C3 map has 120 contacts (CO ≈
0.3509) and includes all the contacts in the C2 map. The folding temperature
T_f of a free protein without the tunnel is defined as the temperature of the
maximum of the specific heat peak (Fig. S1 of the supplementary material) and
equal to 0.866, 0.888 and 1.004 ϵ/k_B for the models with C1, C2 and
C3 contact maps, respectively.  We consider also two
homopolymers of the same length as the GB1 protein (N=56). The first one is a
self-repulsive homopolymer with a repulsive potential of
ϵ (σ/r)^12 for the interaction between any pair of
non-consecutive beads. The second homopolymer is a self-attractive one with the
12-10 LJ potential, given by Eq.  (<ref>), for the attraction between
the beads. Note that the self-repulsive homopolymer can be considered as
representing an intrinsically disordered protein, with regards to an important
class of proteins that do not fold in vivo <cit.>.




Fig. <ref>a shows the dependence of the median escape time,
t_esc, on temperature for the GB1 protein with the three native
contact maps. It is shown that for temperatures roughly larger than T_f, all
three contact maps lead to almost the same escape times. For T<T_f,
differences in the escape times are found among the models, even though the
dependences of t_esc on T are of similar shape for all the three
models. The model with the C3 map has the smallest escape times, indicating
that the larger number of native contacts the faster is the escape of the
protein. On the other hand, the model with C2 map has smaller t_esc
than the model with the C1 map, despite that they have the same number of
native contacts. This result indicates that the escape time also depends on the
details of the native contact map, and a protein with more long-range contacts
would have a faster escape from the tunnel. 

Fig. <ref>b compares the escape times of the GB1 protein with the C3
native contact map with the two homopolymers. It shows that for T>T_f, the
protein has the escape time slightly larger but close to that of the
self-repulsive homopolymer, as expected for an unfolded chain. For T<T_f, the
protein escapes faster than the homopolymer with self-repulsion, reconfirming
the favorable effect of
folding on the escape process. The self-attractive homopolymer shows a
very different behavior of the escape time than the self-repulsive one. In
particular, for temperatures lower than an intermediate temperature of about
0.9 ϵ/k_B, the self-attractive homopolymer has a much larger escape
time than the self-repulsive one; while an opposite trend is seen for
T>0.9 ϵ/k_B, for which the self-attractive polymer escapes faster
than the other one. We find that for T < 0.9 ϵ/k_B, 
the full-length homopolymer starts the escape process with a collapsed
conformation completely fitted inside the tunnel. From this conformation, the
polymer diffuses very slowly in the tunnel until a part of it emerges from
the tunnel. For T > 0.9 ϵ/k_B, the polymer begins to escape with a
conformation having a small part found outside the tunnel. We have checked that
the self-attractive homopolymer has a collapse transition temperature ≈
2.2 ϵ/k_B, thus below this temperature but above 0.9 ϵ/k_B,
its escape process is accelerated by the collapse of the chain.  Above the
collapse transition temperature, the escape time of the self-attractive
homopolymer is smaller but approaching that of the self-repulsive one as
temperature increases.
Note that below the collapse transition temperature, the size of the
collapsed polymer still depends on temperature. Thus, the temperature of 0.9
ϵ/k_B, at which a rapid change in the escape time is seen, should
be understood as specific to the polymer length and the tunnel length
considered. At this temperature, the typical size of the self-attractive
homopolymer along the tunnel axis approximately matches that of the tunnel.

The result of the self-attractive homopolymer shows that the collapse of
the chain accelerates the escape process only when the chain has a part
found outside the tunnel. It indicates the relative size of the polymer
to the tunnel length and also temperature are relevant to the escape behavior
of the polymer. We find that protein behaves similarly on increasing the
tunnel length, as will be shown in the next subsection. 

Only for the self-repulsive homopolymer, the escape time is proportional to
T^-1 for the whole range of temperature. As β k is approximately
constant on changing temperature (see Fig. <ref> of this study and also
Ref. <cit.>), it follows from Eq. (<ref>) that the diffusion
coefficient D of the self-repulsive homopolymer is proportional to T,
consistent with the Einstein's relation for Brownian particle (Eq.
(<ref>)). Deviation from this Brownian behavior on changing
temperature thus is observed for the protein and the self-attractive
homopolymer due to the fact that they adopt different compact conformations
during the escape process at temperatures below their folding or collapse
transition temperatures.






