Gauging nanoswimmer dynamics via the motion of large bodies Ashwani Kr. Tripathi1and Tsvi Tlusty1 2 3 1Center for Soft and Living Matter Institute for Basic Science IBS Ulsan 44919 Republic of Korea

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Gauging nanoswimmer dynamics via the motion of large bodies

Ashwani Kr. Tripathi1and Tsvi Tlusty1, 2, 3, ∗

1Center for Soft and Living Matter, Institute for Basic Science (IBS), Ulsan, 44919, Republic of Korea

2Department of Physics, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea

3Department of Chemistry, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea

(Dated: October 17, 2022)

Nanoswimmers are ubiquitous in bio- and nano-technology but are extremely challenging to measure due to

their minute size and driving forces. A simple method is proposed for detecting the elusive physical features

of nanoswimmers by observing how they affect the motion of much larger, easily traceable particles. Modeling

the swimmers as hydrodynamic force dipoles, we ﬁnd direct, easy-to-calibrate relations between the observable

power spectrum and diffusivity of the tracers and the dynamic characteristics of the swimmers—their force

dipole moment and correlation times.

Introduction.— In recent years, nanoscale swimmers attracted

much interest as a basic physical phenomenon with promis-

ing potential in biomedical and technological applications [1–

3]. Examples include artiﬁcial swimmers, such as chemi-

cally powered nanomotors [2,4–10], bio-molecules that ex-

hibit enhanced diffusion [11], and bio-hybrid swimmers [1,

3,12,13]. Because of their minute size, the motion of nano-

and micro-swimmers is deep in the low-Reynolds regime

where viscous forces dominate over inertia [14–16]. But

the swimmers also experience stochastic forces from the sur-

rounding solvent molecules, and at the nanoscale, these ther-

mal ﬂuctuations become comparable to the typical driving

forces. Hence, measuring the properties of nanoswimmers

using traditional techniques, such as ﬂuorescence correlation

spectroscopy (FCS) and dynamic light scattering (DLS) [5,

11,17], is extremely difﬁcult, leaving core questions in the

ﬁeld—particularly, whether enzymes and small catalysts self-

propel—open and a matter of lively debate [18–25].

An alternative path to characterize nanoswimmers is by ob-

serving how they affect the motion of large, micron-size par-

ticles [26,27], such as silica beads [28], or vesicles [29].

Tracer motion has been extensively investigated in suspen-

sions of micro-swimmers, especially microbes [30–43] whose

size is comparable to the spherical and ellipsoidal tracers used.

These studies typically report a many-fold enhancement of

the tracers’ diffusion compared to thermal diffusion [30–37].

Theoretical models that explain the observed enhancement are

based on the hydrodynamic interactions induced by the swim-

mers’ motion over long, persistent trajectories [38–41]. The

enhancement is proportional to the volume fraction of swim-

mers, their self-propulsion speed [31–33], and geometrical

factors, such as the average run-length of the swimmer before

it changes direction [38–41]. Here, thermal ﬂuctuations have

a negligible effect compared to self-propulsion, as indicated

by a many-fold increase in diffusivity.

While similar experimental studies of tracer motion in a

suspension of nano-swimmers are much fewer [26,27], they

suggest a common mechanism of momentum transfer from

swimmers to tracers which may operate at the molecular scale

of organic reactions [18,19,44]. Unlike the motion of the

nano-swimmers, the motion of the micro-sized tracer particles

is easy to track, for example, by video microscopy, and one

could therefore, in principle, use tracers to probe the forces

generated by the swimmers. But the application of this po-

tentially advantageous method is hindered by the lack of un-

derstanding and rigorous computation of the hydrodynamic

coupling between nano-swimmers and tracers.

