Propulsion of a molecular machine by asymmetric distribution of reaction--products

0.15 MB·English

by  Ramin Golestanian

Propulsion of a molecular machine by asymmetric distribution of reaction–products
Ramin Golestanian,1,2,3 Tanniemola B. Liverpool,1,3 and Armand Ajdari4
1Isaac Newton Institute for Mathematical Sciences, Cambridge, CB3 0EH, UK
2Institute for Advanced Studies in Basic Sciences, Zanjan 45195-159, Iran
3Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
4Laboratoire de Physico-Chimie Th´eorique, UMR CNRS 7083,
ESPCI, 10 rue Vauquelin, F-75231 Paris Cedex 05, France
7 (Dated: February 6, 2008)
0
0 A simple model for the reaction-driven propulsion of a small device is proposed as a model for
2 (part of) a molecular machine in aqueous media. Motion of the device is driven by an asymmetric
distribution of reaction products. The propulsive velocity of the device is calculated as well as the
n
scale of thevelocity fluctuations. Theeffectsof hydrodynamicflowas well as anumberof different
a
scenarios for the kinetics of thereaction are addressed.
J

9 PACSnumbers: 07.10.Cm,82.39.-k,87.19.St

]
t
f Molecular motors are machines that convert chemi-
o y
cal energy to mechanical work [1, 2]. Examples are
s a
. the cytoplasmic motors that move along biological (pro-
t
a tein) tracks in the cell by converting the energy re-
m leased upon ATP hydrolysis into mechanical work [3, 4]. θ=π
z
- These complex machines act as the inspiration for the
d
design of macromolecular devices [5] with the ability to R
n
sort,senseandtransportmaterialinchip-sizedlaborato- x
o ENZYMATIC SITE
c ries [6]. Consequently a major area of current chemical
[ research is the construction of much simpler molecular

machines for nanotechnologicalapplications [7]. FIG. 1: The spherical particle with a reaction site.
1
v In this spirit, we study in this Letter a simple model
9
of a self-propelling device driven by chemical reactions an available substrate into two product particles. If one
6
1 on its surface. It is simple enough that not only is the of the product particles is similar in size to the original
1 construction using chemical techniques feasible, but it substrate (see below), we can assume that the reaction
0 should also be possible to change parameters in order to site effectively acts as a source of excess particles that
7 optimize particular features or functions. are being released at the reaction site at a rate dNp(t),
0 dt
We consider a spherical particle (colloid or vesicle) of and the diffusion equation for the density of the excess
/
at radius R that has a single enzymatic site located on its product particles can be written as
m surface at a fixed position, as sketched in Fig. 1. In
dN (t)
- the presence of a reactive substrate in a non-equilibrium ∂tρ(r,t) D 2ρ(r,t)= p δ3(r rs), (1)
d state,theenzymepromotesthe reactionrateinits vicin- − ∇ dt −
n
ityandproducesadynamicandasymmetricdistribution where D is the diffusion constant of these particles. The
o
of product particles of (hydrodynamic) diameter a R density profile can be obtained from Eq. (1) subject to
c ≪
: which exert osmotic or interfacial forces (depending the the boundary condition of vanishing normal current on
v
boundary properties) and hence propel the sphere in a the surface of the sphere,ˆr ρ =0.
i sphere
X fixed direction. We consider both uniform and peri- The resulting distribution·∇of p|roduct particles around
r odic reactions and calculate the propulsive velocity of the sphere is asymmetric, with a non-zero first moment
a the sphere. We find that the velocity of propulsion is of ρ (t) = πsinθdθ cosθ ρ(r = R,θ,t), leading to
1 0
setbythe sizeofthe productparticles,the propertieson phoretic motion of the sphere. It is well known that
R
the boundary, and the reaction rate. Upon considering colloidal particles in externally imposed solute gradients
the variations in the rate particle release(and taking ac- will be set in motion by a variety of phoretic mecha-
count of density fluctuations), we find that the ratio of nisms [8]. Here in contrast, the gradient (of products)
the mean-squarevelocity fluctuationsto the meanveloc- is self-generated [9] by the device itself. We thus ob-
itydependsontheratioofthetimeforaproductparticle tainageneralexpressionforthevelocityofpropulsionin
to diffuse a distance R to the typical time between the the z-direction v = kBT λ2 ρ , where η is the viscos-
production of successive product particles. ity of the solvent and−thηe leRngth1-scale λ depends on the
The reaction site on the sphere, located at r = ˆzR particular phoretic mechanism[8, 10]. The other com-
s
−
(see Fig. 1), is an enzyme catalyzing the break up of ponents of the velocity vanish due to symmetry. Two
2
mechanisms valid for non-ionic product particles are the v(t)
v
“diffusiophoresis” of a totally impermeable sphere [11], 0
1
because the depletion of the product particles near its
surface causes a lateral slip velocity that results in net 0.8
motionofthesphere,andthe“osmiophoresis”ofaspher- 0.6
icalshell whichis permeable to solventbut impermeable 0.4
to product particles which develops a non-zero velocity
0.2
due to osmotic forces that cause radial flows of solvent Dt
across the membrane[12]. 1 2 3 4 5 R2
Experimental estimates for the diffusiophoretic, λ
D
FIG.2: Transientresponseofthevelocityof propulsionafter
andtheosmiophoretic,λ lengthscalesareavailablefrom
O
thereaction has been switched on.