We have calculated the escape time for a number of small single-domain
proteins other than GB1 with different native state topologies. Fig.
<ref> shows the dependence of the median escape time on temperature for
the Z domain of Staphylococcal protein (SpA) of length N=58. The native state
of SpA is a three-helix bundle (Fig. <ref>, inset). It is shown that at
high temperatures, the escape time of SpA is higher than the escape time of a
same-length self-repulsive homopolymer. However, for T < T_f, the relative
difference between the two escape times decreases with temperature, and for
T<0.4 ϵ/k_B, the protein escape faster than the homopolymer. Thus,
the folding of SpA enhances its escape process. We have found that
another helix bundle with the PDB code 2rjy also escapes faster then the 
homopolymer at temperatures lower than T_f (see Fig. S2 of supplementary
material).  On the other hand, a single α-helix escapes more
slowly than the self-repulsive homopolymer at all temperatures (see Fig. S3 of
supplementary material). These results suggest that local
interactions stabilizing the α-helix slow down the escape process and the
latter is accelerated only by the non-local interactions.

Note that the Go-like model includes local potentials on the bond angles
and dihedral
angles favoring native conformation while the homopolymer model does not. Even
at temperatures higher than T_f, these interactions still have some effect on
the local conformations. For α-helical proteins, they make the
escaping protein conformations less extended and more rigid than those of the
self-repulsive homopolymer. This effect explains why α-helical proteins
escape more slowly than the self-repulsive homopolymer for 
T>T_f, at least for T up to 3ϵ/k_B as shown in Fig.
<ref>.  It can be expected that for much higher temperatures, at which
the local potentials become unimportant, the escape time of protein approaches
that of the homopolymer.


The different effects of local and non-local interactions on the escape
time of proteins can also be seen in Fig. <ref>. Fig.
<ref>b shows that at a temperature favorable for folding,
T=0.4 ϵ/k_B, the escape time of protein to a considerable degree is
correlated with the relative contact order.
Fig. <ref>a shows that the escape time of protein is uncorrelated
with the chain length, whereas that of the self-repulsive homopolymer is almost
independent on the chain length. Fig. <ref>a also shows that the
α/β and all-β proteins have smaller
escape time than the same-length homopolymers, whereas the all-α proteins
may have smaller or larger escape time than the homopolymers. 
We have checked that among the all-α proteins considered, the larger
the number of non-local contacts, the faster the protein escapes.




 §.§ Effect of tunnel length


We study now the dependence of the escape time on the length of the
ribosomal tunnel, which is considered as an adjustable parameter in our model.
For this investigation, we have carried out simulations for the GB1 protein
with the tunnel length L varied between 10 and 130 Å, and analyzed the
statistics of the escape times using the insights from the diffusion model.
From here on, for simplicity, we consider only the Go-like model with the C3
native contact map for protein GB1.

The diffusion model predicts that the ratio between the standard deviation
of the escape time, σ_t, and the mean escape time, μ_t, is
given by

    σ_t/μ_t = √(2/Lβ k) ,

and thus depends only on L and β k. Fig. <ref> shows that the
dependence of σ_t L^1/2 on μ_t for GB1 obtained by the simulations
at two different temperature below T_f is almost linear for L ≤
110 Å. This linear dependence indicates that β k is constant on
changing L, for L ≤ 110 Å. The fits of the simulation data to Eq.
(<ref>) show that β k = 0.269 Å^-1 for
T=0.8ϵ/k_B and β k = 0.294 Å^-1 for T=0.4
ϵ/k_B. Thus, the values of β k are not the same but quite close
for the two temperatures considered. Fig. <ref> shows that for L >
110 Å, the dependence of σ_t L^1/2 on μ_t strongly
deviates from the linear dependence obtained for smaller L, indicating
that β k quickly decreases on increasing L. Thus, the diffusion
properties of the protein changes qualitatively at the tunnel length of L
≈ 110 Å. We call the latter the cross-over length for
the diffusion of protein at the tunnel.







