Protein escape at the ribosomal exit tunnel: Effects of native interactions,tunnel length and macromolecular crowding

Phuong Thuy Bui1 and Trinh Xuan Hoang2, 3, a)1)Duy Tan University, 254 Nguyen Van Linh, Thanh Khe, Da Nang, Viet Nam2)Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi,Vietnam3)Graduate University of Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet,Cau Giay, Hanoi, Vietnam

(Dated: 27 July 2018)

How fast a post-translational nascent protein escapes from the ribosomal exit tunnel is relevant to its foldingand protection against aggregation. Here, by using Langevin molecular dynamics, we show that non-localnative interactions help decreasing the escape time, and foldable proteins generally escape much faster thansame-length self-repulsive homopolymers at low temperatures. The escape process, however, is slowed downby the local interactions that stabilize the α-helices. The escape time is found to increase with both the tunnellength and the concentration of macromolecular crowders outside the tunnel. We show that a simple diffusionmodel described by the Smoluchowski equation with an effective linear potential can be used to map outthe escape time distribution for various tunnel lengths and various crowder concentrations. The consistencybetween the simulation data and the diffusion model however is found only for the tunnel length smaller thana cross-over length of 90 A to 110 A, 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 theefficient folding and efficient escape of single domain proteins. We show that macromolecular crowders lead toan increase of the escape time, and attractive crowders are unfavorable for the folding of nascent polypeptide.

I. INTRODUCTION

Partially folded protein conformations can be found atthe ribosome during protein translation and after transla-tion before the full release of a nascent protein from theribosomal exit tunnel. The folding during translation,namely cotranslational folding, occurs when a nascentpolypeptide chain undergoes elongation due to biosyn-thesis (for reviews, see e.g. Refs.1–3), while the post-translational folding is associated with a full-length pro-tein. In both cases, the behavior of the nascent polypep-tide is strongly influenced by the ribosome, especiallyby the ribosomal exit tunnel through which the nascentchain traverses to the cytosol or to another cellular com-partment. The length of the ribosomal exit tunnel spansfrom 80 A to 100 A, depending on where the exit end isdefined, whereas its width varies between 10 A and 20 A4.Such a geometry allows for the formation of an α-helix ora β-hairpin inside the tunnel5, and also promotes the α-helix formation6, but would hardly accommodate even asmall tertiary structure7. The tunnel was also suggestedto have a recognitive function leading to a translationarrest of certain amino acid sequences, such as the SecMsequence8. Cotranslational folding has been character-ized with vectorial folding1, i.e. the folding events pro-ceed from the N- to the C-terminus; the non-equilibriumeffect of a growing chain9; and the impact of the vary-ing codon-dependent translation rate10–13. Furthermore,folding of nascent proteins is assisted by the action of

a)Electronic mail: hoang@iop.vast.vn

ribosome-associated molecular chaperones14,15. All theseeffects are indicative of a highly conditional and coordi-nated folding of nascent protein at the ribosome, which isclearly different from refolding16 of a denatured proteinin aqueous solvent. There have been experiments17–19

as well as simulations20–22 showing that the folding effi-ciency of proteins is improved under biosynthesis condi-tions. It was also suggested that the impact of cotransla-tional folding is evolutionarily imprinted on the proteinnative states, as seen with an increased helix propensity9

and a decreased compactness23 of the chain near the C-terminus in the statistical analyses of protein structuresfrom the protein data bank (PDB).

While cotranslational folding is progressively under-stood, little is known about post-translational folding atthe ribosome. The latter is considered to take place af-ter the protein C-terminus is released from the peptidyltransferate center (PTC), where the peptide bonds areformed. Certainly, protein must escape from the ribo-somal tunnel to fully acquire the native conformation.A too slow escape would decrease the productivity ofthe ribosome, while a too fast escape would make thenascent protein vulnerable to aggregation24, as the par-tially folded protein may still have a large exposure ofhydrophobic segments. In a recent study22, by using MDsimulations, we have shown that post-translational fold-ing at the exit tunnel is concomitant with the escapeprocess and that the tunnel induces a vectorial foldingof the full-length protein. Such a folding has a greatlyreduced number of pathways and leads to an improvedfolding efficiency. Interestingly, it has been also shown22

that the escape time distribution of protein can be cap-tured by a simple one-dimensional diffusion model of a

arX

iv:1

807.

1003

0v1

[q-

bio.

BM

] 2

6 Ju

l 201

8

2

particle in a linear potential field with an exact solutionof the Smoluchowski equation.

The purpose of the present study is to explore the pro-tein escape at the ribosomal exit tunnel with a focuson several effects, namely the role of native interactions,the impact of tunnel length and the influence of macro-molecular crowders25,26. We use the same approach asgiven in our previous work22, that is to consider sim-ple coarse-grained models for the protein, the exit tunneland the crowders, which enable multiple simulations ofprotein growth and escape by using the Langevin equa-tion. The diffusion model for protein escape previouslyintroduced22 is improved in this study by considering anabsorbing boundary condition. We find that escape timereflects well the changes in the system properties, such asthe native contact map, the tunnel length and the crow-der concentration with a remarkable consistency betweensimulations and the theoretical diffusion model. Interest-ingly, the dependence of the escape time on the tunnellength suggests an explanation for the observed length ofthe exit tunnel in real ribosomes. Our results obtainedwith attractive crowders provide an insight into the ef-fect of ribosome-associated chaperones on the escape andfolding of nascent proteins at the ribosome.

II. METHODS

A. Models of nascent protein, ribosomal exit tunnel andmacromolecular crowders

As a nascent protein we will focus on the B1 domainof protein G of length N = 56 amino acids with the PDBcode of 1pga, denoted as GB1. The protein is consid-ered in a Go-like model27–30, in which each amino acid isconsidered a single bead centered at the position of theCα atom. We adopt the same Go-like model as given inour previous work22 except that with a 10-12 Lennard-Jones (LJ) potential for native contact interactions. Inaddition, we consider three types of native contact maps,denoted as C1, C2 and C3, for the model. The C1 mapis defined by a cut-off distance of 7.5 A between the Cαatoms in the native conformation. The C2 and C3 mapsare obtained based on an all-atom consideration31 of theprotein PDB structure: contact between two amino acidsis identified if there are least two non-hydrogen atomsbelonging to the two amino acids, found at a distanceshorter than λ times the sum of their atomic van derWaals radii. C2 map has λ = 1.27 whereas C3 hasλ = 1.5. The choice of λ = 1.27 is such that the C2map has the same number of native contacts and the C1for the GB1 protein. The interaction between a pair ofamino acids forming a native contact takes the form of a10-12 LJ potential30

V (rij) = ε[5(r∗ij/rij)

12 − 6(r∗ij/rij)10], (1)

where ε is an energy unit in the system correspondingto the strength of the LJ potential, rij is the distance

d

R

x

L

l

FIG. 1. Sketch of the models of a ribosomal exit tunnel witha partially folded nascent protein inside and macromolecularcrowders outside the tunnel. The peptidyl transferate center(PTC), where the protein is grown, is shown as a red circle.

between residues i and j, and r∗ij is the correspondingdistance in the native state. The use of 10-12 LJ potentialmakes the folding transition more cooperative32 than the6–12 LJ potential (used in previous work22) as indicatedby the height of the specific heat peak (see Fig. S1 of thesupplementary material). Additionally, we will consideralso a number of small single-domain proteins to studythe effects of native interactions and the tunnel lengthon the escape process.

