Can mass change the diffusion coefficient of DNA-coated colloids?

Sophie Marbach1, 2 and Miranda Holmes-Cerfon1

1Courant Institute of Mathematical Sciences, New York University, NY, 10012, U.S.A.2CNRS, Sorbonne Universite, Physicochimie des Electrolytes et Nanosystemes Interfaciaux, F-75005 Paris, France ∗

Inertia does not generally affect the long time diffusion of passive overdamped particles in fluids.Yet we have discovered a surprising property of particles coated with ligands, that bind reversibly tosurface receptors – heavy particles diffuse more slowly than light ones of the same size. We show thisby simulation and by deriving an analytic formula for the mass-dependent diffusion coefficient in theoverdamped limit. We estimate the magnitude of this effect for a range of biophysical ligand-receptorsystems, and find it is potentially observable for micronscale DNA-coated colloids.

It is well known that inertia does not affect eitherthe equilibrium probabilities or dynamics in overdampedsystems [1], and especially that it does not affect thelong time diffusion of micron-scale particles in liquidsat equilibrium [2]. In fact, momentum relaxation formicronscale particles occurs over a timescale of or-der 1 µs [3]. Particles are generally observed on muchlonger timescales, where the motion at equilibrium ap-pears overdamped and diffusive, with diffusion coefficientindependent of mass for large enough particles [4, 5]. In-ertia can only affect the short-time mobility of a particle[2, 5], and it can play a role for active particles where τmis comparable to the diffusive rotational timescale, whichis only accessible experimentally for active particles in air(not in liquids) [6]. Therefore, to our knowledge, thereis currently no proposed physical system where inertiacould affect long time diffusion of micronscale particlesin liquids at equilibrium.

Yet, as soon as another timescale, originating froman additional complexity of the system, is of the sameorder as the inertial relaxation timescale, one mightask whether the mobility of the system could dependon its mass [2, 4–12]. One widespread example of anoverdamped system with an additional fast timescale isparticles with ligand-receptor contacts, such as colloidsfunctionalized with DeoxyriboNucleic Acid (DNA) stick-ers [13–15], viruses [16–19] or white blood cells [20–22].Such particles are coated with sticky ligands that bindand unbind to receptors on an opposing surface (Fig. 1),which changes the relative mobility between the parti-cle and the surface [14, 16, 23, 24]. The ligand bindingand unbinding rates can be fast, so one might speculatethat when they occur on the same timescale as the re-laxation of the ambient fluid’s momentum, the couplingbetween these two effects, which both act on the parti-cle’s momentum, could induce inertial effects. For exam-ple, bimolecular reactants with inertia can show differentsurvival probability decay functions depending on theirmass [25, 26]. Furthermore, we have recently pointedout that models of DNA-coated colloids find differentlong-time diffusion coefficients when they start with theunderdamped Langevin equation for particle motion, as

∗ sophie@marbach.fr

in Ref. [27], or from the overdamped equation, as inRef. [28]. For stiff ligands, where motion is strongly sup-pressed when the ligand is bound to a surface receptor,the particle’s diffusion coefficient is significantly reducedwhen it is underdamped compared to when it is over-damped (Fig. 1). We therefore ask: could inertia affectthe long-time diffusion of particles with ligand-receptorcontacts? Can we quantify the magnitude of the effectand predict its onset for existing biophysical systems?

100 101 102 103 104

Mass m ( 2/k)

10-2

10-1

100

Effe

ctiv

e di

ffusi

on D

eff/D

0

This workSimulationMarbach et al. 2021Lee-Thorp et al. 2018

Mobile sticky endFixed to the particle

Mobile particleUniformly sticky surface γ

kΓ qon

qoff

m Deff(m)?

overdamped

underdamped

FIG. 1. Mass changes the diffusion coefficient of a par-ticle with 1-ligand. Numerically obtained long time diffu-sion coefficient Deff of a plate of mass m with bare frictioncoefficient Γ, with a single ligand - called here leg. The leghas friction coefficient γ and spring constant k. It binds andunbinds with rates qon and qoff to a uniformly sticky sur-face. Limit behaviors derived in previous works from theoverdamped Langevin equation for the particle in Ref. [28](dashed) and from the underdamped Langevin equation inRef. [27] (dotted). “This work” refers to Eq. (5). Error barscorrespond to one standard deviation for 20 independent sim-ulations. Parameters for the simulations are qon = 0.01k/Γ,qoff = 0.008k/Γ, γ = Γ.