the literature. Measurements on latex particles in gradi-
ents of Dextran by Staffeld and Quinn [13] obtained val-
ues of λD 38nm. Studies of micron sized lipid vesicles t = 0; i.e. dNp(t) = 1ϑ(t), where τ is the average in-
ofradiusR≃ingradientsofsucrosebyNardietal [14]have dt τf f
verse reaction rate or “firing time” of the product par-
obtainedanosmiophoreticlengthscaleλ 0.3R. Theo-
O ticles, and ϑ(t) is the Heaviside step function. Using
∼
reticaltreatmentsofphoreticpropulsionimplementlocal
Eq. (2), we find the average velocity of the sphere as
momentum conservation by solving the Stokes’ equation v(t) = v ϑ(t) Dt , where the stationary velocity is
for the solvent (Re 1) taking account of the product 0 G R2
given by
≪
particles and force balance on the sphere [8]. For dif- (cid:0) (cid:1)
fusiophoresis, Anderson and Prieve [15] have calculated ℓ
v = , (4)
λ2D = 0∞dl l 1−e−W(l)/kBT , where W(l) is the inter- 0 τf
action energy between the solute particles and the rigid
wall ofRthe sp(cid:2)here at a norma(cid:3)l separation l [11]. This is and (s)=1 cos2s sin2s+4 s F 1;3,5; s2 ,
G − − π 1 2 4 4 −
consistent with the experiments above [13] for an inter- where F 1;3,5; s2 is a generalized hypergeometric
1 2 4 4 − p (cid:0) (cid:1)
action range given by the hydrodynamic radius of Dex- function [17]. The time response of the force is plotted
(cid:0) (cid:1)
tran. For osmiophoresis, Anderson [12] has obtained, in Fig. 2. The function starts at t = 0 with an infinite
λ2 = R2 ηLp/R , where L is the filtration coef- slopeandasymptotestoitsfinalvaluearoundt=R2/D.
O 2+20ηLp/R p We can estimate the steady state propulsive velocity
ficient of hthe membraine. For a R = 2µm lipid vesicle,
using η =10 3 Pa.s and L =10 7 (m/s)/atm [16], one for a spherical device of radius R = 2µm with the en-
− p −
obtains λO 0.01 significantly lower than the experi- zymeonits’surfacecatalyzingafastreactionsuchasthat
R ∼ ofacetylcholinesterase, whichhydrolyzesacetylcholinein
ments above [14]. This discrepancy between theory and
the synaptic cleft into components acetate and choline
experiment remains an open problem [10].
at the rate of 1/τ 25000s 1 [18]. The corresponding
Solving Eq. (1) subject to the appropriate boundary f ≃ −
product particle will be acetate with a hydrated diam-
condition mentioned above, we find ρ , which together
1
eter of a = 0.8 nm. Using the estimates above [13, 15]
with the above expression for the velocity of propulsion
for λ we find a steady-state propulsive velocity from
yields D
diffusiophoresis of a hard sphere as v 1 nm/s. This
D
∼
rather small value could be improved by “engineering”
e iωtf(ω) 1 i iωR2
dω − − D the strength and range of the interaction of products
v(t)=ℓ (cid:20) (cid:18)q (cid:19)(cid:21) , (2) with the surface. The high experimental value of λ
2π 2 O
Z 1 i iωR2 1 iωR2 measured[14]forclosedvesiclesimply amuchhigherve-
" − (cid:18)q D (cid:19)− 2(cid:18)q D (cid:19) # locity of propulsion vO ∼ 1 µm/s for a semipermeable
shell. Theoretical estimates [12] however predict a much
whereℓ= kBT λ2,andf(ω)isthe Fouriertransformof smaller velocity v 1nm/s.