The ribosomal tunnel is modeled as a hollow cylinderof repulsive walls with diameter d = 15 A and length L(Fig. 1). It has been shown22 that this diameter allowsfor the formation of an α-helix and a β-hairpin insidethe tunnel but not tertiary structures. In the presentstudy, L is allowed to changed between 0 and 140 A. Thecylinder has one of its circular bases open and attachedto a repulsive flat wall mimicking the ribosome’s outersurface. Macromolecular crowders are modeled as softspheres of radius R = 10 A (R is chosen approximatelyequal to the radius of gyration of GB1, Rg = 10.2 A).Assume that the x axis is the tunnel axis, the crowdersare confined between the ribosome’s wall and anotherwall parallel to it at a distance l = 100 A along the xdirection. Periodic boundary conditions are applied forthe y and z directions with a box size equal to l. Thecrowders’ volume fraction is given by φ = M(4π/3)R3/l3,with M the number of crowders.

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

Vrep(r) =

{4ε[(σ/r)12 − (σ/r)6

]+ ε , r ≤ 21/6σ

0 , r > 21/6σ,

(2)where σ = 5 A is a characteristic length equal to the typ-ical diameter of an amino acid; r is the distance betweenan amino acid and the nearest virtual residue22 of diam-eter σ, embedded under the surface a wall or a crowder,

3

FIG. 2. Virtual residues (red dotted spheres) in the interac-tions between an amino acid and a wall (1), between a crowderand a wall (2), between an amino acid and a crowder (3), andbetween two crowders (4). The virtual residues are of thesame size as amino acid (blue) and can be at any positionembedded under the surface of a wall (solid line) or a crowder(cyan). The interaction potentials involving a wall or a crow-der are defined based on the distance to the nearest virtualresidue (as in 1 and 3) or between the two nearest virtualresidues (as in 2 and 4) belonging to these objects.

or between two such virtual residues (see Fig. 2 for thedefinition of virtual residue).

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

Vatt(r) = 4ε1[(σ/r)12 − (σ/r)6

], (3)

with energy parameter ε1 < ε.The motions of the protein and the crowders are simu-

lated by using the Langevin equations22,28. One assumesthat all amino acids have the same mass, m, while thecrowders has a molecular mass mc equal to the mass ofthe protein, i.e. mc = Nm. Similarly, the friction co-efficient of amino acid is ζa, whereas that of crowder isζc = Nζa. The Langevin equations are integrated byusing a Verlet algorithm introduced in Ref.22 with timestep ∆t = 0.002τ , where τ =

√mσ2/ε is the time unit in

the system. In the simulations we use ζa = 5mτ−1, forwhich the dynamics of the system are in the overdampedlimit22,33. Fig. 3 shows that the employed dynamics leadto the same diffusion characteristics for a folded proteinand for crowders in the solution. These characteristicsare also close to that of a Brownian motion for timeslarger than τ .

Following previous work22, the nascent protein isgrown inside the tunnel at the PTC from the N termi-nus to the C terminus with the growth time per aminoacid tg = 100τ . This growth speed is sufficiently slow toproduce fully translated conformations of similar struc-tural characteristics to those obtained by a much slowergrowth speed22. The escape time is measured from the

0.1

1

10

100

1000

0.1 1 10 100 1000 10000

T=0.8

<(Δr)2>(Å

2 )

time(τ)

GB1crowders

FIG. 3. Time dependence of the mean square displacementof the protein GB1 (circles) and crowders (crosses) in simula-tions without the tunnel. The simulations are carried out forthe protein and 47 crowders at the volume fraction of φ = 0.2.The simulation temperature is T = 0.8 ε/kB , at which theprotein stays in its native state. The protein displacement iscalculated using its center of mass. The averages are takenover 100 independent trajectories. The solid line has a slopeequal to 1.

moment the protein has grown to its full length until allof its amino acids are escaped from the tunnel.

B. Diffusion model of protein escape

Our previous study22 has shown that the protein es-cape at the exit tunnel in absence of crowders is a down-hill process corresponding to a free energy which mono-tonically decreases along a reaction coordinate associatedwith the escape degree, such as the number of residuesoutside the tunnel or the position of the C-terminus.Such a process is consistent with the diffusion of a par-ticle in an one-dimensional external potential field U(x)where U is a decreasing function of x. This diffusionprocess is described by the Smoluchowski equation34

∂

∂tp(x, t|x0, t0) =

∂

∂xD

(β∂U(x)

∂x+

∂

∂x

)p(x, t|x0, t0),

(4)where p(x, t|x0, t0) is a conditional probability density offinding the particle at position x and at time t, giventhat it was found previously at position x0 at time t0;D is diffusion constant, assumed to be position indepen-dent; and β = (kBT )−1 is the inverse temperature withkB the Boltzmann constant. Assume that the externalpotential field has a linear form, U(x) = −kx, with k aconstant. For a nascent protein at the tunnel, the con-stant k presents an average slope of the dependence ofthe free energy of the protein on the escape coordinate.In such a case, a solution of Eq. (4) for an unconstrained

4

particle is given by

p(x, t) ≡ p(x, t|0, 0) =1√

4πDtexp

[− (x−Dβkt)2

4Dt

].