We address these questions by investigating a minimalmodel that includes the essential ingredients of (a) in-ertial relaxation and (b) stochastic dynamics of bindingand unbinding. We consider an N -legged particle of massm, with spring-like legs representing the ligands [29–31],and a sticker at the tip of each leg that may transientlyattach to a uniformly sticky surface – see Fig. 1. Legsattach independently to the surface with rate qon and

arX

iv:2

112.

0526

6v1

[co

nd-m

at.s

oft]

10

Dec

202

1

2

detach with rate qoff , where both rates are here takenconstant for simplicity. Inertia of the legs may be ne-glected as in general legs are much lighter than the par-ticle (see also Supplementary 2.4 for a detailed justifi-cation). The particle’s motion is investigated in 1D, onthe lateral dimension parallel to the surface axis. Thedynamical equations for the unbound leg lengths lj are

dljdt

= −kγ

(lj − l0) +

√2kBT

γηj , (1)

where k is a spring constant, l0 is a rest length and γ is thefriction coefficient of each leg. The ηj are uncorrelatedwhite gaussian noises, such that 〈ηj(t)ηk(t)〉 = δkjδ(t−t′)and 〈ηj(t)〉 = 0 where 〈·〉 is an average over realizationsof the noise. The bound legs li are constrained to moveat the same speed as the particle, v = dx/dt, where x isthe particle position:

dx

dt=dlidt

= v. (2)

Finally, the particle’s velocity is governed by Newton’slaw, including friction and stochastic forces on the parti-cle induced by the ambient fluid, as well as friction, recalland stochastic forces originating from the bound legs

mdv

dt= −Γv +

√2kBTΓηx

+∑

i∈ bound

(−γv − k(li − l0) +

√2kBTγηi

).

(3)

Here the ηi and ηx are further uncorrelated white gaus-sian noises and i is a running index over currently boundlegs. Note that for a particle in fluid the relevant massscale in Eq. (3) is m→ m+mf/2 where mf is the massof the displaced volume of fluid [2, 32]. We remark thatcompared to a starting point with overdamped equationsfor both the leg and the particle, here it is not necessaryto project the unbound dynamics to obtain the bounddynamics [28, 33]; Newton’s law is sufficient.

The Langevin dynamics in Eq. (3) are our startingpoint to investigate the effect of inertia [11, 34]. We re-call that in fluids of similar density as the particle – asis mostly envisioned here, except for e.g. gold colloids inwater or viruses in air – momentum relaxes algebraicallyinstead of exponentially and thus the effective decay of in-ertial effects is even slower than predicted with Langevindynamics [2]. Capturing this algebraic decay generallyrequires solving for the full flow field of the solvent andis therefore not suitable for analytic investigation, how-ever, we expect our model will give a lower bound onthe effect of inertia, and is therefore suited to explore theonset of inertial effects.

Stochastic simulations of our model show that the longtime diffusion strongly depends on mass. As an exam-ple, we show in Fig. 1, the long time diffusion coefficientcalculated as

Deff =t�τmax

〈x2(t)〉2t

(4)

of a particle with N = 1 leg, where τmax is the longestphysical timescale in the model (see Supplementary 1).Deff continuously decreases with particle mass, betweenthe two limit regimes identified respectively in Ref. [28](overdamped) and Ref. [27] (underdamped), by morethan an order of magnitude.

To further understand how this decrease depends onthe model parameters (N, k, qon, qoff , γ,Γ), we derive ananalytic expression for Deff by considering the over-damped limit of the combined particle and leg dynamics.We begin by introducing the 5 nondimensional scales

x→ Lxx, li − l0 → Lli, t→ τ t, v → V v, m→Mm.

Here L =√kBT/k is the characteristic length of leg fluc-

tuations, while Lx and τ are respectively the lengthscaleand timescale for the long-time motion of x. In con-trast with other works [35, 36], the latter two scales arenot determined a priori by any intrinsic scales. Rather,they are chosen large enough that coarse-graining will beappropriate over such scales and will lead to diffusive dy-namics [28]. Hence we choose Lx = L/ε where ε � 1 isa small nondimensional number, and τ = L2

x/D0, whichcorresponds to τ = 1

ε2Γk . Velocity fluctuations are fast

compared to diffusive motion such that V = Lx/τε.Importantly, we must specify the scale of mass by

comparing the velocity auto-correlation time τv to τ .One choice is to relate τv to the recall spring force, asτv = MV/(Lk) [27]. If we require τv to be much smallerthan the observation timescale, say τv = ε2τ then weobtain τm = M/Γ = Γ/kε2. This implies the iner-tial relaxation τm is very large compared to the relax-ation of leg fluctuations, typical of an underdamped mo-tion, and is therefore not the appropriate scale to con-sider. Instead we consider another choice, i.e. the veloc-ity auto-correlation time in the absence of recall forcesτv = τm = M/Γ and consider that it is small comparedto the observation time τv = τm = ε2τ . Note that thelatter scaling is the usual scaling to obtain overdamped,or more generally long time diffusive, dynamics [37].