4πηD R2 O ∼
dNp(t). Using Einstein’s formula for the diffusion coeffi- Periodic Particle Release. If the reactions oc-
dt cur at well defined time intervals of τ , then
cient D =k T/(3πηa), we find f
B f(ω) = +n=∞ eiωnτf = 2τπf +m∞= δ(ω − 2πm/τf)
3λ2 in Eq. (2−)∞ and the velocity−∞reads v(t) =
ℓ=a . (3) P P
4R2 +∞ R
(cid:18) (cid:19) v0 e−i2πmt/τf m , where m(x) =
A √Dτ A
The time dependence of the velocity will depend on the m= (cid:18) f(cid:19)
X−∞
1+(1 i)√πmx
particlereleaserate. Twoimportantcasesofuniformand
− .Theaboveexpressioncan
periodic particle release are considered. 1+(1 i)√πmx iπmx2
(cid:20) − − (cid:21)
Uniform Particle Release. Consider the case where besimplifiedintwolimitingcases. For R 1,theex-
√Dτf ≪
a particle source with a uniform rate is switched on at pressioninthebracketscanbeapproximatedas1andthe
3
v(t) fluctuations of the velocity about its mean value. We
v
0 find the long-time behavior of the velocity correlations
3 as [v(t) v ][v(t) v ] ℓ2 δ(t t), where the over-
2.5 h − 0 ′ − 0 i∼ τf − ′
line denotes averaging with respect to the time-interval
2
distribution. This result implies diffusive behavior, and
1.5
we obtain an effective diffusion coefficient ℓ2/τ ,
zz f
1 D ∼
for the sphere in the propulsion direction. Similar re-
0.5
t sults are obtained for the effective diffusion coefficients
1 2 3 τf xx and yy in the lateral directions. We can also es-
D D
timate the relative importance of the velocity fluctua-
FIG.3: Timeevolutionofthepropulsivevelocityforthecase tions as compared to the average velocity by calculating
of periodic particle release, corresponding toR/√Dτf =1. (∆v)2 [v(t) v ]2 . In the long-time limit, we find
0
(∆v)2 ≡Dhτ /R−2, foriR a. The velocity fluctuations
v02 ∼ f ≫
seriescanbesummeduptov(t)≃v0 +n=∞ δ(t/τf−n). are thus controlled by the ratio between the firing time
This corresponds to the case where the−d∞iffusion time and the time it takes for the product particles to diffuse
P
around the sphere is much less than the firing time. In across the sphere and homogenize their profile. For ro-
this case we expect the initial inhomogeneous profile of bustpropulsion,thefiringtimemustbeconsiderablyless
the product particles to lead to instantaneous impulses than the diffusion time.
that are immediately screened due to fast homogeniza- Effect of Hydrodynamic Flow. To make the calcula-
tion. In the opposite limit of R 1, however, the tions self-consistent we should also take account of the
√Dτf ≫
rapid release of the product particles maintains a stabi- factthatthe diffusionofthe productparticleswillbe af-
lized inhomogeneous profile and the velocity is approxi- fectedbythehydrodynamicflowaroundthesphere. The
mately constant: v(t) v0. The time evolution of the velocity profile of the solvent around the sphere can be
≃
velocity is plotted in Fig. 3 for an intermediate case of shown to have a 1/r3 dependence, which is character-
R = 1. We note the that the relative magnitude of istic of phoretic flows [12]. A corresponding convective
√Dτf
thevelocityfluctuationsandtheaveragevelocityissetby term should then be added to Eq. (1), which suggests
the ratio between the firing time and the diffusion time. a systematic solution of the equations as a perturbative
Velocity Fluctuations. The velocity calculated in Eq. series in the P´eclet number Pe = vR/D. Previous work
(2) should be considered as the average of a fluctuat- has shown that the first correction term is of the order
ing quantity. There are two sources of fluctuations that of Pe2 for osmiophoresis [12], and Pe (λ /R) for diffu-
D
needbetakenintoconsiderationforaquantitativeassess- siophoresis [15]. These corrections are negligible for the
ment of the velocity fluctuations, namely, product parti- cases considered here (see below).
