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simple model for structured objects
Armand Ajdari
To cite this version:
Armand Ajdari. Force-free motion in an asymmetric environment: a simple model for structured objects. Journal de Physique I, EDP Sciences, 1994, 4 (11), pp.1577-1582. �10.1051/jp1:1994208�.
�jpa-00247015�
J. Phys. I France 4 (1994) 1577-1582 NOVEMBER 1994, PAGE1577
Classification Physics Abstracts
05.40 05.60 87.lo 82.80
short communication
Force.IFee motion in an asymmetric environment: a simple model for structured objects
Armand Ajdan (*)
Laboratoire de Physico-Chimie Théorique (**), ESPCI, la rue Vauquelin, F-75231 Paris Cedex os, France
(Received 3 August 1994, accepted 12 September 1994)
Abstract. Recent models have shown how dissipation of energy can set into directed motion
a point-like particle in an environment of vectorial symmetry, even in the absence of macroscopic forces. This statement is extended to the case of
an object with an internai structure, focusing
on the simplest model:
an elastic dumb-bell. We briefly show that specific mechanisms then
come into play, able to set the object into directed motion even ifthermal diffusion is neghgible.
l. Introduction.
Recent theoretical works have shown how partides m a periodic environment can be set into motion in trie absence of any macroscopic force or gradient provided that: (1) trie structure
possesses vectorial symmetry, (ii) energy dissipation takes place [1-9]. Trie latter, granting
that trie system is maintained out of equilibrium, can manifest itself in various forms, e-g- trie variation in time (periodic or stochastic) of trie eoEect of trie structure on trie particle il-à, 9]
or trie presence of a non-white noise acting on trie partiale [6-8]. These models may be of some reievance to analyze the motion of motor protein assemblies which bave trie above mentioned symmetry [3-10], but also help designmg artificial selective pumping devices for various kinds of abjects Ill, 12].
However, these models addressed trie case of point-like partides described by one spatial degree of freedom z, with in some cases an additional parameter describing trie interaction
between trie particle and its environment, this parameter changing in time periodically [1-3, 9] or stochastically [3-5, 9]. Trie atm of trie present note is to briefly point out that (and how) motion can also result from the internai structure of trie object placed in the asymmetric
environment.
For this purpose we take the simple model of a "dumb-bell" two point-like partides linked by an elastic spnng (an object with a single internai degree of freedom). Focusing on specific
regimes, we then show that motion can be obtained due to the presence of the corresponding
internai mode in two cases: (A) trie object is immersed in a sawtooth potential switched on and off periodically, (B) each of trie two partides stochastically undergoes transitions between two states, which results in a variation of its sensitivity to a stable asymmetric environment.
In both cases, motion persists in trie limit of zero dioEusivity of trie particles. This feature
clearly exhibits trie dioEerence between trie mechanism described here and those proposed for
a single point-like partide under trie saine conditions [1-6]. Trie present mechanism is also (*) present address: Physics Dept., Harward University, Cambridge MA 02138, U-S-A-
(**) URA CNRS f1382
dioEerent from that at work in a very recent two-head motor protein model [13], although serve
ingredients are similar.
2. Model.
We consider trie one-dimensional motion along a direction z of an object represented as a
"dumb-bell". two identical particles Pi and P2, interacting with each other through a potential
U;nt = 1/2K(z2 xi (o)~> (l)
where xi and x2 are the positions of Pi and P2 respectively (xi > x2)> and to gives a measure of the equilibrium size of the object.
To model an environment of vectorial symmetry we will consider a simple periodic sawtooth
potential, defined by its value on a period p:
Uasym = Uo + W(xla) for 0 < x < a
Uasym = Uo + W(p z)16 for a < z < p, (2)
where a, b
= p a are trie lengths of trie two sides of a "tooth" (a < b).
As in previous studies [1-5], we will here focus on two cases:
. case A: trie source of trie potential U~~ym is switched periodically on (dunng T~n) and off (dunng r~w). So both particles are simultaneously periodically submitted to Uasym.
. case B: the source of trie potential U~~ym is stable but each partide independently under-
goes transitions between a state (+) where it feels U~~ym and a state (-) where it does not. Life times T+ and r- in trie two states are taken equal for trie two partiales and inde- pendent of z, which implies that some externat action bas moved trie system away from equilibrium and trie ratio r+ (z) /r- (z) from its detailed balance value exp(-U~~ym(z)/kT)
[3].