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

〈x〉 = (Dβk)t , (6)

with a diffusion speed equal to Dβk. For a Brownianparticle, D depends on the temperature T and on thefriction coefficient ζ according to the Einstein’s relation

D =kBT

ζ. (7)

The escape time of nascent protein at the tunnel cor-responds to the first passage time (FPT) of a diffusedparticle subject to the initial condition at x = 0 andan absorbing boundary condition at x = L. The lattercondition is given as

p(L, t) = 0 . (8)

The FPT distribution for this absorbing boundary con-dition can be found in Ref.35 and is given by

g(t) =L√

4πDt3exp

[− (L−Dβkt)2

4Dt

]. (9)

Using the distribution in Eq. (9) one obtains the meanescape time

µt ≡ 〈t〉 =

∫ ∞0

t g(t) dt =L

Dβk, (10)

and the standard deviation

σt ≡ (〈t2〉 − 〈t〉2)12 =

√2βkL

D(βk)2. (11)

It follows that the ratio σt/µt is independent of D. Notethat both µt and σt diverges when k = 0, for which g(t)becomes the heavy-tailed Levy distribution. It can beexpected that D and βk may depend on L and on thecrowders’ volume fraction φ. These dependences will beinvestigated in the present study.

III. RESULTS AND DISCUSSION

A. Effect of native interactions

Our previous study22 has shown that the folding ofnascent proteins speeds up their escape process at the ri-bosomal tunnel. Here, we investigate how this enhance-ment is sensitive to the details of native interactions andhow the escape of a protein is different from that of ahomopolymer. For this investigation, we fix the length

of the tunnel to be L = 80 A and consider the proteinwithout crowders.

We consider the protein GB1 with three different na-tive contact maps, C1, C2 and C3, as described in theMethods section. Both the C1 and C2 maps have 102 na-tive contacts, but of which only 72 contacts are common.The C2 map has more long-range contacts than the C1one. The relative contact order (CO)36 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 includesall the contacts in the C2 map. The folding temperatureTf of a free protein without the tunnel is defined as thetemperature of the maximum of the specific heat peak(Fig. S1 of the supplementary material) and equal to0.866, 0.888 and 1.004 ε/kB for the models with C1, C2and C3 contact maps, respectively. We consider also twohomopolymers of the same length as the GB1 protein(N = 56). The first one is a self-repulsive homopolymerwith a repulsive potential of ε (σ/r)12 for the interactionbetween any pair of non-consecutive beads. The secondhomopolymer is a self-attractive one with the 12-10 LJpotential, given by Eq. (1), for the attraction betweenthe beads. Note that the self-repulsive homopolymer canbe considered as representing an intrinsically disorderedprotein, with regards to an important class of proteinsthat do not fold in vivo37.

Fig. 4a shows the dependence of the median escapetime, tesc, on temperature for the GB1 protein with thethree native contact maps. It is shown that for tempera-tures roughly larger than Tf , all three contact maps leadto almost the same escape times. For T < Tf , differencesin the escape times are found among the models, eventhough the dependences of tesc on T are of similar shapefor all the three models. The model with the C3 maphas the smallest escape times, indicating that the largernumber of native contacts the faster is the escape of theprotein. On the other hand, the model with C2 maphas smaller tesc than the model with the C1 map, de-spite that they have the same number of native contacts.This result indicates that the escape time also dependson the details of the native contact map, and a proteinwith more long-range contacts would have a faster escapefrom the tunnel.

Fig. 4b compares the escape times of the GB1 pro-tein with the C3 native contact map with the two ho-mopolymers. It shows that for T > Tf , the protein hasthe escape time slightly larger but close to that of theself-repulsive homopolymer, as expected for an unfoldedchain. For T < Tf , the protein escapes faster than thehomopolymer with self-repulsion, reconfirming the favor-able effect of folding on the escape process. The self-attractive homopolymer shows a very different behaviorof the escape time than the self-repulsive one. In partic-ular, for temperatures lower than an intermediate tem-perature of about 0.9 ε/kB , the self-attractive homopoly-mer has a much larger escape time than the self-repulsiveone; while an opposite trend is seen for T > 0.9 ε/kB , forwhich the self-attractive polymer escapes faster than the

5

100

1000

10000

0.2 0.4 0.6 1 2 3

(b)

Tf (C3)

homopolymerself−attractive

homopolymerself−repulsive

GB1(C3)t e

sc (

τ)

T (ε/kB)

100

1000

0.2 0.4 0.6 1 2 3

(a)

Tf (C3)

GB1t esc (

τ)

C1C2C3

FIG. 4. Dependence of the median escape time, tesc, on tem-perature, T , at the tunnel of length L = 80 A: (a) For theGB1 protein in Go-like models with the C1 (triangles), C2(open circles) and C3 (filled circles) native contact maps; (b)For GB1 with the C3 map (filled circles), the self-repulsive ho-mopolymer (filled squares) and the self-attractive homopoly-mer (open squares). The escape times for the self-repulsivehomopolymer are fitted with a T−1 dependence (dashed line).Arrow indicates the folding temperature Tf = 1.004 ε/kB forthe protein with the C3 map. The native state of GB1 isshown as inset in (a).

other one. We find that for T < 0.9 ε/kB , the full-lengthhomopolymer starts the escape process with a collapsedconformation completely fitted inside the tunnel. Fromthis conformation, the polymer diffuses very slowly inthe tunnel until a part of it emerges from the tunnel. ForT > 0.9 ε/kB , the polymer begins to escape with a confor-mation having a small part found outside the tunnel. Wehave checked that the self-attractive homopolymer has acollapse transition temperature ≈ 2.2 ε/kB , thus belowthis temperature but above 0.9 ε/kB , its escape processis accelerated by the collapse of the chain. Above thecollapse transition temperature, the escape time of theself-attractive homopolymer is smaller but approachingthat of the self-repulsive one as temperature increases.Note that below the collapse transition temperature, thesize of the collapsed polymer still depends on temper-ature. Thus, the temperature of 0.9 ε/kB , at which arapid change in the escape time is seen, should be under-stood as specific to the polymer length and the tunnellength considered. At this temperature, the typical sizeof the self-attractive homopolymer along the tunnel axis

100

1000

10000

0.2 0.4 0.6 1 2 3

Tf

SpA (C3)

self−repulsive

homopolymer

t esc (

τ)

T (ε/kB)

FIG. 5. Dependence of the median escape time, tesc, ontemperature, T , for the SpA protein (circles) and the self-repulsive homopolymer (squares). The homopolymer has thesame length (N = 58) as SpA. The SpA is considered in theGo-like model with the C3 native contact map, and its nativeconformation is shown as inset. The folding temperature ofSpA, Tf = 0.952 ε/kB , is indicated by an arrow.

approximately matches that of the tunnel.The result of the self-attractive homopolymer shows

that the collapse of the chain accelerates the escape pro-cess only when the chain has a part found outside thetunnel. It indicates the relative size of the polymer tothe tunnel length and also temperature are relevant tothe escape behavior of the polymer. We find that pro-tein behaves similarly on increasing the tunnel length, aswill be shown in the next subsection.

