A simple self-organized swimmer driven by molecular motors

0.29 MB·English

by  Stefan Gunther

epl draft
A simple self-organized swimmer driven by molecular motors
S. Gu¨nther1,2 and K. Kruse1,2
9
01 Saarland University, Department of Physics, Campus E2 6, 66123 Saarbru¨cken, Germany
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2 Max Planck Institute for the Physics of Complex Systems, No¨thnitzer Str. 38, 01187 Dresden, Germany
2

n
a
J PACS 87.16.-b–Subcellular structureand processes

1 PACS 47.15.G-–Low-Reynolds-number(creeping) flows
2 PACS 05.45.Xt–Synchronization; coupled oscillators

] Abstract.-Weinvestigateaself-organized swimmeratlowReynoldsnumbers. Themicroscopic
h
p swimmer is composed of three spheres that are connected by two identical active linker arms.
- Each linker arm contains molecular motors and elastic elements and can oscillate spontaneously.
o We find that such a system immersed in a viscous fluid can self-organize into a state of directed
i
b swimming. The swimmer provides a simple system to study important aspects of the swimming
. of micro-organisms.
s
c
i
s
y
h
p
[ Introduction. – Micro-organisms often use flagella a) x x
or cilia to move autonomously in an aqueous environ- l r

1
ment [1]. In eukaryotes these hair-like appendages are 1 2 3
v x
internally driven by molecular motors. The motors set
4 b)
3the appendage into oscillatory motion leading to two- or y
K
2three-dimensional beating patterns, which propel the or-
3ganism. Duetothecomplicatedstructureofciliaandflag- k
.
1ella, our understanding of swimming through these inter- ω
u
0nallydrivenfilamentsisstillfarfrombeingcomplete. This ω
9holds even more for effects caused by hydrodynamic in- b
0 v
teractionsbetweenswimmingmicro-organisms,whichcan
: x
vleadto turbulent motion in bacterialsuspensions [2] or to
ithe formation of vortex arrays for ensembles of sperma-
X
tozoa [3]. In the latter case, the flagella propelling the Fig.1: a)Schematicrepresentationofthethree-spheresystem.
r
aspermatozoa have been observed to beat synchronously. b) Schematicrepresentation of thedynamiclinkerelement. It
Synchronous beating can lead to metachronal waves [4] consistsofafilament,amotorfilament,andalinearspringwith
whichareofbiologicalimportancefortheswimmingofthe elastic modulus K. Motors bind to the filament with rate ωb
and unbind with rate ωu. Motors bound to the filament move
single-celled freshwater ciliate paramecium or the trans-
with velocity v so as to shorten the linker length x. Motors
port of mucus in our lungs.
are connected to the backbone by springs of stiffness k and
extension y along thex-axis.
Physical studies of the swimming of micro-organisms
have focused on the motion resulting from a given se-
quence of shape changes. As the Reynolds numbers asso-
ciatedwith the flow generatedby such swimmers are low, The mechanisms underlying these shape changes, how-
inertial effects do not play a role. Consequently, swim- ever, have received less attention in the physical com-
ming requires shape changes of more than one degree of munity. Exceptions are provided by works on the os-
freedom[5]. Differentswimmergeometrieshavebeencon- cillatory beating of flagella [14–16]. There, the periodic
sidered in this context, see, for example, Refs. [5–10] the motion is assumed to be generated by spontaneous me-
simplestbeingprobablytheswimmerintroducedbyNajafi chanical oscillations of molecular motors coupled to elas-
andGolestanian[9,11–13],whichconsistsofthreelined-up tic elements [17,18]. Combining the action of motor
spheres with periodically varying distances, see Fig. 1a. molecules and the bending elasticity of a flagellum, wave
p-1
S. Gu¨nther1,2 K. Kruse1,2
patternssimilartothoseobservedexperimentallyaregen-
erated [14–16].
a)x x x
In this work, we introduce a simple dynamical system r r r
consisting of three spheres that are arranged in a line - +
and that are connected by linkers. The linkers consist + +
of molecular motors and elastic elements and their length
can oscillate spontaneously. While the swimmer is sym- xS=0 xl xS=0 xl xS=0 xl
b) c) d) e)
metric with respect to space-inversionthe system can dy- ~xr
namically break this symmetry and spontaneously swim 112
directionally. We investigate the system’s bifurcation di- 110
agram and provide evidence that the transition to direc- 20 40 60 ~t 110 112 ~x l 20 40 60 ~t 110 112 ~x l
tional swimming shares striking similarities to transitions
occuring in integrate-and-fire neurons.