Introducing trie internai structure of trie object adds to trie preexisting typical scales of energy
(W and kT), time (r~ (1 E (+,
-, on, off)), p~(/kT and p~(/W), and
size (p), those related to the internai mode: Kt(, (/K, and (o (where ( is trie friction coefficient of trie partides).
Solving the complete set of equations corresponding to cases A and B therefore becomes somewhat comphcated, and we here restrict trie analysis to the followmg regimes as to exhibit
simply mechanisms able to induce motion:
(Ri The energy scale of trie potential is taken much larger than that of trie interaction between trie two partides: W » Kil.
(R2) Thermal diffusion is neglected.
(R3) Relaxation of trie internai mode of trie object is faster than trie time involved in trie cycles
"particlelasymmetric environment" (/K < r~.
(R4) Relaxation under trie eoEect of U~sym is also faster: (p~/W « T~.
(Ri), (R2) and (R4) imply that whenever a particle feels Uasym it gets quickly trapped at trie bottom of trie corresponding valley. (R2) and (R4) imply that whenever (at least) one partiale
ceases to feel U~sym, trie dumb-bell relaxes quickly (Gt instantaneously) to its equilibrium size
(~_ Within this frame of hypothesis it is easy to prove that trie partide is set into motion
provided (o and (p, a) are appropriately chosen.
N°11 FORCE FREE MOTION OF A STRUCTURED OBJECT 1579
3. Case A: asymmetric potential applied periodically.
In case A, trie sawtooth potential U~sym dots periodically and simultaneously on trie two par- ticles. When it is applied PI and P2 get trapped at trie bottom of trie valley where they sit.
When it is tumed off the spring relaxes back symmetrically to to.
A way to get directed motion is that upon retrapping, trie dumb-bell gets successively stretched and compressed, and that in some of the resulting relaxation steps, PI or P2 manages to move to the neighbour valley of Uasym. Figure 1 illustrates this by showmg trie dumb-bell at the end of "on" and "ofll' periods: net motion towards the nght of the picture is dear. This
holds because the periodicity and asymmetry of trie structure and to have been chosen in such
a way that p + 2a < to < p + (b a), which implies b > 3a. More generally, a similar motion
is obtained if:
np + 2a < ÉD < np + (b a), (3)
trie dumb-bell cycling between compression (z2 xi " np) and extension (z2 xi " (n +1)p)
at trie end of trie "on" penods. Under such conditions, trie obtained velocity is:
v = p/2(~~~ + ~~~). (4)
This steady state picture is reached whatever the initial conditions. Note also that for this pro-
cess to be ellicient, the object has to be larger than some characteristic size of the environment:
2a < to at least case n
= 0).
~ asym
b P -Î ~
1 ON
@ OFF
£ ON
1 OFF
£ ON
Fig. l. Case A. The potential Umym is represented on top of the picture. (a) to (e) Positions of the dumb-bell at the end of successive on and off penods are exhibited. The dumb-bell has relaXed to its natural length io in (b) and (d), is extended in (a) and (e), and compressed in (c). It has moved
one period to the right of the picture in two consecutive on /off cycles.
4. Case B: two-state model for tl~e particles.
Let us now tutu to a picture in which PI and P2 transit independently between state (+)
and (-). Within hypothesis R(3) and R(4) the system genencally relaxes between transitions.
Therefore, defining n by np < (o < (n +1)p, the state of the system before one of trie partides undergoes a transition belongs to trie following list (see also Fig. 2 for trie case n
= 1):
. A~: Both partides are m state (+). PI is trapped at xi " iv and P2 is trapped at
x2 " (1+ n +1)p. Trie spnng is extended in this state as z2 xi > (o.
. B~: Both partides are in state (+). PI is trapped at xi " iv and P2 is trapped at
x2 " (1+ n)p. Trie spring is compressed in this state as x2 xi < (o.
. C): Pi in state (+) is trapped at xi " iv and P2 in state (-) gets to the position that relaxes trie dumb-bell: z2
" iv + (o.
. Cl Both partides are in state (-), xi
" iv and z2
" iv + to.
. D): P2 in state (+) is trapped at x2 = (1+ n)p and Pi in state (-) gets to trie position
that relaxes trie dumb-bell: xi
" (1 + n)p ÉD
. Dl Both partides are in state (-), xi
" (1+ n)p to and x2
" (1+ n)p.