Only for the self-repulsive homopolymer, the escapetime is proportional to T−1 for the whole range of tem-perature. As βk is approximately constant on changingtemperature (see Fig. 12 of this study and also Ref.22),it follows from Eq. (10) that the diffusion coefficient Dof the self-repulsive homopolymer is proportional to T ,consistent with the Einstein’s relation for Brownian par-ticle (Eq. (7)). Deviation from this Brownian behavioron changing temperature thus is observed for the pro-tein and the self-attractive homopolymer due to the factthat they adopt different compact conformations duringthe escape process at temperatures below their folding orcollapse transition temperatures.

We have calculated the escape time for a number ofsmall single-domain proteins other than GB1 with differ-ent native state topologies. Fig. 5 shows the dependenceof the median escape time on temperature for the Z do-main of Staphylococcal protein (SpA) of length N = 58.The native state of SpA is a three-helix bundle (Fig. 5,inset). It is shown that at high temperatures, the es-cape time of SpA is higher than the escape time of asame-length self-repulsive homopolymer. However, forT < Tf , the relative difference between the two escapetimes decreases with temperature, and for T < 0.4 ε/kB ,the protein escape faster than the homopolymer. Thus,

6

800

1000

1200

1400

1600

40 50 60 70 80 90

(a)

T=0.4

t esc (

τ)

N

800

1000

1200

1400

1600

0.15 0.2 0.25 0.3 0.35 0.4

(b)

T=0.4

t esc (

τ)

CO

FIG. 6. (a) Dependence of the median escape time, tesc, onthe chain length, N , of proteins (filled symbols) and the self-repulsive homopolymers (crosses). The data shown are ob-tained at T = 0.4 ε/kB for 12 small single-domain proteinswith PDB codes 1iur, 2jwd, 2rjy, 1wxl, 2spz, 1wt7, 2erw,1pga, 2ci2, 1f53, 2k3b, and 1shg, classified as all-α (circles),α/β (triangles) and all-β (squares), in the Go-like model withthe C3 native contact map, and for corresponding homopoly-mers of the same lengths as the proteins. The average escapetime of the homopolymers is indicated by horizontal line. (b)Dependence of tesc on the relative contact order (CO) forthe proteins (filled symbols) with an average trend shown asdashed line.

the folding of SpA enhances its escape process. We havefound that another helix bundle with the PDB code 2rjyalso escapes faster then the homopolymer at tempera-tures lower than Tf (see Fig. S2 of supplementary mate-rial). On the other hand, a single α-helix escapes moreslowly than the self-repulsive homopolymer at all temper-atures (see Fig. S3 of supplementary material). Theseresults suggest that local interactions stabilizing the α-helix slow down the escape process and the latter is ac-celerated only by the non-local interactions.

Note that the Go-like model includes local potentialson the bond angles and dihedral angles favoring nativeconformation while the homopolymer model does not.Even at temperatures higher than Tf , these interactionsstill have some effect on the local conformations. Forα-helical proteins, they make the escaping protein con-formations less extended and more rigid than those of theself-repulsive homopolymer. This effect explains why α-helical proteins escape more slowly than the self-repulsive

homopolymer for T > Tf , at least for T up to 3 ε/kB asshown in Fig. 5. It can be expected that for much highertemperatures, at which the local potentials become unim-portant, the escape time of protein approaches that of thehomopolymer.

The different effects of local and non-local interactionson the escape time of proteins can also be seen in Fig. 6.Fig. 6b shows that at a temperature favorable for folding,T = 0.4 ε/kB , the escape time of protein to a considerabledegree is correlated with the relative contact order. Fig.6a shows that the escape time of protein is uncorrelatedwith the chain length, whereas that of the self-repulsivehomopolymer is almost independent on the chain length.Fig. 6a also shows that the α/β and all-β proteins havesmaller escape time than the same-length homopolymers,whereas the all-α proteins may have smaller or larger es-cape time than the homopolymers. We have checked thatamong the all-α proteins considered, the larger the num-ber of non-local contacts, the faster the protein escapes.

B. Effect of tunnel length

We study now the dependence of the escape time onthe length of the ribosomal tunnel, which is consideredas an adjustable parameter in our model. For this in-vestigation, we have carried out simulations for the GB1protein with the tunnel length L varied between 10 and130 A, and analyzed the statistics of the escape times us-ing the insights from the diffusion model. From here on,for simplicity, we consider only the Go-like model withthe C3 native contact map for protein GB1.

The diffusion model predicts that the ratio betweenthe standard deviation of the escape time, σt, and themean escape time, µt, is given by

σtµt

=

√2

Lβk, (12)

and thus depends only on L and βk. Fig. 7 shows thatthe dependence of σtL

1/2 on µt for GB1 obtained bythe simulations at two different temperature below Tf is

almost linear for L ≤ 110 A. This linear dependence indi-cates that βk is constant on changing L, for L ≤ 110 A.The fits of the simulation data to Eq. (12) show thatβk = 0.269 A−1 for T = 0.8ε/kB and βk = 0.294 A−1

for T = 0.4ε/kB . Thus, the values of βk are not the samebut quite close for the two temperatures considered. Fig.7 shows that for L > 110 A, the dependence of σtL

1/2 onµt strongly deviates from the linear dependence obtainedfor smaller L, indicating that βk quickly decreases on in-creasing L. Thus, the diffusion properties of the proteinchanges qualitatively at the tunnel length of L ≈ 110 A.We call the latter the cross-over length for the diffusionof protein at the tunnel.