cle density fluctuations and randomness in the reaction
ReactionKinetics. Itisilluminatingtoexaminemore
that leads to the particle release.
closely the conditions under which the above simplified
To take account of the density fluctuations, we can go
picture holds, taking into account the reaction kinetics.
backtotheLangevindynamicsoftheparticlesdescribed
We have a solution of substrate at a relatively high
as ∂ r (t) = u (t) for the velocity of the i-th particle, S
t i i concentration C that has a natural slow tendency to
whereu (t)isarandomnoisewithadistributionP[u]= S
i dissociate, + where and represent the
exp 1 dtu (t)2 . We canconstruct a stochastic S −→ S′ P S′ P
densi−ty4Das ρˆi(r,t) =i δ3(r r (t)), and show that it product particles that are assumed to exist in vanish-
(cid:2) P R i(cid:3) − i ingly small concentrations in the bulk. This means that
satisfies Eq. (1) with an additional noise term Qˆ(r,t) theentirebulk ofthe systemisinanon-equilibriumcon-
P
added to its right hand side, whose moments can be dition that will not relax to equilibrium in laboratory
calculated using the above distribution as Qˆ(r,t) = 0 time scale. The presence of an enzyme catalyzes this
and Qˆ(r,t)Qˆ(r′,t′) =2D( 2)δ3(r r′)δ(ht t′)ρi(r,t), reaction, so that it takes place considerabEly faster in the
wherhe ρ(r,t) = ρˆ(ir,t) . T−o∇incorpo−rate th−e random- vicinity of the enzyme, which leads to an increased pro-
h i
ness of the reaction, we write the particle release rate as duction of and with corresponding concentrations,
′
tdiNmdpte(t)in=tervanl τnδ(bte−tweejn<nthτej)r,ewlehaseereofwtehheanve+d1e-fithneadnda oCfS′, ainndthCeSPnereigshpbePcotrihvoeolyd. oTfhtehree wenilzlyamlseo. bTehaedperpoldeuticotns
P P S
the n-th particles. We further assume that the reaction will slow down the reaction by moving it towards equi-
leading to the product release is a Poisson process, in librium using the backward path + that is
′
S P −→ S
which case the probability distribution of the time in- inevitably present, but one can show that it is only a
tervals betweentwo consecutiveparticle releaseevents is small perturbation to the reaction condition, because all
given by P(τ)= τ1f e−τ/τf. theconcentrationshavetomatchsmoothlywiththoseof
Usingthe formulationoutlinedabove(detailsofwhich thebulkthataremaintainedoutofequilibrium. Thenet
will be presented elsewhere [10]), we can calculate the propulsion of the sphere depends on all three concentra-
4
tions. If the diffusion constants (i.e. the hydrodynamic size can travel of the order of ten times its own size be-
radii) and other properties of the and the particles foreitlosessenseofitsoriginalorientation. Theeffective
′
S S
are not that different, the sum of the two concentrations diffusionconstantsderivedfromthevelocityfluctuations
C +C ′ remainsconstantandtheircontributionstothe above contributes negligibly to the orientational mem-
S S
phoretic propulsioncancel [19]. This leads to the simpli- oryloss,givingarotationaldiffusiontime ofthe orderof
fied picture presentedabove of a diffusive process with a τ (R/ℓ)2τ 1000 s. We also note that the typical
vF f
∼ ≃
localized source where ρ=C in Eq. (1). time that the system spends in a transient regime after
To gain further insight onPwhat controls the effective a change is given by τ =R2/D (see Fig. 2), which is of
0
particle release rate, we consider the following general the order of 5 ms. The effect of the hydrodynamic flow
multi-stage reaction pathway can be seen to be unimportant, because Pe 10−2.
∼
In conclusion, we have proposed a model design for
+ k1 k2 + k3 + + , (5) a molecular machine that can propel a vessel in aque-
S E −−−→ SE −−−→ P2E P1 −−−→ P1 P2 E
ous media with a mechanism that involves asymmet-
where the two products (1st) and (2nd) repre- ric release of reaction products. The motor may be
1 2
P P
sent and , the order depending on which of them thought of as a diffusive equivalent of the jet engine: it
′
S P
( or ) is released first. The backward reaction paths releases asymmetrically the reaction products in a vis-
′
S P
in Eq. (5) have been eliminated for simplicity, and cous medium, lets them diffuse and takes advantage of
they can be shown to have negligible effect provided their thermodynamic forces, instead of gaining inertial
the nonequilibrium workingcondition describedabove is thrust by ejecting the exhaust.
maintained [10]. Under steady-state reactionconditions, We acknowledge the support of the Royal Society.
the diffusion–reaction equations take on the form of Eq.