The active three-sphere swimmer. – Consider Fig. 2: a) Schematic representation of possible orbits in the
three spheres, each of radius R moving in a fluid of vis- (xl,xr)-plane of periodic motions of the three-sphere system.
cosity η along the x-axis, with neighboring spheres being Left: reciprocal motion, middle: non-reciprocal motion with
linked by elements as illustrated in Fig. 1b. Their struc- x¯s = 0, right: non-reciprocal motion with x¯s 6= 0. b), c)
ture resembles that of sarcomeres, which are the elemen- Left and right linker length as a function of time in the non-
tary force generating units of skeletal muscle, and consist swimming state O1, see Fig. 4, with ξ & ξs and d), e) in the
of motors moving along a polar filament combined with swimmingstateSwwithξ.ξs. OtherparametersasinFig.3.
a linear spring. Such structures are known to be able to
oscillate spontaneously [18], a phenomenon that has been
toneglectinghydrodynamicinteractionsbetweendifferent
observed experimentally for muscle sarcomeres [19]. Be-
spheres. In this case, C(xl,xr) = 0, implying that hy-
foreinvestigatingthedynamicsofthethreeactivelylinked
drodynamicinteractionsareessentialforswimming ofthe
spheres, let us first describe the three-sphere swimmer
three-sphere system.
with externally imposed changed of the spheres’ mutual
If motion is reciprocal, that is, if the corresponding
distances.
curveinthe(xl,xr)-planeretracesitself,thenx¯s =0illus-
The three spheres arelocated atpositions x , i=1,2,3
i trating Purcell’s Scallop theorem, see Fig. 2a left. Note,
along the x-axis. At low Reynolds numbers, the forces f
j thatatrajectorywhichis invariantunderthe exchangeof
acting on sphere j with j = 1,2,3 result in velocities x˙
i left and right, i.e., space inversion, is necessarily recipro-
according to [20]
cal. Examples of other periodic motions with vanishing
net displacement are given by xl(t) = xr(t + T/2), see
x˙ = H f . (1)
i ij j Fig. 2a middle. These solutions are invariantunder space
Xj inversionandasimultaneousshift intime byT/2. A sim-
Here,the mobility tensor H depends onthe fluid’s viscos- ple form of xl and xr with x¯s 6=0 was proposed by Najafi
and Golestanian [9], see Fig. 2a right.
ity,thesizeofthespheres,aswellasthedistancesbetween
Wenowdescribethedynamicsofthelinkerelementsde-
them, and satisfies H = H . Using Eq. (1) as well as
ij ji
picted in Fig. 1b, consisting of motors and elastic compo-
the globalforcebalance equationf1+f2+f3 =0, the ve-
nents. Similar systems have been studied in the contexts
locity of the center of mass of the system, dxs/dt ≡ x˙s =
of spontaneous spindle [21] and muscle oscillations [22].
(x˙1 +x˙2 +x˙3)/3, can be expressed in terms of the dis-
Alinkerelementconsistsofmotorsmovingdirectionally
tances between the middle and the outer spheres, xl and
along a polar filament and a linear spring of stiffness K.
