Molecular Dynamics Simulation of Folding and Diffusion of Proteins in Nanopores

0.17 MB·English

by  Leili Javidpour

Molecular Dynamics Simulation of Folding and Diffusion of Proteins in Nanopores
8
0
0 Leili Javidpour,a Muhammad Sahimi,b,† and M. Reza Rahimi Tabara,c
2 aDep. of Physics, Sharif University of Technology, Tehran 11365, Iran

n bMork Family Department of Chemical Engineering & Materials Science, University of Southern California, Los Angeles,
a California 90089-1211, USA
J cCNRS UMR 6529, Observatoire de la Coˆte d’Azur, BP 4229, 06304 Nice Cedex 4, France

5
2 A novel combination of discontinuous molecular dynamics and the Langevin equation, together
withanintermediate-resolution model,areusedtocarryoutlong(severalµs)simulation andstudy

] folding transition and transport of proteins in slit nanopores. Both attractive (U+) and repul-
ft sive (U−) interaction potentials between the proteins and the pore walls are considered. Near the
o folding temperature T and in the presence of U+ the proteins undergo a repeating sequence of
f
s
folding/partially-folding/ unfolding transitions, while T decreases with decreasing pore sizes. The
. f
t oppositeistruewhen U− ispresent. Theproteins’effectivediffusivityD iscomputedasafunction
a
m oftheirlength(numberoftheaminoacidgroups),temperatureT,theporesize,andtheinteraction
potentials U±. Farfrom T ,D increases (roughly) linearly with T,but dueto thethermal fluctua-
- f
d tionsand theireffect on theproteins’structurenearTf,thedependenceof D onT in thisregion is
n nonlinear. Under certain conditions, transport of proteins in smaller pores can be faster than that
o in larger pores.
c PACS: 87.15.Aa, 83.10.Mj, 87.15.Cc, 87.15.Vv, 87.83.+a
[

2 Proteins’ importance to biological systems cannot be invivofoldingisnotalwaysspontaneous;rather,asubset
v overstated [1]: as enzymes they catalyze and regulate of proteins may require molecular chaperones.
7 cells’ activities; tissues are made of proteins, while as Protein (enzyme) immobilization using porous solid
8
antibodies proteins are a vital part of the immune sys- support,viaadsorption,encapsulation,andcovalentlink-
3
tem. Proteins with globular structure fold into compact ing, has been used for a long time [13,14]. Such practi-
3
0 and biologically active configurations, and an important cal applications as biocatalysis [15] and biosensors also
7 problemisunderstandingthe mechanismsbywhichthey entail not only better understanding of the folding in
0 attain their folded structure, and factors that contribute confined media, but also transport of proteins in such
t/ to the folding [2–4]. Such understanding is important media. At the same time, protein purification using
a
due to debilitating illnesses, such as Alzheimer’s and nanoporous membranes is also gaining attention [16].
m
Parkinson’sdiseases,thatarebelievedtobethe resultof SiC nanoporous membranes [17] allow [18] diffusion of
-
accumulationoftoxicproteinaggregates [5–8],aswellas proteins up to 29000 Daltons, but exclude larger ones.
d
n to the industrial production of enzymes and therapeutic Despite the fundamental and practical significance of
o proteins based on the DNA recombinant method [9]. transportofproteininconfinedmedia,thereis currently
c While the three-dimensional (3D) structure of native little understanding of the phenomenon.
:
v proteinsiscontrolledbytheiraminoacidsequence [2–4], The goalof this paper is twofold. First, we use molec-
i theirtransportpropertiesandthekineticsoftheirfolding ular dynamics (MD) simulation to study protein folding
X
depend on the local environment. But, whereas protein andstabilityinslitnanopores. Second,weutilizeanovel
r
a foldingindilutesolutionsunderbulkcondition,typically combination of MD simulation and the Langevin equa-
usedinin vitrostudies,isrelativelywell-understood,the tion (LE) to study protein transport in the nanopores.
more important problem of protein folding in a confined To our knowledge, our combination of the MD simula-
medium is not. The environment inside a cell in which tion and the LE has never been proposed before, nor
proteins fold is crowded, with the volume fraction of the has there been any simulation of transport of proteins
crowding agents (such as RNA) may be 0.2-0.3. Thus, in nanopores. For such important practical applications
even in the absence of interactions between proteins and as membrane purification, biocatalysis, and sensors, the
othercellularmolecules,theirmovementinsidethecellis transport of proteins in nanopores is of utmost impor-
limited. Thelimitationaffectsproteins’stability. Exper- tance. Aslitnanoporeisareasonablemodelforthetype
iments indicated [10] that confinement often stabilizes ofporesthatoneencountersinsuchapplications [15–18]
the proteins’ native structure [11], denatures them in and, despite its simplicity, it might also be a reasonable
the limited space of the cage model, first suggested by model for the pores in biological membranes.