~asym
ip 'ç+llp ~>+2]>P ~
~i' (~ (+)
Cj;+ (+) f-
cj7 (_'' '-
O~; j_j ~+
OP r-1 '(-
Fig. 2. Case B. The potential Umym is represented on top of the figure. The six states described
in section 4 are displayed for a case where p + a < io < p + b. Cyclmg between these states along equation (6) leads to net motion to the right.
N°il FORCE FREE MOTION OF A STRUCTURED OBJECT 1581
Let us consider a dumb-bell such that
np+a<io<np+b. (5)
Then, with trie help of figure 2, it is easy to see that upon PI or P2 undergoing a transition, trie system moves according to:
A~ - Dl+i Or Cl; B~ - Dl
Or Cl;
Cl- CT Or A~; CT - Cl or Dt+i;
Dl- B~
or Dl Dl - Cl
or D) (6)
It is clear that index1 never regresses and sometimes progresses during stochastic wandenng
between states along these rules: net motion to trie right is obtained.
To get a numerical estimate of trie velocity, one can simplify trie problem further with trie
hypothesis that trie partiales spend most of their time in state (+): T- < T+, so that typical
sequences are PI (P2) transits to state (-) and then quickly back to (+) before something happens to P2 (PI). In this limit equation (6) reduces to:
A~ - B~+i or A~
B~ - B~ or A~ (7)
The four corresponding transition rates are equal to 1/T+. Trie object spends half of its time
in state B and half in state A, where the index1 progresses by 1 on average alter time T+. Trie
average velocity is thus:
V l% p/2T+. (8)
5. Discussion.
. Through a very simple model and focusing on given regimes, we have shown that land how) an object can be set into directed motion in an asymmetric environment, in trie ab-
sence of macroscopic externat forces, provided a certain fit between its internai structure
and trie environment characteristics le-g- Eqs. (3) and là)).
. Trie determiuistic motion obtained here is in trie same direction as that predicted for
a point-like partide m similar circumstances [2-4], and trie resulting average velocities span the same ranges of magnitude (typically a fraction of the penod p per cycle). Note
however that the underlying mechanism is structurally dioEerent as it survives thermal diffusion going to zero. From this point of view, the present model shares some similarity with that presented in reference [9], m which a point-hke particle is successively under trie influence of two asymmetric potentials of saine period, but with minima spatially shifted by some distance. Here in some sense, trie partiale carnes trie analog of a "shift"
ils size ÉD.
. It is clear that trie introduction of thermal diffusion will modify trie picture, introducing stochasticity in case A aud developing it in case B. Solving in a general way trie model of section 2 without restrictions (Rl-4) would lead to a frame encompassing borin trie limit presented here and those of references [2-4] corresponding to ÉD < a. These more
elaborate calculations are needed to determme trie conditions leading to optimal velocities
or abiiities of these mater/pumps to resist externat forces. Note that trie additionnai force scale Kto introduced in this model should show up in such an analysis.
. On symmetry grounds, one should be able to gel motion in a symmetric periodic structure with an asymmetric dumb-bell: Pi and P2 dioEerent e-g- in their friction coefficients
transition rates, etc.. However, this requires thermal diffusion to act, once agam similarly
to the model of two shifted potentials [9] in trie case where trie two potentials are chosen symmetric [14]. Note that under such conditions tumbling will stochastically occur (P2 moving to trie left of Pl), after which trie object will start moving in trie reverse direction.
Trie motion would then appear a5 directed up to a certain (possibly very long) time scale and random at longer times.
. In a recent more elaborate model [13], Peskin and Oster consider a symmetric two-head
motor protein, where trie interaction with the polar substrate (microtubule) induces an
asymmetry in trie "unbinding" transition (somewhat analogous to trie present (+) - (-))
rate of trie two heads (partides). When one head (partiale) unbinds, it can rebind only
at one of trie neighbour trapping sites of trie still attached head: it either rebinds at the position it has just left or binds on the other side of the attached head (the front and back head exchanging their rotes). If trie "back" head detaches more often than trie "front"
head, the resulting waltz leads the object to progress "forwards". This model gives a very good agreement with experimental data for kinesin motors [13]. It relies however
on mechanisms somewhat dioEerent from those presented here (e,g. dioEusional tumbling
is permanently at work and trapping occurs ouly on specific sites).
. Molecular models for biological motor processes [13, 15] could make use of bases similar to those of trie present simple model, but we emphasize that trie latter may also be a
useful guide to develop selective pumping devices sensitive to internai parameters Ill,
12].