xr, respectively. For periodically varying distances, one
Themotorsareattachedtoacommonbackbonebysprings
finds for the displacement x¯s after one period T,
of stiffness k and extension y. In a sarcomere,the motors
T would be ensembles of myosin II that move on an actin
x¯s = x˙s dt= C(xl,xr)dxl∧dxr , (2) filament, while elastic elements are provided by various
Z0 ZO
structural elements, e.g., the protein titin. Motors bound
where O denotes the oriented surface encircled in the tothefilamentmovesuchthatthelinkershortens. Anun-
(xl,xr)-plane duringone periodofthe motion, seeFig.2a bound motor binds to the filament at rate ωb. Attached
and also Refs. [5,7]. In this expression, C is a field with motors unbind from the filament at rate ωu. These rates
C(xl,xr) = C(xr,xl) that only depends on the mobility generally depend on the force applied to the motor. We
tensor H. Thus the displacement is independent of the assume that the force dependence is restricted to the un-
oscillation period. For the calculation of C, it is conve- bindingrate. MotivatedbyKramers’ratetheorywewrite
nient to employ the Oseen approximation, which consists ωu = ωu0 exp{|f|a/kBT}, where f = ky and a is a mi-
in expanding H in terms of R/xl and R/xr and retaining croscopic length scale. Also the velocity of bound motors
leadingordertermsonly. Inlowestorder,thiscorresponds depends on the applied force. For simplicity, we assume a
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A simple self-organized swimmer driven by molecular motors
linearforce-velocityrelationv(f)=v0(1−f/f0). Here,f0 will evolve as δx˜(t) ∝ exp(st) and analogously for δQ
isthestallforceandv0 thevelocityofanunloadedmotor. and δy˜. It grows if ℜ(s) > 0. This condition can be
The dynamic equation for the length x of the linker expressedin terms of the system parameters1 and implies
′
elementconnectingtwospheres,say1and2,x=x2−x1,is y0ωu(y0) > ωb + ωu(y0). The force dependence of the
givenbyEq.(1),wheretheforcef onspherej isthesum binding kinetics is thus essential for an instability of the
j
oftheelasticandthemotorforcesgeneratedbythelinker. stationary state. The system oscillates if ℑ(s) 6= 0 at
The elastic force fe is given by fe = −K(x−L0) with the instability. This implies Q0ky0 < K∆, that is, the
L0 being the spring’s rest length. To calculate the force increase of the motor force by adding one motor in the
exerted by the motors, fm, consider first a single motor overlapregionneeds tobe smallerthanthe corresponding
bound to the polar filament. If y denotes the extension increase of the restoring force by the elastic element to
n
of the spring linking this motor to the backbone then the produce an oscillatory instability.
forceitexertsis−ky . The valueofy changesaccording The oscillation mechanism can be understood intu-
n n
to itively as follows: As the motors shorten the linker, the
y˙ =v +x˙ , (3) elastic restoring forces increases and, therefore, the force
n n
onthemotors. Thisinturnincreasestheunbindingrateof
wherev isthevelocityofthismotoronthefilament. The
n N motors. Whenafew motorsdetachfromthefilament,the
total motor force then is fm = −k n=1σnyn. Here, N remaining motors experience an even higher force, such
is the overall number of motors andPσn = 1 if motor n is that an avalanche of motor unbinding events occurs. The
bound and 0 otherwise.We employ a mean-field approxi-
elastic element then stretches the linker and the motors
mation,whichconsistsinassumingthatallbound motors
rebind, repeating the cycle.
have the same spring extension, y = y for all n. We de-
n This behaviour shares similarities with the dynamics of
termine the mean position by the product of the average
integrate-and-fire model neurons [23]. There, the electric
motor speed multiplied with the average time they stay
membrane potential of a neuron increases monotonically
attached, y =y˙/ωu. For unbound motors, we assume fast
as a consequence of an input current. Upon reaching a
relaxation,such that their distribution equals the equilib-
thresholdvalue,the potentialis instantaneouslyresetand
rium distribution implying y =0 for these motors. Under
the loading process restarts. In the case of the linker ele-
these conditions, the fraction Q of bound motors evolves
ment, the mechanicalanalogofneuron’selectricpotential
according to [21]
istheinternalstressofthelinker. Loadingoccursthrough
Q˙ =ωb−(ωb+ωu)Q , (4) the action of motors and resetting is accomplished by the
spring K after a detachment avalanche of the motors.
while the total motor force is Letusnowreturntothethree-spheresystem,seeFig.1,
where the spheres are connected by dynamic linker ele-
fm =−N(x)Qky . (5) ments. The forces acting on the three spheres are f1 =
l l r r
−fe −fm, f2 = −f1 −f3, and f3 = fe +fm, where the
Here, N(x)=(ℓf+ℓm−x)/∆ is the number of motors in superscripts l and r distinguish between the left and the
theoverlapregionofthefilamentandthemotorbackbone right linker, respectively.