Anfinsen [2–4], and accelerates folding relative to that Some MonteCarlo [19]andMD [5,20]simulationsof
in bulk solutions. Studies of proteins of different native proteins’behaviorin nanoporeswere reportedbefore. In
architecturesincylindricalnanporesindicated [12]that, particular, Lu et al. [5] and Cheung et al. [20] studied
folding of proteins in spherical pores of different radii. between side chains and backbone N and C UAs to hold
Cheung et al. studied the phenomenon as a function the side-chain beads fixed relative to the backbone. All
of the volume fraction of a crowding agent, which they of this keep the interpeptide group in the trans configu-
modeledby a bed ofhardspheres withrepulsive interac- ration, and all the residues as L isomers,as required.
−
tion with the proteins. While a spherical pore may be a The potential U between a pair ij of bonded beads,
ij
suitable model for the cavity of GroEl-GroES complex, separated by a distance r , is given by, U = , for,
ij ij
∞
itisnotsoforthe poresofmembranes,biocatalysts,and r l(1 δ), and r l(1 + δ), and, U = 0 for
ij ij ij
≤ − ≥
sensors that are of prime interest to us. Instead, the slit l(1 δ)<r <l(1+δ). Here,l isthe idealbondlength,
ij
−
(and cylindrical) pores are more appropropriate. More- and δ = 0.02375 is the tolerance in the bond’s length
over, for the types of applications that we consider, the [23–26]. Thehydrophobic(HP)interactionsbetweenthe
pore space consists of interconnected channels, which is side chains and the H in the sequence, if there are at
completely different from what Refs. [5,20] considered. least 3 intervening residues between them, is given by,
In addition, the protein model that we use (see be- U = , ǫ , and 0 for, r σ , σ < r
HP HP ij HP HP ij
∞ − ≤ ≤
low) is, in our opinion, much more realistic than what 1.5σ , andr >1.5σ , respectively,where σ is the
HP ij HP HP
the previous investigations [5,20] utilized. For example, HP side-chains’ diameter.
they used a simplified model for the amino acids that The HB interaction may occur between the N and C
was based on two united atom (UA) beads. Moreover, beadswithatleast3interveningresidues,buteachbead
thesidechainsoftheamino-acidresidueswerenotexplic- may not contribute to more than one HB at any time,
itly considered. The modelthatwe utilize representsthe with the range of the interaction being about 4.2 ˚A.
amino acids using four UA beads (see below), while the The HBs are stable when the angles in N-H-O and C-
side chains arealso consideredexplicitly, hence honoring O-H, controlled by a repulsive interaction between each
the proteins’ structure much more realistically. of the N and C beads with the neighboring beads of the
We simulate de novo-designed α family of proteins other one, are almost 180◦. Thus, if a HB is formed be-
[21], which consists of only 4 types of amino acids in tween beads N and C , a repulsive interaction between
i j
their 16-residuesequence,simplified further [22]to a se- neighbor beads of N , namely, C and C , with C
i i−1 αi j
quenceofhydrophobic(H)andpolar(P)residues. Using is assumed, and similarly for the neighbor beads of C ,
j
periodicityinthe H-Psequenceofthe16-residuepeptide namely, N and C , with the N bead.