By fitting the distribution of the escape time obtainedfrom the simulations to that given by Eq. (9) with the

7

0

5000

10000

15000

20000

0 1000 2000 3000 4000

(b)

T=0.4

GB1

βk=0.294σt L

1/2

µt (τ)

0

2500

5000

7500

10000

0 500 1000 1500 2000

(a)

T=0.8

GB1

βk=0.269σt L

1/2

FIG. 7. Dependence of the standard deviation of the es-cape time multiplied by the square root of the tunnel length,σtL

1/2, on the mean escape time, µt, for protein GB1 withthe C3 contact map at two temperatures, T = 0.8ε/kB (a)and T = 0.4ε/kB (b). The data points shown are obtainedfor various tunnel length L between 10 and 130 A. The pointsassociated with L ≤ 110 A are fitted to a linear function cor-responding to the diffusion model with βk = 0.269 A−1 (a)and βk = 0.294 A−1 (b).

values of βk as given in Fig. 7, one obtains the effec-tive diffusion constant D of the protein at the tunnel forL ≤ 110 A. Fig. 8a shows that D decreases with L.This dependence reflects the facts that the protein hasa changing shape when escaping from the tunnel, andthat the shape depends on L. When L is increased, theinitial conformation of the full-length protein at the tun-nel becomes more extended leading to a slower diffusion.Fig. 8b shows that the mean escape time increases withL. As indicated by Eq. (10), the growth of the escapetime on increasing L is due to both the longer diffusiondistance (which is equal to L) and the slower diffusionspeed. Fig. 8b also shows that for L > 110 A, the escapetime increases with L much faster than for L ≤ 110 A, inconsistency with the change in diffusion properties shownin Fig. 7.

The cross-over in the diffusion properties and the es-cape time observed at L ≈ 110 A for GB1 is related to therelative size of a tunnel compared to that of a protein.If the tunnel length is such that the protein, presum-ably with most of the secondary structures formed, canbe found completely inside the tunnel, then the escapeof the protein is much slower than the case of a shortertunnel length, in which the protein cannot fit itself en-

0

0.5

1

1.5

0 20 40 60 80 100 120 140

(a) GB1

T=0.4

T=0.8

D (

Å2 /τ

)

0

1000

2000

3000

0 20 40 60 80 100 120 140

(b)GB1

T=0.4

T=0.8

µt (

τ)

L (Å)

FIG. 8. Dependence of the diffusion constant, D, (a) andthe mean escape time, µt, (b) on the tunnel length, L, forthe GB1 protein with C3 contact map at T = 0.8 ε/kB (filledcircles) and T = 0.4 ε/kB (open circles). The values of D(data points) are obtained by fitting the escape time distri-bution obtained from simulations to the distribution functiongiven by Eq. (9) using the βk values as given in Fig. 7. In(a), the dependence of D on L is fitted by the function ofD = D∞ + aLL

−2 (solid and dashed) with D∞ and aL thefitting parameters, for L ≤ 110 A and for the two tempera-tures as indicated. In (b), the fitting curves are obtained byusing Eq. (10) and the corresponding fitting functions foundin (a).

tirely in the tunnel. The tunnel length of 110 A thus isrelated to the size of GB1, such that it can merely have asmall part outside the tunnel at the moment the chain isreleased from the PTC. The escape process is acceleratedonly by the folding of the escaped part of the protein atthe tunnel. Fig. 9a shows that in typical escape pro-cesses, the number of amino acid residues escaped fromthe tunnel, Nout, has similar trends in the time evolutionfor different tunnel lengths L, except that Nout has dif-ferent values at t = 0, the moment a full-length proteinbegins the escape process. Fig. 9b shows that the dis-tribution of Nout at t = 0 strongly depends on L. ForL = 130 A, the protein is mostly found completely insidethe tunnel, i.e. Nout = 0. On the other hand, for L = 80A, the protein always has a significant part outside thetunnel with Nout essentially ranging from 18 to 26. Forthe cross-over length L = 110 A, Nout varies between 0and 15. The cross-over length approximately correspondsto the smallest tunnel length for which Nout can have azero value.

8

0

10

20

30

40

50

60

70

0 500 1000 1500 2000 2500 3000

GB1

T=0.4

(a)

L=80 L=110 L=130

Nout

t (τ)

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

GB1

T=0.4

L=80L=110

L=130

(b)

t=0

norm

aliz

ed h

isto

gra

m

Nout

FIG. 9. (a) Dependence of the number of amino acid residuesoutside the tunnel, Nout, on the time, t, in typical escape pro-cess of protein GB1 at T = 0.4 ε/kB for the tunnel length ofL = 80 A (red), L = 110 A (magenta) and L = 130 A (blue),as indicated. The processes are complete when Nout = 56(dashed). (b) Histograms of Nout at the moment the full-length protein begins the escape process (t = 0), obtainedfrom multiple simulations of the growth process for the threetunnel lengths as considered in (a). The protein conforma-tions at t = 0 corresponding to the trajectories shown in (a)are shown as insets.

In consistency with the above mechanism, we find thata similar cross-over of the diffusion properties and the es-cape time on increasing the tunnel length is observed forthe three-helix bundle protein SpA. For SpA, the cross-over occurs at L ≈ 90 A (see Figs. S4 and S5 of supple-mentary material), quite close to the cross-over length forGB1, and is consistent with the fact the both proteins aresingle domain and of similar size. The shorter cross-overlength for SpA is a little shorter than for GB1 due to thefact that SpA can form more α-helices inside the tunnel,leading to a shorter size than GB1. Interestingly, thereal length of ribosomal exit tunnel falls between 80 Aand 100 A, very close to our estimates of the cross-overlength for the GB1 and SpA proteins. Note that the lat-ter are among the smallest single domain proteins. Itis suggested that the ribosome’s tunnel length has beenselected to facilitate an efficient escape of small single do-main proteins. Our study indicates that the cross-overtunnel length increases with the protein size, thus largeproteins would have no problem of escaping the riboso-mal tunnel from the viewpoint of diffusibility.

0

0.001

0.002

0 1000 2000 3000 4000

(a)

φ=0.0

0

0.001

0.002

0 1000 2000 3000 4000

(b)

no

rma

lize

d h

isto

gra

m

φ=0.2

0

0.001

0.002

0 1000 2000 3000 4000

(c)

escape time (τ)

φ=0.4

FIG. 10. Distribution of protein escape time without (a) andin presence of crowders at the volume fraction φ = 0.2 (b)and φ = 0.4 (c). The histograms are obtained for proteinGB1 with the C3 contact map with repulsive crowders atT = 0.8 ε/kB for the tunnel length of L = 80 A.

C. Effect of macromolecular crowding

We proceed now study the escape of nascent proteinin the presence of a crowd of macromolecules outside theribosomal tunnel. For this investigation, we fix the tun-nel length to be L = 80 A and consider various volumefractions φ of the crowders. A snapshot of an escap-ing protein molecule entering the solution of crowders isshown in Fig. S6 of supplementary material.