Acknowledgments.
I am very grateful to Professor G. Oster for sending me reference [13] while this work was
almost completed, and for a fruitful correspondence. This work is part of trie project "Pompage Moléculaire", Contrat Coopératif Institut Cune/Association de la Montagne Sainte-Geneviève,
m cooperation with P. Silberzan, D. Chatenay, L. Peliti and J. Prost.
References [Ii Ajdari A., PhD Thesis, Université Paris 6 (April, 1992).
[2] Ajdan A., Prost J., C. R. Acad. Sci. Paris II 315 (1992) 1635-1639.
[3] Prost J., Chauwin J.-F., Peliti L., Ajdari A., Phys. Rev. Lent. 72 (1994) 2652-2655.
[4] Astumian R-D-. Bien M., Phys. Rev. Lent. 72 (1994) 1766-1769.
[si Peskin C.S., Ermentrout G-B-, Oster G-F-, "Cell Mechanics and Cellular Engineering", V. Mow et al. Eds. (Spnnger, New York, 1994).
[6] Magnasco M.O., Phys. Rev. Lent. 71 (1993) 1477-1481.
I?i Magnasco M.O., Phys. Rev. Lent. 72 (1994) 2656-2659.
[8] Doering C.R., Horsthemke W., Riordan J., Phys. Rev. Lent. 72 (1994) 2984-2987.
[9] Chauwin J-F-, Ajdari A., Prost J., Europhys. Lent. 27 (1994) 421-426.
[loi see e.g.: "Guidebook to Cytoskeletal and Motor Proteins", T. Kreis and R. Vale Eds. (Oxford
University Press, New York, 1993) chap. 4;
Walker R-A-, Sheetz M.P., Ann. Rev. Biochem. 62 (1993) 429-452; and reference therein.
[Il) Rousselet J., Salome L., Ajdan A., Prost J., Nature 370 (1994) 446-448.
[12] Ajdan A., Lewiner J., Prost J., Viovy J.L., French patent f 9311346 (1993).
[13] Peskin C.S., Oster G.F., Les Houches 1994, prepnnt.
[14] Chauwin J.-F., unpublished results (1994).
[15] Peskin C.S., Odell G-M-, Oster G-F-, Biophys. J. 65 (1993) 316-324.
J. Phys. I France 4 (1994) 1583-1596 NOVEMBER1994, PAGE 1583
Classification Physics Abstracts
05.40 05.20 75.loN
Manifolds in random media: a variational approach to the spatial probability distribution
Yadin Y. Goldschmidt
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, U-S-A-
(Received 25 April 1994, received in final form 9 August 1994, accepted 12 August 1994)
Abstract. We develop a new variational scheme to approximate the position dependent
spatial probability distribution of a zero dimensional manifold in a random medium. This celebrated 'toy-model' is associated via a mapping with directed polymers in 1+1 dimension,
and also describes features of the commensurate-incommensurate phase transition. It consists of
a pointlike 'interface' in one dimension subject to a combination of a harmonic potential plus a
random potential with long range spatial correlations. The variational approach we develop gives
far better results for the tait of the spatial distribution than the Hamiltonian version, developed
by Mezard and Parisi, as compared with numerical simulations for a range of temperatures.
This is because the variational parameters are determined as functions of position. The replica
method is utilized, and solutions for the variational parameters are presented. In this paper we limit ourselves to the replica symmetric solution.
1. Introduction.
Recently a lot of attention has been devoted to the behavior of manifolds in random media [1-21]. This is partially due to their connection with vortex hne pmning in high T~ supercon-
ductors Ill, 21], but also because of trie intrinsic interest in trie behavior of interfaces between two coexisting phases of a disordered system, like magnets subject to random fields or random
impurities. In addition mappmgs are known to exist between one dimensional manifolds like directed polymers m disordered media and growth problems in trie presence of random noise described by trie KPZ equation [22].
Trie vanational method [8, 13, 16] appears to be an important tool in approximating many
properties of trie system like trie calculation of trie roughness exponent of a wauderiug manifold in a disordered medium. Recently we bave conducted a numencal study of trie spatial proba- bility distribution of directed polymers in 1+1 dimensions in trie presence of quenched disorder [15]. Directed polymers refer to an interface in two dimensions with no overhaugs, lu addition to the width of trie interface, it is important to know what is trie probability that the interface
would wander a certain distance
m the transverse direction. We have then exploited a mapping