j+1 αj i
α , we made 3 other sequences with lengths ℓ = 9, 23 An N or C bead at one end of the protein has only
1B
and30residues. As the four proteins havesimilarnative one neighbor bead in its backbone, instead of 2. Hence,
structures, the differences in their behavior is attributed controlling the HB angles will be limited, causing the
to their lengths. The simulations indicated that they all HBswithoneoftheirterminalconstituentstobelessre-
fold into an α-helix. strictedand,thus,morestablethanthe otherHBs. This
The proteins are modeled by an intermediate- maycauseformationofnon-α-helicalHBsinapartofthe
resolutionmodel [23–26],withseveralchangesdescribed protein between the N and C beads, and of semistable
below. Every amino acid is represented by four UA structures that influence the results. To address this
groups or beads. A nitrogen UA represents the amide problem, assume that the N-terminal bead, N , has a
1
N and hydrogen of an amino acid, a C UA represents HB with C . For i = 1, bead C does not exist to
α j i−1
the α-C and its H, and a C UA for the carbonyl C and havearepulsiveinteractionwithC andhelpcontrolthe
j
O. The fourth bead R represents the side chains, all of HB angles. So, we use C . Not only can we consider
α1
which are assumed to have the same diameter as CH . the repulsion between this bead and C , but also we de-
3 j
Allthebackbonebondlengthsandbondanglesarefixed fine an upper limit for their distance so as to control
at their ideal values, and the distance between consecu- the motion freedom of N and C that constitute the
1 j
tive C UA is fixed according to experimental data. beads in the HB. The potential U of such interactions
α kl
Tocarryoutlongandefficientsimulations,weusedis- isgivenby,U = , ǫ , 0,and forr 1(σ +σ ),
continuous MD (DMD) [27]. This allowed us to carry 1(σ +σ ) <krl ∞d ,HdB < r ∞d , ankdl,≤r 2>kd , rle-
2 k l kl ≤ 1 1 kl ≤ 2 kl 2
out 5 µs MD simulations, one order of magnitude longer spectively.
than the previous simulations. The forces acting on the Two H atoms have chemical bonds with the nitrogen
beads are the excluded-volume effect, and attractionbe- intheproteins’N-terminal,andarefreetorotatearound
tweenbondedandpseudobondedbeads,betweenpairsof the N -C bond, while at the same time satisfying the
1 α1
backbone beads during HB formation, and between hy- constraints on the angles between the chemical bonds of
drophobic side chains. Nearest-neighborbeads alongthe N . Thus, if a HB is formed, one of the two H atoms
1
chainbackbonearecovalentlybonded,asaretheC and lies in the plane of N, O and C, such that the angles
α
RUAs. Pseudobondsarebetweennext-nearestneighbor in N-H-O and C-O-H are as close to 180◦ as possible.
beads along the backbone to keep the backbone angles Hence,weforcethe maximumdistancebetweenC and
α1
fixed; between neighboring pairs of C beads to main- C to be the same as the maximum distance d between
α j 2
tain their distances close to the experimental data, and C and C in the usual HBs, and similarly when the C-
αi j
terminal C has a HB with N . This allows us to control scale over which this effect is important is much differ-
ℓ i
theanglesinaHBthatcontainsN . TheT dependence ent from ∆t, we apply, at the end of the time period
1
−
(T isdimensionless)ofd (in˚A), obtainedfromseparate ∆t, the Langevin equation (LE) to the proteins’ CM to
2
MD simulations (the details will be given elsewhere), is, correcttheir velocity due to the presenceof the solvent’s
d 5.53 0.019/T forN -C ,andd 5.69 0.044/T molecules. To do so, we represent a protein as a particle
2 1 αj 2
≃ − ≃ −
for C -C . with amass m andaneffective radius equalto its radius
ℓ αi
There is also hard-core repulsion between two un- of gyrationR . Then, the force F on its CM is givenby,
g
bonded beads that have no HB and HP interactions. At
F=m(v v )/∆t. (1)
the sametime, interactionsbetweena pairofbeads,sep- e b
−
arated along the chain by 3 or fewer bonds, are more
The discretized LE is given by,
accuratelyrepresentedbythosebetweentheatomsthem-
selves,notthe UAs. Thus,we developeda variantofthe ∆v=v v =F∆t/m ξv ∆t/m+∆R(∆t), (2)
n b b
− −
previousmodels [23–26]toaccountforsuchinteractions:
where,ξ =6πR η,∆RisaGaussianrandomforce(with
the beads are allowed to overlap by up to 25% of their g
zero mean and variance 2k Tξ∆t/m2), and v is the
bead diameters, while for those separated by 4 bead di- B n
speed after applying the LE (acting as the v for the
ameters the allowed overlapis 15% of their diameters. b
next LE application). Thus,
We use a slit nanopore,modeled as the space between
two 2D structureless carbon walls in the xy plane be- dv=v v = ξ∆tv /m+∆R(∆t), (3)
n e b
tween z = h/2, with periodic boundary conditions in − −
thexandy±directions. Theinteractionbetweenthewalls whichyieldsv ,thevelocityoftheproteinscorrectedfor
n
and the protein beads is, U = , ǫ , 0, ǫ , thesolventeffect. TheDMDsimulationisthencontinued
PW PW PW
and for, z (h/2 d ), ∞(h/2− d ) <−z for another time period ∆t using v as the v , the LE is
X 3X 3X X n b
∞ ≤ − − − − ≤
(h/2 d d ), (h/2 d d ) < z < applied againto correctthe proteins’velocity at the end
3X 4X 3X 4X X
− − − − − −
h/2 d d , h/2 d d < z h/2 d , of the period, and so on.