of lengths ℓf and ℓm, respectively, and ∆ is the distance As we have seen above, hydrodynamic interactions be-
on the backbone between adjacent motors. tween different spheres are essential for swimming of a
Beforeconsideringthe active three-sphereswimmer,let three-sphere system. Compared to the two-sphere case,
us present the dynamics of two actively linked spheres. wethusnowexpandthemobilitytensortothenexthigher
The dynamic equations for this system are given by order. This leads to H = (6πηR)−1, i = 1,2,3,
ii
Eqs. (1), (3)-(5), where the forces on the spheres 1 and H12 = (4πηR)−1R/xl, H23 = (4πηR)−1R/xr, and
2 are f1 = −f2 = −fe−fm. To lowest order the Oseen- H13 = (4πηR)−1R/(xl +xr). The equations of motion
expansion of H gives H11 = H22 = (6πηR)−1, and arethen givenby Eqs.(7)and(8)respectivelyforthe left
H12 = 0. Hii is the Stokes friction of an isolated sphere. and right part of the swimmer. Equation (6) is replaced
In dimensionless form, the equations of motion read by
ξx˜˙ = −2·[N(x˜)Qκy˜+x˜−L]≡−2F(x˜,Q,y˜) (6) 3r
Q˙ = 1−[1+ω(y˜)]·Q (7) ξx˜˙l,r = (cid:26)x˜l,r −2(cid:27)F(x˜l,r,Ql,r,y˜l,r) (9)
x˜˙ = y˜·[ω(y˜)+γ]−1 , (8) 3r 1 1 1
+ − − +1 F(x˜r,l,Qr,l,y˜r,l)
(cid:26) 2 (cid:18)x˜l+x˜r x˜l x˜r(cid:19) (cid:27)
where the dimensionless parameters are given by ξ =
6πηRωb/K, L=L0ωb/v0, κ=k/K, ω =ωu/ωb and γ = with r = Rωb/v0. As before, the dynamic equations are
kv0/(f0ωb). Furthermore, x˜ = xωb/v0 and y˜ = yωb/v0, invariant under space inversion, i.e., exchange of left and
while time has been rescaled by ωb. 1Theinstabilityoccursatξ(1+ω(y˜))+1−Qκy˜∆˜+[N(x˜)Qκ(1+
The equations have a stationary state (x˜0,Q0,y˜0). For ω(y˜)−y˜ω′(y˜)]/[ω(y˜)+y˜ω′(y˜)+γ]=0,with∆˜ =v0/∆ωb. Here,x˜,
sufficiently short times, a small perturbation δx˜,δQ,δy˜ Qandy˜aretakeninthestationarystate.
p-3
S. Gu¨nther1,2 K. Kruse1,2
v
s φ ~
x
max L2
14.8 O
118 Sw 1
y
0.006 or 14.6 Sw
t
a
cill T/2 114 14.4 O2
0.004 s
-o 6 7 ξs 8 ξ
n 110
0.002 velocity no O2 O1
phase St
106
0 0 5ξs ξ2 15 25 ξ1 ξ
0.04 0.4 4 ξ ξ ξ
s 1
Fig. 4: Bifurcation scenario represented by the linker ampli-
Fig. 3: State diagram of the dynamic three-sphere swimmer.
For ξ > ξ1 the system is stationary. Filled diamonds indicate tude x˜max for varying parameter ξ. Solid and dotted lines
the mean velocity of the swimmer’s center of mass. For ξs < indicate stable, dashed lines unstable states. For ξ < ξ1 the
stationary state St becomes unstable via a Hopf-bifurcation.
ξ<ξ1 it oscillates, but does not move on average, vs =0. For
ξ<ξs,vs >0. Crossesindicatetherelativephasebetweenthe Theemerging oscillatory state O1 losesstability at ξ=ξs and
theswimmingsolutionSw(dottedline)appears. Furthermore,
oscillations of the two linker elements. Equations (7)-(9) were
solvednumericallywithL=137.5,κ=400,ω0 =0.5,γ =9.1, at ξ = ξ2, a second oscillatory Hopf-mode O2 appears. Inset:
∆˜ =0.16 and r=11. Representation of the swimming bifurcation in terms of the
L2-norm (see text).
right.
state coexists a state moving into the opposite direction.