By fitting the distribution of the escape time obtained from the simulations to
that given by Eq. (<ref>) with the values of β k as given in Fig.
<ref>, one obtains the effective diffusion constant D of the protein at
the tunnel for L ≤ 110 Å. Fig. <ref>a shows that D decreases
with L. This dependence reflects the facts that the protein has a changing
shape when escaping from the tunnel, and that the shape depends on L.
When L is increased, the initial conformation of the full-length protein at
the tunnel becomes more extended leading to a slower diffusion.
Fig. <ref>b shows that the mean escape time increases with L. As
indicated by Eq. (<ref>), the growth of the escape time on increasing
L is due to both the longer diffusion distance (which is equal to L) and
the slower diffusion speed. Fig. <ref>b also shows that for L >
110 Å, the escape time increases with L much faster than for L ≤
110 Å, in consistency with the change in diffusion properties shown in
Fig.  <ref>.





The cross-over in the diffusion properties and the escape time observed
at L≈ 110 Å for GB1 is related to the relative size of a tunnel
compared to that of a protein. If the tunnel length is such that the protein,
presumably with most of the secondary structures formed, can be found
completely inside the tunnel, then the escape of the protein is much slower
than the case of a shorter tunnel length, in which the protein cannot fit
itself entirely in the tunnel. The tunnel length of 110 Å thus is related
to the size of GB1, such that it can merely have a small part outside the
tunnel at the moment the chain is released from the PTC. The escape process is
accelerated only by the folding of the escaped part of the protein at the
tunnel. Fig. <ref>a shows that in typical escape processes,
the number of amino acid residues escaped from the tunnel, N_out,
has similar trends in the time evolution for different tunnel lengths L,
except that N_out has different values at t=0, the moment a
full-length protein begins the escape process.
Fig. <ref>b shows that the distribution of N_out
at t=0 strongly depends on L.  For L=130 Å, the protein is mostly
found completely inside the tunnel, i.e. N_out=0. 
On the other hand, for L=80 Å, the protein
always has a significant part outside the tunnel with N_out 
essentially ranging from 18 to 26. For the cross-over length L=110 Å,
N_out varies between 0 and 15. The cross-over length approximately
corresponds to the smallest tunnel length for which N_out can have a
zero value.


In consistency with the above mechanism, we find that a similar cross-over of
the diffusion properties and the escape time on increasing the tunnel length is
observed for the three-helix bundle protein SpA.
For SpA, the cross-over occurs at L ≈ 90 Å (see
Figs. S4 and S5 of supplementary material), quite close to the cross-over
length for GB1, and is consistent with the fact the both proteins are single
domain and of similar size.
The shorter cross-over length for SpA is a little shorter than for GB1
due to the fact that SpA can form more α-helices inside the tunnel,
leading to a shorter size than GB1.  Interestingly, the real length of
ribosomal exit tunnel falls between 80 Å and 100 Å, very close to our
estimates of the cross-over length for the GB1 and SpA proteins.
Note that the latter are among the smallest single domain proteins.
It is suggested that the ribosome's tunnel length has been selected
to facilitate an efficient escape of small single domain proteins. 
Our study indicates that the cross-over tunnel length increases with the
protein size, thus large proteins would have no problem of escaping the
ribosomal tunnel from the viewpoint of diffusibility.



 §.§ Effect of macromolecular crowding







We proceed now study the escape of nascent protein in the presence of a crowd
of macromolecules outside the ribosomal tunnel. For this investigation, we fix
the tunnel length to be L=80 Å and consider various volume fractions
ϕ of the crowders. A snapshot of an escaping protein molecule entering
the solution of crowders is shown in Fig. S6 of supplementary material.