Here, we present a ﬁrst-principles theory that addresses this

problem by linking the tracer’s motion to the dynamics of

the swimmer suspension. The theory derives the hydrody-

namic ﬂow ﬁeld generated by swimmers, and its effect on the

tracer’s motion, accounting for three physical effects domi-

nant in the nano-regime: (a) Thermal ﬂuctuations – due to

their nanometric size, the swimmers are subjected to strong

thermal forces, giving rise to vigorous stochastic rotation and

translation. (b) Stochastic driving – nano-swimmers are often

propelled by strongly ﬂuctuating chemical reactions, where

intermittent activity bursts are separated by rest periods as, for

example, in enzymatic reactions. (c) Near-ﬁeld hydrodynam-

ics: a micron-sized tracer is effectively a large-scale boundary

and swimmers are in the near-ﬁeld view of the tracer. By com-

puting these three physical effects (see Model section), we

obtain our main results: simple expressions for the observable

force-force autocorrelation, power spectrum, and diffusivity

of the tracer particles from which one can gauge the nano-

swimmer’s dipole moment and persistence time (particularly,

Eqs (9,11,13)).

In the following, we explain the underlying physical

intuition and main steps of the derivation (whereas the details

are given in the Supplemental Material (SM) [45]). We

then perform a Brownian dynamics simulation of the tracer

motion in the swimmer suspension and compare it with our

analytical ﬁndings. Finally, we propose and demonstrate,

using the simulation results, how to use the derived estimates

in experiments, especially as physical bounds for testing

hypothesized self-propulsion mechanisms.

Model.— The swimmer motion is within the highly viscous

regime, at low Reynolds number, and it is force- and torque-

free. Hence, the leading order contribution to the ﬂow ﬁeld of

a swimmer is due to the force dipole [46–49]. Thus, we con-

sider the nano-swimmer suspension as an ensemble of force

dipoles, each consisting of two equal and opposite point forces

(Stokeslets) of strength f=fˆ

f, separated by an inﬁnitesi-

arXiv:2210.07557v1 [physics.bio-ph] 14 Oct 2022

mal distance `D=`Dˆ

e. The resulting force dipole, which

determines the far-ﬁeld ﬂow is the tensor mαβ =mˆ

eαˆ

fβ,

where m=f`Dis the dipole’s magnitude. The veloc-

ity ﬁeld induced at a distance rfrom the dipole is obtained

from the gradient of Gαβ, the hydrodynamic Green func-

tion, vα

D(r) = mβγ ∂γGαβ(r). The Green function, Gαβ =

(δαβ +ˆ

rαˆ

rβ)/(8πηr), is the mobility tensor, deﬁned as the

ﬂow generated by a Stokeslet of unit strength. From a physi-

cal point of view, it is instructive to divide the induced velocity

vDinto symmetric and anti-symmetric parts,

vD(r) = [3 (f·ˆ

r) (`D·ˆ

r)−f·`D]ˆ

8πηr2+(f×`D)×ˆ

8πηr2.(1)

The ﬁrst, symmetric contribution, called stresslet, arises from

the straining motion of the dipole when the forces are par-

allel to the dipole’s orientation. The second, anti-symmetric

term, known as rotlet, corresponds to the rotational motion

of the dipole, arising when the forces and the dipole are not

aligned [50–53].

A tracer subjected to the velocity ﬁeld vDexperiences a

hydrodynamic drag that depends on its size. Tracers are

much larger than the swimmers, move much slower, and

therefore effectively serve as static boundaries. Finding the

ﬂow ﬁeld near the tracer’s surface generally requires calculat-

ing the image system of the force dipole by a multipole ex-

pansion [48,49,54]. However, one ﬁnds that the force F

exerted on a spherical tracer depends only on the leading-

order monopole term, and can be calculated using Fax´

en’s

law [55,56],

F= 6πµa 1 + 1

6a2∇2vD(r)r=0

,(2)

where ais the radius of the spherical tracer whose center is at

r= 0. We notice that the second term arises from the large

scale of the tracer and becomes negligible when the tracer size

is small compared to its distance from the swimmer (ar).

Substituting the dipolar velocity ﬁeld (Eq. (1)) in Fax´

en’s law

(Eq. (2)), we obtain the force exerted on the tracer, F=Fstr +

Frot, where the contributions from the stresslet and rotlet are,

Fstr =3am

4r2"ˆ

r−ˆ

e·ˆ

f+ 3(ˆ

e·ˆ

r)(ˆ

f·ˆ

r)+

r2hˆ

rˆ

e·ˆ

f−5(ˆ

e·ˆ

r)(ˆ

f·ˆ

r)+ˆ

e(ˆ

f·ˆ

r) + ˆ

f(ˆ

e·ˆ

r)i#,

Frot =3am

4r2hˆ

f(ˆ

e·ˆ

r)−ˆ

e(ˆ

f·ˆ

r)i,(3)