Figure 3 summarizes the three-sphere system’s behav- Theactualdirectionofmotionisdeterminedbytheinitial
ior with the active linker elements as the parameter ξ is conditions. Letusalsonotethatchangingtheparameters
changed. This can be achieved, for example, by chang- γ or r can also lead to transitions from the stationary
ing the viscosity of the surrounding fluid or the binding to the non-swimming oscillatory state or from the non-
rate of the motors. Having ξ > ξ1, the system settles swimming oscillatory to the swimming state. We did not
into a stationary state. For ξ < ξ1, the linkers sponta- findtwosubsequentbifurcationsleadingfromthe station-
neously oscillate in length. The oscillations of the two ary to the swimming state, though.
linker elements are identical but shifted with respect to We will now take a closer look at the swimming tran-
each other by half a period. As is illustrated in Fig. 2b,c, sition. The bifurcation diagram of the swimmer in terms
this symmetry implies that, for low Reynolds numbers, of the oscillation amplitude x˜max is presented in Fig. 4.
the swimmer’s center of mass xs remains stationary on The stationary state St loses stability through a super-
average. There is a second critical value ξs, at which the critical Hopf-bifurcation at ξ = ξ1. The newly gener-
relative phase shift between the two oscillating linkers Φ ated oscillatory state O1 satisfies x˜l(t˜) = x˜r(t˜+ T/2),
starts to deviate from half a period, such that the system where T is the dimensionless oscillation period. At ξ =
spontaneously swims if ξ < ξs. In contrast, the form of ξs, O1 loses stability through a pitchfork bifurcation,
the oscillatory motion of each of the two linker elements see Fig. 3, resulting in the swimming state Sw. Here,
essentially does not change. By dynamically breaking the x˜l(t˜) = x˜r(t˜+φ), where the relative phase φ 6= T/2. For
space-inversion symmetry, the corresponding orbit in the ξ < ξs, the states O1 and Sw have almost the same am-
(xl,xr)-plane is not symmetric with respect to the line plitude. Therefore, we display in the inset of Fig. 4 a
xl =xr, see Fig. 2d,e, hydrodynamic interactions allow in representationofthe bifurcationin terms ofthe L2-norm,
averagefor directed motion of the three-sphere system. T−1 T dt˜ x˜2+x˜2+Q2+Q2+y˜2+y˜2 . For a similar
0 l r l r l r
Decreasing the value of ξ further speeds up the swim- valueR of ξ(cid:0)a second non-swimming sta(cid:1)te O2 bifurcates
mer. The increase in the swimming velocity is the conse- from the stationary state at ξ = ξ2. This state, which
quenceoftwoeffects. Ononehand,therelativephaseshift is unstable for all values of ξ, satisfies x˜l(t˜)=x˜r(t˜).
gets larger and, on the other hand, the amplitude of the The appearance of the state O2 together with an in-
oscillationincreases,bothleadingtolargerareasencircled spection of the corresponding isotropy lattice might sug-
by the orbit in the (xl,xr)-plane. For still lower viscosi- gestmode interactionto be atthe originofthe symmetry
ties one expects eventuallya decreasein the velocity such transition. However, the frequencies of O1 and O2 are
that swimming stops at ξ = 0. The dynamic equations different at ξs, which argues strongly against mode inter-
presented here do not capture this effect as their applica- action.
bilityisrestrictedtotheregimeoflowReynoldsnumbers. While we are currently lacking a thorough understand-
By the symmetry of the swimmer, with each moving ingoftheswimmingtransition,wewouldliketopointout
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A simple self-organized swimmer driven by molecular motors
thatthistransitionisreminiscentofaphenomenonoccur- cillatespontaneously. Bydecreasingthedimensionlesspa-
ing for two coupled integrate-and-fire neurons [24]. The rameterξ belowacriticalvalue,whichcanbeachievedei-
membrane potential of an integrate-and-fire neuron is in- therbyreducingthetheviscosityofthesurroundingliquid
creased at a constant rate and resetted when a certain orthemotors’bindingrateorbyincreasingthestiffnessK
threshold voltage is reached. Upon resetting, the neuron oftheelasticelement,thesystemoscillatesspontaneously.