First, we consider the case in which the interaction between the crowders and
amino acids are purely repulsive. Fig. <ref> shows histograms of the
escape time obtained by the simulations for the GB1 protein at
T=0.8 ϵ/k_B for the crowders' volume fraction ϕ=0, 0.2 and 0.4.
It is shown that as ϕ increases the histogram is more spread and shifted
toward higher time values, meaning that the escape time is longer and more
disperse in the presence of crowders. The histograms of the escape time
are found to be consistent with the distribution function given by Eq.
(<ref>) of the diffusion model. The fits to this function give us the
effective values of D and β k for different crowder concentrations. 

Fig. <ref>a shows that both D and β k decrease with ϕ. We
find that both D and β k can be approximately described with a
logarithmic dependence on ϕ in the following forms: 

    D     =     D_0 + a ln(1 - ϕ/ϕ_c)  , 
    
    β k      =    β k_0  + b ln(1 - ϕ/ϕ_c)  ,

where D_0 and k_0 are the values of D and k, respectively, at ϕ=0;
a and b are the fitting parameters; ϕ<ϕ_c and ϕ_c is a cut-off
volume fraction, beyond which the full escape of the protein becomes
impossible. We find that ϕ_c=0.5 is a good estimate. The above logarithmic
dependences suggest that the effect of the crowders on the escape process of
protein has an entropic origin, as (1-ϕ/ϕ_c) can be considered as
the effective volume fraction accessible to the escaping protein in the space
outside the tunnel. Note that entropy loss due to excluded volume is also the
primary effect of crowding and confinement on protein stability
<cit.>. Having the functions given in Eqs. (<ref>,
<ref>), one can calculate the mean and the standard deviation of the
escape time from Eqs. (<ref>,<ref>) of the diffusion model. Fig.
<ref>b shows that the mean escape time and the dispersion of the escape
time obtained from simulations at various ϕ also agree with the diffusion
model.



Fig. <ref> shows that the standard deviation of the escape time,
σ_t, depends almost linearly on the the mean escape time, μ_t, for
both the protein and the self-repulsive homopolymer at various temperatures,
indicating that β k is constant for each system on changing the
temperature. The value of β k however depends on the volume fraction
ϕ of the crowders, as indicated by the slopes of the fits shown in Fig.
<ref>. One finds that β k decreases when ϕ increases
from 0 to 0.3 for both the protein and the polymer, indicating
that diffusion is slower in the presence of crowders. Again here, one 
also finds that for both cases, with and without crowders, the value of β
k for GB1 is larger than for the self-repulsive homopolymer, confirming the
enhancing effect of folding on the escape of protein.



Fig. <ref> shows the dependence of the median escape time on temperature
for the protein GB1 and the self-repulsive homopolymer in the presence of
repulsive crowders at the volume fraction ϕ=0.3. It is shown that
the log-log plot of this dependence for both the protein and the homopolymer
has similar characteristics to that found in Fig. <ref> for the case
without crowders, except that the escape times are longer with the crowders.
For the homopolymer, the escape time decreases with temperature linearly in the
log-log plot with a slope close to -1, indicating that the diffusion
constant of the polymer also depends linearly on temperature like for the case
without crowders. In the presence of repulsive crowders, the escape times of
protein at high temperatures are close to those of the self-repulsive
homopolymer. At low temperatures, favorable for folding, the protein has
significant shorter escape times than the polymer, indicating that the impact
of folding on the escape time is not affected by the crowders. 

Fig. <ref> also shows the escape times of protein in the presence of
attractive crowders with two different interaction strengths, ϵ_1=
0.3ϵ and ϵ_1=0.5ϵ, of the attraction between crowder
and amino acid. It can be seen that the attractive crowders make the escape
faster than the repulsive crowders, but only at high and intermediate
temperatures. At low temperatures (T ≤ 0.4 ϵ/k_B), the protein
escapes more slowly in the presence of attractive crowders than of the
repulsive ones. Furthermore, the escape time also increases when the
attraction strength ϵ_1 increases at low temperatures. The reason for
this increase is that, in contrast to repulsive crowders, attractive crowders
destabilize the native interactions in protein. Thus, folding is less favorable
in the presence of attractive crowders leading to a weaker enhancement of
folding on the escape speed. Fig. <ref> shows that the distributions of
the root mean square deviation (rmsd) from the native state and the radius of
gyration of protein conformations obtained at the moment of full escape from
the tunnel are shifted towards higher values when switching from repulsive
crowders to attractive crowders. 