and ris the distance between the swimmer and sphere center

with the unit vector ˆ

Consider a suspension consisting of an ensemble of Nforce

dipoles of strengths {mi}located at positions {ri}with dipole

and force orientations, {ˆ

ei}and {ˆ

fi}. Due to the linearity

of the Stokes ﬂow and the minute size of the swimmers, one

can neglect higher-order terms and multiple-scattering inter-

actions among the dipoles. Within this approximation, the

total force on the tracer is simply a superposition of the forces

exerted by the individual dipoles,

Ftot =

i=1 F(i)

str +F(i)

rot,(4)

where F(i)

str and F(i)

rot are the contributions due to stresslet and

rotlet from the i-th dipole.

Due to their nanometric size, the dipoles experience strong

stochastic kicks by the solvent molecules and other noise

sources present in the suspension, resulting in two effects.

First, the ﬂuctuations induce diffusive motion, translational

and rotational. The translational diffusivity scales inversely

with the swimmer’s size , D=kBT /(3πη`D), whereas ro-

tational diffusion has an inverse cubic dependence, Dr=

kBT/(πη`3

D). As a result, during a typical rotational timescale

τr= 1/(2Dr), a particle will diffuse to a distance ∼`D

while rotating about one radian. Since the separation between

tracers and dipoles is typically much larger than the dipole

size (r`D), the change in dipole position due to transla-

tional diffusion has a negligible effect on the hydrodynamic

force it exerts on the tracer (Eq. 3). In contrast, within the

same period, a swimmer performs, on average, a full rotation,

thus strongly affecting the force on the tracer. This stochastic

wandering of the orientations ˆ

eand ˆ

fon the surface of a unit

sphere is captured by a rotational diffusion equation [57–59],

from which we obtain probability moments for orientations

that are required for the calculating moments of the force Ftot

(see details in SM [45]).

The second stochastic effect stems from internal ﬂuctua-

tions of the force dipole that vary its magnitude, m(t)[60].

Certain force dipoles, particularly those fueled by chemical

cycles, will work in bursts with ﬁnite persistence time τceach

stroke. Such swimmers are additionally characterized by τp,

the typical cycle period between strokes, which in catalysts

is the inverse of the turnover rate. We take for simplicity,

the bursts of duration τcthrough which the force dipole is

constant, m(t) = m. These square bursts occur on average

every τp, and in between the bursts, the force dipole is idle

m(t)=0. The resulting autocorrelation of the moment is

hm(t)m(0)i=m2bhb+ (1 −b)e−t/τmi,(5)

where b=τc/τpis the relative fraction of the bursts during

the cycle, and τm=τc(1 −b)is the timescale of moment

ﬂuctuations (See SM [45]).

Results.— The ﬂuctuations in the dipoles’ orientation and

moment render the force Ftot stochastic and to calculate its

statistics, we consider an arbitrarily oriented force dipole with

axisymmetric (ˆ

e=ˆ

f) and transverse (ˆ

e⊥ˆ

f) components,

f=fkˆ

e+f⊥ˆ

f. For the axisymmetric dipole, only the stresslet

contributes to the total force, whereas for the transverse dipole

both stresslet and rotlet contribute (Eqs. (3,4)). Thus the total

force can be expressed as the sum of axisymmetric and trans-

verse components, Ftot(t) = Ftot

k(t) + Ftot

⊥(t)). To ﬁnd the

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GaugingnanoswimmerdynamicsviathemotionoflargebodiesAshwaniKr.Tripathi1andTsviTlusty1,2,3,1CenterforSoftandLivingMatter,InstituteforBasicScience(IBS),Ulsan,44919,RepublicofKorea2DepartmentofPhysics,UlsanNationalInstituteofScienceandTechnology,Ulsan,44919,RepublicofKorea3DepartmentofChemistry,UlsanNa...

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Gauging nanoswimmer dynamics via the motion of large bodies Ashwani Kr. Tripathi1and Tsvi Tlusty1 2 3 1Center for Soft and Living Matter Institute for Basic Science IBS Ulsan 44919 Republic of Korea

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