3X 4X 3X 4X X 3X
− − − − ≤ −
and z h/2 d , respectively, where z is the dis- We now present the results of our simulations. Con-
X 3X X
≥ −
tance between the center ofa beadX and the walls. For sider the case of attractive interaction potential U+ be-
all the beads, ǫ = 1ǫ , so chosen to represent re- tween the proteins and the pore’s walls. A folded state
PW 8 HB
alistically the competition between protein folding and attaches itself to the walls only through its end groups,
its beads’ interaction with the walls. To estimate d while unfolded ones may completely attach themselves
3X
and d , the energy and size parameters between the C to the walls. Thus, with U+ the decrease in the average
4X
atoms in the walls andvarious beads were calculatedus- potential energy ofthe unfolded states is largerthan the
ing Lorentz-Berthelot mixing rules, σ = 1(σ +σ ), corresponding decrease for the folded one. Hence, com-
CX 2 C X
andǫCX =√ǫCǫX,whereX =N,Cα,CandR.Then,us- pared to the bulk, the unfolded states in the pore with
ingseparatesimulations(detailswillbegivenelsewhere), a U+ are more stable than the folded one, in qualitative
the interaction potential U between different beads agreement with Refs. [4,20] for spherical cavities.
CX
was estimated. The distances at which U and its sec- Figure 1 shows a sequence of events for a protein of
CX
ondderivativewerezeroweretakenasd andd +d . size ℓ = 16 in a pore of size h = 1.75 nm at T 0.13.
3X 3X 4X f
≃
The results (in ˚A) are, d =2.85, 3.02, 3.14, and 3.31, The protein changes its state from completely folded to
3X
and d =0.96, 1.01, 0.98, and 1.12, for X =N, C , C, apartiallyfoldedto anunfoldedonewhichiscompletely
4X α
and R, respectively. attached to the pore’s wall (frame D). Due to U+, the
We now describe how the effective diffusivity D of the transitions occur easily and repeatedly, even after a long
proteins is computed. To do so, one must take into ac- time. Note that, in moving from B to C, the set of de-
count the effect of the solvent on the motion of the pro- formed α helical HBs changes, hence indicating rapid
−
teins. In the previous works the solvent’s effect on the dynamics ofthe HB formationanddeformationnear T .
f
proteins’ motion was included only implicitly by the HP Also shown in Figure 1 are protein configurations
attraction between the side chains. While this might be (frames E and F with pore size h=1.5 nm) for ℓ=9 at
appropriateforstudyingthe folding,itisnotsoforcom- T = 0.08. In these pores, a protein of length ℓ = 9 does
puting D, since the solvent’s viscosity η strongly and not attain its native state at low T. Instead, it has a U
directly affects D. To explicitly include the solvent ef- shape with its two sides attached to the walls; it has 4
fect, we have developed the following model which, to HBs,onlyoneofwhichisα helical(the nativestatehas
−
our knowledge, is new. 5 α helical HBs), and more of its atoms are close to the
−
We first carry out DMD simulation for a time period walls than those in the folded state. Although the po-
∆t. Suppose that the speeds of the proteins’ center of tentialenergyofsuchunfoldedstatesisroughlythesame
mass (CM) at the beginning and end of the period ∆t as one in the folded one, entropic effects which favor the
are, respectively,v and v . Since the solvent’s viscosity unfolded states are also important. Upon further cool-
b e
affects the proteins’ velocity in the pore, but the time ing at T below the apparentfolding temperatureT , the
f
proteinbecomes trapped in the U shape without enough stronglyvarieswithT. Consequently,D nolongervaries
kinetic energytoovercomethe energybarriertoattaina linearlywithT. Increasingtheproteins’lengthdecreases
folded state. Thus, such configurations do not represent D, as expected.
truly folded states. Figure4presentsthe diffusivities foraproteinoffixed
WealsofindthatT decreaseswiththeporesize,which length, ℓ = 16, in four different pores, and compares
f
is due to the attractive interaction energy between the the results with those in the bulk. Note that, the T
f
protein and the walls. The decrease in T is indicative for the bulk state and those for the pores are not the
f
of the more stable unfolded states (or less stable folded same. Therefore, due to the strong fluctuations of R
g
state). and D in the region around T , the pore and bulk dif-
f
The opposite (namely, increasing T with decreasing fusivities cannot be directly compared. The comparison
f
pore size) is true if the interaction is purely repulsive, is possible mainly attemperatures far fromT (both be-
f
U−. In that case, proteins are even more confined in low and above T ). As Figure 4 indicates, at least for
f
the pores. Thus, the number of possible unfolded states the D D (taking into account the numerical
bulk pore
≥
and,hence,theirtotalentropy,willbesmaller [28]. But, uncertainties.