(1), with the same rate (as a sink for the substrate and
sourcefortheproducts)appearingontherighthandside
of these equations. The mean rate constant is given by
[1] M. Schliwa and G. Woehlke,Nature 422, 759 (2003).
the Michaelis–Menten rule [20]
[2] F.Julicher,A.Ajdari,andJ.Prost,Rev.Mod.Phys.69,
1269 (1997); R.D. Astumian, Science276, 917 (1997).
1 k k C ( )
= 2 3 S E , (6) [3] J. Howard, Mechanics of Motor Proteins and the Cy-
τf (cid:18)k2+k3(cid:19) KM +CS(E) toskeleton (Sinauer, New York,2000).
[4] see e.g. D. MacKay and R. MacKay , unpub-
where K 1 k2k3 , is the Michaelis constant and lished, proposing a model of trapped phosphate
M ≡ k1 k2+k3 as a mechanism for force transduction in myosin:
C ( ) is the sub(cid:16)strate c(cid:17)oncentration at the position of (http://www.inference.phy.cam.ac.uk/mackay/dynamics/myosin/)
thSe Eenzyme. [5] V.Balzanietal.,Angew.Chem.Int.Ed.39,3348(2000).
The concentration of the substrate at the position of [6] R.K. Soong et al., Science 290, 1555 (2000).
the enzyme, which we expect to be depleted in compari- [7] D.A. Leigh et al., Nature424, 175 (2003).
[8] J.L. Anderson, Ann.Rev. Fluid Mech. 21, 61 (1989).
sontothebulkconcentrationC ,canbefoundbysolving
0
[9] P. Lammert, J. Prost, and R. Bruinsma, J. Theor. Biol.
the corresponding reaction–diffusion equation with the
178, 387 (1996).
appropriate boundary condition. We find
[10] R.Golestanian,T.B.Liverpool,andA.Ajdari,tobepub-
lished.
1 3k ηa 2 [11] B.V. Derjaguin, S.S.Dukhin,A.A.Korotkova, Kolloidn.
C ( ) = C0 1+ 1 S KM +4KMC0 Zh. 23, 53 (1961).
S E 2s(cid:20) −(cid:18) kBTaE (cid:19) (cid:21) [12] J.L. Anderson, Phys.Fluids 26, 2871 (1983).
1 3k ηa [13] P.O. Staffeld and J.A. Quinn, J. Coll. Int. Sci. 130, 88
1
+ 2 C0− 1+ k TaS KM , (7) (1989).
(cid:20) (cid:18) B E (cid:19) (cid:21) [14] J. Nardi, R. Bruinsma, and E. Sackmann, Phys. Rev.
Lett. 82, 5168 (1999).
where a is the hydrodynamic radius of the substrate
S [15] J.L. AndersonandD.C. Prieve,Sep.Purif.Methods 13,
and a is a typical size of the enzyme.
E 67 (1984).
Discussion. The molecular locomotive device will be
[16] M. Bloom, E. Evans, and O.G. Mouritsen, Q. Rev. Bio-
useful if it can perform directed motion over a distance phys.24, 293 (1991).
that is large compared to its own size. A very im- [17] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Se-
portant limiting factor for directional motion is rota- ries, and Products, (Academic Press, New York,2000).
tional diffusion, whose time scale for a spherical colloid [18] L. Stryer,Biochemistry, (Freeman, NewYork,1995).
isτ =8πηR3/(k T),whichimpliesthatwecanachieve [19] Thisrestrictionisonlyimposedtosimplifythepresenta-
R B tion. If neither of the product particles is similar to the
moredirectedmotionbyincreasingthesizeofthesphere.
substrate, the net velocity of the sphere is given by the
Therotationaldiffusiontimeturnsouttobeoftheorder
algebraic sum of thevarious contributions.
of50s,forR=2µm,whichmeansthatgivenanoptimal [20] P. Nelson, Biological Physics: Energy, Information, Life
propulsivevelocityofv0 0.5µm/s,alocomotiveofthat (Freeman, New York,2003).
∼