It is believed that ribosome-associated chaperones, such as the trigger factor
(TF) in prokaryotes or the Hsp70 Ssb and NAC (nascent chain-associated complex)
in eukaryotes, are of particular importance in guiding nascent proteins to fold
correctly <cit.>. The binding of these chaperones to the ribosome
effectively leads to a very high concentration of chaperones near the exit
tunnel <cit.>, promoting their interaction with nascent polypeptide.
As a result, the chaperones quickly bind to unfolded, hydrophobic segments of
the polypeptide before these segments can fold or misfold, keeping the nascent
chain unfolded. Our simulations with attractive crowders show a similar effect,
as indicated in Fig. <ref>, that attractive crowders make the fully
released protein conformation less native-like and less compact than in the
case with repulsive crowders. Our simulations predict that the escape time of
nascent protein increases in the presence of chaperones due to both their
crowding effect and their attractive interaction with hydrophobic segments.



§ CONCLUSION


Post-translational escape of nascent protein at the ribosome is a stochastic
process governed by protein native interactions, the geometry of the
ribosomal exit tunnel and macromolecular crowders outside the tunnel. We
have shown that non-local native interactions
speed up the escape process at temperatures favorable for folding,
while the local interactions responsible for the formation of
α-helices slow it down. As a consequence, proteins with 
a content of β-sheets tend to
escape faster than those with only α-helices in the native state.
Increasing the tunnel length or the concentration of crowders also slows down
the protein escape. In the view that the concomitant folding and escape
of nascent protein at the exit tunnel are beneficial for both
the productivity of the ribosome and the protection of nascent protein against
aggregation, it can be conjectured that the protein synthesis machinery has
been evolved to facilitate both the folding and the escape of nascent proteins.
In support of this conjecture, we have shown that real ribosomal exit tunnel
has adopted the length that is close to a cross-over length of the tunnel,
beyond which the protein escape falls into a regime of a much slower diffusion
for small single domain proteins. 

Our study shows that repulsive crowders outside the tunnel induce an entropic
effect on the diffusion properties of protein at the tunnel, leading to
increased escape times but does not change the enhancing effect of folding on
the escape process. The latter effect is changed only in
the case of attractive crowders, whose attraction to amino acids competes with
native interactions in the nascent polypeptide. Due to this competition the
fully escaped protein conformation is more extended and less native-like. The
unfavorable effect of attractive crowders on the folding of nascent protein is
also reflected on the increased escape times at low temperatures, as shown in
our study. It is suggested that the ribosome-associated chaperones induce
similar effects on nascent polypeptides as found with attractive crowders. 

Low-dimensional diffusion models have been successfully applied to study
complex dynamics <cit.>.
Our work proves that the simple diffusion model considered is useful for
understanding the escape of protein at the exit tunnel. The results suggest
that intrinsically disordered proteins, considered as the self-repulsive
homopolymer in our study, have longer escape time than foldable proteins
with a significant number of long-range contacts,
and their diffusion is the most akin to that of a Brownian particle.

This research is funded by Vietnam National Foundation for Science and
Technology Development (NAFOSTED) under Grant No. 103.01-2016.61.



§ SUPPLEMENTARY MATERIAL

See supplemental material for the specific heats for the Go-like model of GB1
with different native contact maps, the dependence of the escape
time on temperature for the helical protein 2rjy and a single
α-helix, the dependence of the dispersion of the escape time on the
mean escape time for SpA, the dependence the diffusion constant and the mean
escape time on the tunnel length for SpA, and a snapshot of an escaping protein
entering a solution of macromolecular crowders.






















§ SUPPLEMENTARY MATERIAL FOR:
“PROTEIN ESCAPE AT THE RIBOSOMAL EXIT TUNNEL: EFFECTS OF NATIVE INTERACTIONS,
TUNNEL LENGTH AND MACROMOLECULAR CROWDING”, P.T. BUI AND T.X. HOANG




