duetoitscompactconfiguration,confinementaffectsthe The results shown in Figures 3 and 4 were for an at-
entropy of the folded state less strongly, implying that, tractivepotentialU+betweentheproteinsandthepores’
inthepresenceofU− thefoldedstateismorestablethan walls. Figure5comparesthe diffusivities fora proteinof
that in the bulk. length ℓ = 16 in a pore of size h = 1.75 nm for both at-
Figure2presentstheaverageinteractionenergy U tractive and repulsive interaction potentionals U±, and
PW
h i
oftheproteinswiththewalls,computedbytheweighted compares them with the bulk values. Generally speak-
histogramanalysismethod [29]. Coolingthe proteins at ing, for T T the diffusivities with U− are larger than
f
≤
T > T increases U , as well as the average n those with U+.
f PW α
|h i| h i
of the α helical HBs (which is, however, very small). Inourcurrentwork,wearecarryingoutextensivesim-
−
Near T n is nonnegligible,and the proteins canonly ulations for computing the effective diffusivity D as a
f α
h i
laterallyattachthemselvestothewalls,hencedecreasing functionofthestrengthofthe interactionpotentialU
PW
U . ByloweringT further,nearlytheentireα-helix between the proteins and the walls, its sign (attractive
PW
|h i|
is formed, and U increases again. Thus, Figure versus repulsive), and other relevant parameters. The
PW
|h i|
2 indicates that, not only does the interaction with a results will be presented in a future paper.
nanopore disturb folding, but also folding to a definite We thank M. R. Ejtehadi, C. K. Hall, and M. D. Niry
structuredisturbstheproteins’interactionwiththepore. for useful discussions. The work of LJ was supported by
Before describing the results for the effective diffusivi- Iranian Nanotechnology Initiative.
ties,weshouldpointoutthat,overthetemperaturerange
that we have simulated, the proteins resemble a prolate
ellipsoid. Thus, they become elongated in very small
nanopores, while they can “stand up” in larger pores,
such that their ends touch the pores’ walls, if they are
long enough. Such configurations are also shown in Fig-
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FIG. 1. Various configurations (A-D) that a protein of length ℓ = 16 takes on in a pore at T ≃ 0.13. Blue, gray,
f
green, and pink spheres show, respectively, the N bead, Cα, C, and the side chains. In between the walls and the thin
lines at a distance d3, UPW =∞,beyond which the U+ acts for a distance d4. Times t are in ps. Frames E and F show
a short protein (lipid) with ℓ = 9 that has not reached its native state. Frames G and H show the configurations of a
protein parallel and perpendicular to thewalls.
−1
〉
W
P
U −2
〈
−3
0.08 0.1 0.12 0.14 0.16
T
FIG.2. The average interaction energy hUPWi for a pore of size h=1.75 nm and protein of length ℓ=16.
7
ℓ=9
ℓ=16
h/R = 4.10
ℓ=23 g
ℓ=30
5
7
0
1
×
)
s
2/
m
c
(
D
3
1
0.08 0.1 0.12 0.14 0.16 0.18
T
FIG. 3. Dependence of the effective diffusivity D on the temperature T for various protein length ℓ, in a pore with
h/Rg =4.1. Arrows indicate the location of the folding temperature Tf.
h=1.75
h=3
h=7 ℓ=16
h=12
4.5 Bulk
7
0
1
× 4
)
s
2/
m
c
(
D
3.5
3
0.08 0.1 0.12 0.14 0.16
T
FIG.4. Dependenceof theeffectivediffusivityD, foraprotein ofsize ℓ=16, onthetemperatureT andtheporesize
h. The radius of gyration Rg is constant, computed at T =0.08 underbulk condition (Rg/h=0).
5
Attractive pore
ℓ=16
Repulsive pore
Bulk h =1.75nm
4.5
7
0
1
×
)
s 4
2/
m
c
(
D
3.5
3
0.08 0.1 0.12 0.14 0.16
T
FIG.5. Comparisonoftheeffectivediffusivitiesinaporewithattractiveandrepulsiveinteractionsbetweentheprotens
and the pore’s walls. The arrows indicate the location of T .
f