First, we consider the case in which the interaction be-tween the crowders and amino acids are purely repulsive.Fig. 10 shows histograms of the escape time obtained bythe simulations for the GB1 protein at T = 0.8 ε/kB forthe crowders’ volume fraction φ = 0, 0.2 and 0.4. It isshown that as φ increases the histogram is more spreadand shifted toward higher time values, meaning that theescape time is longer and more disperse in the presence ofcrowders. The histograms of the escape time are found tobe consistent with the distribution function given by Eq.(9) of the diffusion model. The fits to this function giveus the effective values of D and βk for different crowderconcentrations.

Fig. 11a shows that both D and βk decrease withφ. We find that both D and βk can be approximatelydescribed with a logarithmic dependence on φ in the fol-lowing forms:

D = D0 + a ln

(1− φ

φc

), (13)

βk = βk0 + b ln

(1− φ

φc

), (14)

9

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5

(a)

T=0.8

GB1D

βk

D (

Å2 /τ

)

βk

(Å−

1 ) 0

1000

2000

3000

0 0.1 0.2 0.3 0.4 0.5

T=0.8

(b)

GB1

µt

σt

µt,

σt (

τ)

φ

FIG. 11. (a) Dependence of the diffusion constant, D, (filledsquares) and the potential parameter, βk, (open squares) onthe volume fraction φ of crowders. (b) Dependence of themean, µt, (filled circles) and the standard deviation (open cir-cles) σt, of the escape time on φ. The data shown are obtainedfor the GB1 protein with C3 contact map at T = 0.8 ε/kB forthe tunnel length L = 80 A with repulsive crowders. The val-ues of D and βk are obtained by fitting the escape time dis-tribution from the simulations to that of the diffusion model.The fits in (a) (solid and dashed lines) have a logarithmic de-pendence on φ (see text), whereas the smooth lines in (b) arecalculated from the fitting functions shown in (a).

where D0 and k0 are the values of D and k, respectively,at φ = 0; a and b are the fitting parameters; φ < φcand φc is a cut-off volume fraction, beyond which thefull escape of the protein becomes impossible. We findthat φc = 0.5 is a good estimate. The above logarith-mic dependences suggest that the effect of the crowderson the escape process of protein has an entropic origin,as (1 − φ/φc) can be considered as the effective volumefraction accessible to the escaping protein in the spaceoutside the tunnel. Note that entropy loss due to ex-cluded volume is also the primary effect of crowding andconfinement on protein stability25,26. Having the func-tions given in Eqs. (13, 14), one can calculate the meanand the standard deviation of the escape time from Eqs.(10,11) of the diffusion model. Fig. 11b shows that themean escape time and the dispersion of the escape timeobtained from simulations at various φ also agree withthe diffusion model.

Fig. 12 shows that the standard deviation of the escapetime, σt, depends almost linearly on the the mean escapetime, µt, for both the protein and the self-repulsive ho-mopolymer at various temperatures, indicating that βkis constant for each system on changing the temperature.

0

250

500

750

0 500 1000 1500 2000

GB1 (C3)

(a)

σt (τ

)

φ=0.0φ=0.3

0

500

1000

1500

0 1000 2000 3000 4000

(b)

self−repulsivehomopolymer

σt (τ

)

µt (τ)

φ=0.0φ=0.3

FIG. 12. Dependence of the standard deviation (σt) on themean (µt) of the escape time in the cases without crowders(circles) and with repulsive crowders at volume fraction φ =0.3 (squares) for the GB1 protein with the C3 native contactmap (a) and for the self-repulsive homopolymer (b). The datapoints, obtained for various temperatures between 0.3 and 2ε/kB for the tunnel length of L = 80 A, are fitted by a linearfunction for φ = 0 (solid) and φ = 0.3 (dashed). The fitscorrespond to βk = 0.264 A−1 and 0.204 A−1 for GB1, andβk = 0.219 A−1 and 0.153 A−1 for the self-repulsive polymer,at φ = 0 and φ = 0.3, respectively.

The value of βk however depends on the volume fractionφ of the crowders, as indicated by the slopes of the fitsshown in Fig. 12. One finds that βk decreases when φincreases from 0 to 0.3 for both the protein and the poly-mer, indicating that diffusion is slower in the presenceof crowders. Again here, one also finds that for bothcases, with and without crowders, the value of βk forGB1 is larger than for the self-repulsive homopolymer,confirming the enhancing effect of folding on the escapeof protein.

Fig. 13 shows the dependence of the median escapetime on temperature for the protein GB1 and the self-repulsive homopolymer in the presence of repulsive crow-ders at the volume fraction φ = 0.3. It is shown thatthe log-log plot of this dependence for both the proteinand the homopolymer has similar characteristics to thatfound in Fig. 4 for the case without crowders, except thatthe escape times are longer with the crowders. For thehomopolymer, the escape time decreases with tempera-ture linearly in the log-log plot with a slope close to −1,indicating that the diffusion constant of the polymer alsodepends linearly on temperature like for the case with-

10

1000

10000

0.2 0.3 0.4 0.6 0.8 1 1.2 1.6 2

φ=0.3self−repulsivehomopolymer

GB1

t esc (

τ)

T (ε/kB)

FIG. 13. Log-log dependence of the median escape time,tesc, on temperature, T , in presence of crowders. The datapoints shown are for a self-repulsive homopolymer with re-pulsive crowders (squares), the GB1 protein with repulsivecrowders (filled circles), GB1 with attractive crowders withinteraction strength ε1 = 0.3ε (open circles), and GB1 withattractive crowders with ε1 = 0.5ε (triangles). The C3 nativecontact map is used for GB1. In all cases, the tunnel length isL = 80 A and the crowders’ volume fraction is φ = 0.3. Theescape times of the homopolymer are fitted by a straight linewith a slope equal to −1.05.

out crowders. In the presence of repulsive crowders, theescape times of protein at high temperatures are close tothose of the self-repulsive homopolymer. At low temper-atures, favorable for folding, the protein has significantshorter escape times than the polymer, indicating thatthe impact of folding on the escape time is not affectedby the crowders.

Fig. 13 also shows the escape times of protein in thepresence of attractive crowders with two different interac-tion strengths, ε1 = 0.3ε and ε1 = 0.5ε, of the attractionbetween crowder and amino acid. It can be seen thatthe attractive crowders make the escape faster than therepulsive crowders, but only at high and intermediatetemperatures. At low temperatures (T ≤ 0.4 ε/kB), theprotein escapes more slowly in the presence of attractivecrowders than of the repulsive ones. Furthermore, theescape time also increases when the attraction strengthε1 increases at low temperatures. The reason for thisincrease is that, in contrast to repulsive crowders, at-tractive crowders destabilize the native interactions inprotein. Thus, folding is less favorable in the presenceof attractive crowders leading to a weaker enhancementof folding on the escape speed. Fig. 14 shows that thedistributions of the root mean square deviation (rmsd)from the native state and the radius of gyration of pro-tein conformations obtained at the moment of full escapefrom the tunnel are shifted towards higher values whenswitching from repulsive crowders to attractive crowders.