sends an electric pulse to other neurons it is connected Whilethisstateisnotassociatedwithanaveragedisplace-
to and influences their membrane potentials. For broad ment,asubsequentbifurcationencounteredwhendecreas-
pulses,thetwocoupledneuronssettleintoastateinwhich ingξ further,leadstodirectedswimming. Changingother
theyfireperiodicallywitharelativephaseshiftofhalfthe system parameters can either lead to spontaneous oscilla-
period. Forshortenoughpulses,however,thisstateisun- tions or to swimming but has not been found to induce
stableandanewvalueofthephase-shiftisgenerated[24]. both transitions subsequently.
For the swimmer, in analogy to the electrical pulses for Couldthe self-organizedthree-sphereberealizedexper-
the neurons, mechanical pulses are transmitted netween imentally? One approach could be based on muscle sar-
the two linkers. A mechanical pulse is released by one comeres as linker elements. As mentioned above, sarcom-
linker when it relaxes as a consequence of a detachment erescanoscillatespontaneously[19]. Usingparameterval-
avalancheof the motors. In that case,the duration of the ues appropriate for sarcomeres2 and assuming that they
pulsechanges,becauseofalterationsofthedurationofthe linksphereswithadiameterofR=0.1µm,wefindswim-
linkersexpansionphasethateffectivelytransmittstheme- ming velocities on the order of vs ≈ 1µm/min. This has
chanical pulse. Decreasing ξ leads to faster stretching of to be compared to the diffusion constant. For a sphere
the linkers. In terms of physical parameters, decreasing ξ with a diameter of 0.1µm we find in water at room tem-
eithercorrespondstoreducingfrictionforcesonthebeads, perature D ≈ 2µm2/s. On a distance comparable to the
to decreasing the binding rate of motors, or to strength- swimmer size L ≈ 3µm, diffusion thus dominates the
ening the elastic elements. As for the two-neuron system, motion Lvs/D = 0.025. In order to reach bigger swim-
a sharper pulse can be sufficient to cause dephasing [25], ming speeds, the oscillationfrequency could be increased.
which implies swimming. To this end stiffer springs and stronger motors would be
The equations of motion (7)-(9) are of the form of a necessary. Another way to increase the swimming speed
singularly perturbed system with ξ as the small parame- would be through modifications of the swimmer design.
ter. Such systems are known to present strong deforma- For example, it has been argued that the pushmepullyou
tions of a limit cycle close to a Hopf-bifurcation, where swimmer [10] is faster than the three-sphere swimmer.
the deformation occurs for exponentially small parameter Ourself-organizedswimmercanbeusedtostudytheef-
changes[26]. Incertainparameterregions,we indeedfind fects of hydrodynamiccoupling between micro-swimmers.
that the Hopf-bifurcation is followed by such a deforma- So far, studies of this subject have largely neglected the
tion which is called a Canard explosion. Accompanying mechanismusedbymicro-swimmerstogeneratetheshape
with the Canard phenomenon, we find also excitable dy- changesnecessaryfor swimming. Preliminaryresults sug-
namics for the swimmer. gest interesting synchronization effects that greatly alter
In addition to the Canard explosion, the singularly the system’s behavior compared to swimmers with fixed
perturbed system (7)-(9) shows dynamic behavior that is sequences of shape changes. Obviously, it would also be
reminiscent of the dynamics associated with global bifur- interesting to extend our approachto study the dynamics
cations. In this case, the spontaneous oscillations of the of cilia and flagella.
linker are still generated by a Hopf bifurcation. The dy-
namicsclosetothebifurcation,however,showscharacter- ∗∗∗
isticsofthedynamicsclosetoasaddle-nodeinfiniteperiod
bifurcation (SNIPER). A SNIPER bifurcation is charac- We thank A. Hilfinger and E. M. Nicola for useful dis-
terized by two fixed points, a saddle and a node, that cussions.
collide, which leads to oscillations around a third, unaf-
fected fixed point, namely, an unstable node [27]. In the
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2ℓf = 0.6µm, ℓm = 0.75µm, ∆ = 56nm, k = 4pN/nm,
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