It is believed that ribosome-associated chaperones,such as the trigger factor (TF) in prokaryotes or the

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10 12 14 16 18

(a)

φ=0.3T=0.3

rmsd (Å)

repulsive crowdersattractive crowders

0

0.2

0.4

0.6

0.8

9 10 11 12 13 14 15 16 17 18

(b)

Nor

mal

ized

his

togr

am

Rg (Å)

repulsive crowdersattractive crowders

FIG. 14. Histogram of the root mean square deviation(rmsd) from the native state (a) and the radius of gyration(Rg) (b) of the protein conformations at the moment of fullescape from the exit tunnel. The data shown are obtainedfrom 500 independent simulation trajectories at T = 0.3 ε/kBfor protein GB1 with either repulsive crowders (solid) or at-tractive crowders (dashed) at volume fraction φ = 0.3. Theattractive crowders have the interaction strength of ε1 = 0.3ε.

Hsp70 Ssb and NAC (nascent chain-associated complex)in eukaryotes, are of particular importance in guidingnascent proteins to fold correctly15. The binding of thesechaperones to the ribosome effectively leads to a veryhigh concentration of chaperones near the exit tunnel38,promoting their interaction with nascent polypeptide. Asa result, the chaperones quickly bind to unfolded, hy-drophobic segments of the polypeptide before these seg-ments can fold or misfold, keeping the nascent chain un-folded. Our simulations with attractive crowders showa similar effect, as indicated in Fig. 14, that attractivecrowders make the fully released protein conformationless native-like and less compact than in the case withrepulsive crowders. Our simulations predict that the es-cape time of nascent protein increases in the presence ofchaperones due to both their crowding effect and theirattractive interaction with hydrophobic segments.

IV. CONCLUSION

Post-translational escape of nascent protein at the ri-bosome is a stochastic process governed by protein nativeinteractions, the geometry of the ribosomal exit tunnel

11

and macromolecular crowders outside the tunnel. Wehave shown that non-local native interactions speed upthe escape process at temperatures favorable for folding,while the local interactions responsible for the formationof α-helices slow it down. As a consequence, proteinswith a content of β-sheets tend to escape faster thanthose with only α-helices in the native state. Increas-ing the tunnel length or the concentration of crowdersalso slows down the protein escape. In the view that theconcomitant folding and escape of nascent protein at theexit tunnel are beneficial for both the productivity of theribosome and the protection of nascent protein againstaggregation, it can be conjectured that the protein syn-thesis machinery has been evolved to facilitate both thefolding and the escape of nascent proteins. In support ofthis conjecture, we have shown that real ribosomal exittunnel has adopted the length that is close to a cross-overlength of the tunnel, beyond which the protein escapefalls into a regime of a much slower diffusion for smallsingle domain proteins.

Our study shows that repulsive crowders outside thetunnel induce an entropic effect on the diffusion proper-ties of protein at the tunnel, leading to increased escapetimes but does not change the enhancing effect of fold-ing on the escape process. The latter effect is changedonly in the case of attractive crowders, whose attractionto amino acids competes with native interactions in thenascent polypeptide. Due to this competition the fullyescaped protein conformation is more extended and lessnative-like. The unfavorable effect of attractive crowderson the folding of nascent protein is also reflected on theincreased escape times at low temperatures, as shown inour study. It is suggested that the ribosome-associatedchaperones induce similar effects on nascent polypeptidesas found with attractive crowders.

Low-dimensional diffusion models have been success-fully applied to study complex dynamics39–41. Our workproves that the simple diffusion model considered is use-ful for understanding the escape of protein at the exittunnel. The results suggest that intrinsically disorderedproteins, considered as the self-repulsive homopolymerin our study, have longer escape time than foldable pro-teins with a significant number of long-range contacts,and their diffusion is the most akin to that of a Brown-ian particle.

This research is funded by Vietnam National Founda-tion for Science and Technology Development (NAFOS-TED) under Grant No. 103.01-2016.61.

SUPPLEMENTARY MATERIAL

See supplemental material for the specific heats for theGo-like model of GB1 with different native contact maps,the dependence of the escape time on temperature for thehelical protein 2rjy and a single α-helix, the dependenceof the dispersion of the escape time on the mean escapetime for SpA, the dependence the diffusion constant and

the mean escape time on the tunnel length for SpA, anda snapshot of an escaping protein entering a solution ofmacromolecular crowders.

1A. N. Fedorov and T. O. Baldwin, J. Biol. Chem. 272, 32715(1997).

2L. D. Cabrita, C. M. Dobson, and J. Christodoulou, Curr. Opin.Struct. Biol. 20, 33 (2010).

3D. V. Fedyukina and S. Cavagnero, Ann. Rev. Biophys. 40, 337(2011).

4N. Voss, M. Gerstein, T. Steitz, and P. Moore, J. Mol. Biol. 360,893 (2006).

5J. Lu and C. Deutsch, Nat. Struct. Mol. Biol. 12, 1123 (2005).6G. Ziv, G. Haran, and D. Thirumalai, Proc. Natl. Acad. Sci.USA 102, 18956 (2005).

7A. Kosolapov and C. Deutsch, Nat. Struct. Mol. Biol. 16, 405(2009).

8H. Nakatogawa and K. Ito, Cell 108, 629 (2002).9D. Marenduzzo, T. X. Hoang, F. Seno, M. Vendruscolo, andA. Maritan, Phys. Rev. Lett. 95, 098103 (2005).

10G. Zhang, M. Hubalewska, and Z. Ignatova, Nat. Struct. Mol.Biol. 16, 274 (2009).

11E. Siller, D. C. DeZwaan, J. F. Anderson, B. C. Freeman, andJ. M. Barral, J. Mol. Biol. 396, 1310 (2010).

12E. P. O’Brien, M. Vendruscolo, and C. M. Dobson, Nat. Comm.5, 2988 (2014).

13D. A. Nissley and E. P. O’Brien, J. Am. Chem. Soc. 136, 17892(2014).

14J. Frydman, Ann. Rev. Biochem. 70, 603 (2001).15R. D. Wegrzyn and E. Deuerling, Cell. Mol. Life Sci. 62, 2727

(2005).16C. B. Anfinsen, Biochem. J. 128, 737 (1972).17M. S. Evans, I. M. Sander, and P. L. Clark, J. Mol. Biol. 383,

683 (2008).18K. G. Ugrinov and P. L. Clark, Biophys. J. 98, 1312 (2010).19C. M. Kaiser, D. H. Goldman, J. D. Chodera, I. Tinoco, and

C. Bustamante, Science 334, 1723 (2011).20M. Chwastyk and M. Cieplak, J. Chem. Phys. 143, 045101

(2015).21M. Chwastyk and M. Cieplak, J. Phys. Cond. Matt. 27, 354105

(2015).22P. T. Bui and T. X. Hoang, J. Chem. Phys. 144, 095102 (2016).23N. Alexandrov, Prot. Sci. 2, 1989 (1993).24C. M. Dobson, Nature 426, 884 (2003).25A. P. Minton, J. Biol. Chem. 276, 10577 (2001).26H.-X. Zhou, G. Rivas, and A. P. Minton, Annu. Rev. Biophys.37, 375 (2008).

27N. Go, Ann. Rev. Biophys. Bioeng. 12, 183 (1983).28T. X. Hoang and M. Cieplak, J. Chem. Phys. 112, 6851 (2000).29T. X. Hoang and M. Cieplak, J. Chem. Phys. 113, 8319 (2000).30C. Clementi, H. Nymeyer, and J. N. Onuchic, J. Mol. Biol. 298,

937 (2000).31M. Cieplak and T. X. Hoang, Int. J. Mod. Phys. C 13, 1231

(2002).32H. Kaya and H. S. Chan, Prot. Struct. Func. Bio. 40, 637 (2000).33D. Klimov and D. Thirumalai, Phys. Rev. Lett. 79, 317 (1997).34N. G. Van Kampen, Stochastic Processes in Physics and Chem-istry, 3rd ed. (Elsevier, 1992).

35D. R. Cox and H. D. Miller, “The theory of stochastic processes,”(Chapman and Hall, 1965) pp. 219–223.

36K. W. Plaxco, K. T. Simons, and D. Baker, J. Mol. Biol. 277,985 (1998).

37P. E. Wright and H. J. Dyson, Nat. Rev. Mol. Cell Biol. 16, 18(2015).

38S. N. Witt, Prot. Pep. Lett. 16, 631 (2009).39H. A. Kramers, Physica 7, 284 (1940).40R. Zwanzig, Proc. Natl. Acad. Sci. USA 85, 2029 (1988).41B. Peters, P. G. Bolhuis, R. G. Mullen, and J.-E. Shea, J. Chem.

Phys. 138, 054106 (2013).

1

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

0

300

600

900

1200

1500

0.4 0.6 0.8 1 1.2 1.4

GB1

C1

C1(LJ 6-12)

C2C3

C (

kB)

T (ε/kB)

FIG. S1. Dependence of the specific heat, C, on the temperature, T , for the GB1 protein in Go-like models with threedifferent native contact maps, C1 (magenta), C2 (green) and C3 (red), with the 10-12 Lennard-Jones (LJ) potential for thenative contacts (see Methods section of the main text); and for the C1 contact map with the 6-12 LJ potential for nativecontacts (dashed). The folding temperature Tf , defined as the temperature of the maximum of the specific heat, is equal to0.866, 0.888 and 1.004 ε/kB for the models with the C1, C3, and C3 maps, respectively, with the 10-12 potential; and 0.922ε/kB for the model with the 6-12 potential.

100

1000

10000

0.2 0.4 0.6 1 2 3

Tf

2rjy (C3)

self−repulsivehomopolymer

t esc (

τ)

T (ε/kB)

FIG. S2. Dependence of the median escape time, tesc, on the temperature, T , for the villin headpiece protein (PDB code:2rjy) (circles), a helical protein of length N = 64 amino acids, and a same-length self-repulsive homopolymer (squares). Theescape times are calculated for the tunnel of length L = 80 A and without the crowders. The protein is considered in theGo-like model with the C3 native contact map. The protein native state is shown as inset and the arrow indicates its foldingtemperature Tf .

2

100

1000

10000

100000

0.2 0.4 0.6 1 2 3

5omd (C3)

self−repulsivehomopolymer

t esc (

τ)

T (ε/kB)

FIG. S3. Same as Fig. S2 but for a single α-helix of length N = 66 amino acids from the S. cerevisiae Ddc2 N-terminalcoiled-coil domain (PDB code: 5omd).

0

5000

10000

15000

20000

0 1000 2000 3000 4000

(a)

T=0.8

βk=0.251

SpA

σt L

1/2

0

10000

20000

30000

0 1000 2000 3000 4000 5000 6000

(b)

T=0.4

βk=0.257

SpA

σt L

1/2

µt (τ)

FIG. S4. Dependence of the standard deviation of the escape time, multiplied by the square root of the tunnel length, σtL1/2,

on the mean escape time, µt, for the Z domain of Staphylococcal protein (SpA) in the Go-like model with the C3 native contactmap. The panels show the simulation data for two temperatures, T = 0.8 ε/kB (a) and T = 0.4 ε/kB (b). Different data pointscorresponding to different tunnel length L, considered to take values between 30 A and 120 A. The data points of L ≤ 100 Aare fitted to the diffusion model (dashed line) with a constant βk, equal to 0.251 A−1 (a) and 0.257 A−1 (b), as indicated.

3

0

0.5

1

1.5

0 20 40 60 80 100 120

(a)

T=0.8T=0.4

SpA

D (

Å2 /τ

)

0

2000

4000

6000

0 20 40 60 80 100 120

(b)SpA

T=0.8T=0.4

µt (

τ)

L (Å)

FIG. S5. Dependence of the diffusion constant, D, (a) and the mean escape time, µt, (b) on the tunnel length L, for proteinSpA at T = 0.8ε/kB (filled circles) and T = 0.4 ε/kB (open circles). The values of D for L between 30 A and 90 A are obtainedby fitting the escape time distribution obtained from simulations to the diffusion model with the βk values given in Fig. S2.The dependence of D on L is fitted by the function D = D∞ + aLL

−2, with fitting parameters D∞ and aL, for T = 0.8 ε/kB(solid) and T = 0.4 ε/kB (dashed). The dependence of µt on L obtained by the diffusion model is shown as solid and dashedlines for the two temperatures, as indicated.

FIG. S6. Snapshot of the protein GB1 (red) escaping into a solution of crowders (cyan) with the crowders’ volume fractionφ = 0.3.