Finally, we observe the system at sufficiently long timessuch that the remaining timescales are much shorter:γ/k ∼ Γ/k ∼ q−1

on ∼ q−1off . We thus can take qon → τ

ε2 qon

and similarly for qoff .We use our hierarchy of scales to coarse-grain the

dynamics of the system specified through Eqns. (1-3).The equations have an associated backward Kolmogorovequation ∂tf = Lf , where f is a vector-valued functionwith 2N components corresponding to all the possiblebond states and L is the generator of the system. In-ferring the long-time dynamics of f is sufficient to in-fer the long-time dynamics of the particle. The non-dimensional backward equation is (dropping the · forsimplicity) ∂tf = Lf = 1

ε2L0f + 1εL1f . We seek a so-

lution to this equation in the form of an expansion in ε,f = f0 + εf1 + ε2f2 + .... Standard coarse-graining steps(Supplementary 2) allow us to establish long time over-damped dynamics on f0 [27, 28, 35, 37]. We find that thesystem diffuses with a diffusion coefficient, independent

3

of ε, as

Deff(m) =kBT

Γeff(m)=

N∑n=0

pnkBT

Γn(m)(5)

where pn =(Nn

) qnonqN−noff

(qon+qoff )Nis the probability to have n

bonds and Γn(m) are the effective friction coefficientscontributing to a state with n bonds. The {Γn} satisfya linear system of equations involving the mass of theparticle m that is reported in Supplementary 2. Impor-tantly, Eq. (5) predicts up to order of magnitude changeson the effective diffusion Deff depending on the specificmicroscopic parameter values.

Let us analyze in detail a N = 1-legged particle. Thesystem determining {Γn} can be solved analytically toobtain friction coefficients for the bound and unboundstates

Γ0(m)

Γ= 1 +

mqon

Γ

γeff

γeff + Γ +m(qon + qoff),

Γ1(m)

Γ= 1 +

γeff

Γ

Γ +mqon

Γ +m(qon + qoff).

(6)

Here γeff = γ + k(

1qoff

+ γkqonqoff

)is the effective friction

from the leg, including the leg’s bare friction γ and recallforces from the tethered spring. The coefficients satisfyΓ0 ≤ Γ1 as, when it is bound, the leg exerts additionalrecall forces on the particle. We compare our analyticresult for Deff = kBT/Γeff with direct stochastic simula-tions over a wide range of parameters and find excellentagreement (Figs. 1 and 2A). Overall, the effective fric-tion Γeff increases with mass, and therefore the particle’sdiffusive motion slows down with increased mass.

Eq. (6) gives insight into what controls both the on-set of the diffusion slow down, and the magnitude of theeffect. Diffusion begins to decrease when the bindingand unbinding times τon = q−1

on , τoff = q−1off become small

enough to be comparable with the relaxation time ofinertia, τm = m/Γ. This is also apparent in Fig. 2Awhere the transition between limit regimes systemati-cally occurs for mqon/Γ ∼ 1. For shorter binding times,τon∼ τoff � τm, inertial effects matter, and the frictioncoefficients converge to (using the notation m = ∞ form� Γ/qon,Γ/qoff)

Γm=∞0

Γ=

Γm=∞1

Γ= 1 + p1

γeff

Γ. (7)

The friction coefficients contributing to each state are thesame, regardless of the state (bound or unbound) of theparticle. This is coherent: since the particle has signif-icant inertia, it does not have the time to accelerate ordecelerate to a different dynamical regime upon changingstate. Binding and unbinding happen too rapidly for theparticle to sense differences. Eq. (7) was also obtained inRef. [27] which started with the underdamped equationsand the scaling τv = MV/(Lk).

10-2 10-1 100 101

Inertia to binding ratio m q on/

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Effe

ctiv

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ffusi

on D

eff/D

0

100 101

Number of legs N

10-2

10-1

100

Effe

ctiv

e di

ffusi

on D

eff/D

0

10-2 10-1 100 101

Inertia to binding ratio m q on/

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Effe

ctiv

e di

ffusi

on D

eff/D

0

SimulationAnalytic

= = 10

q off = 0.08 k/q off = 0.8 k/

m/ << 1/q onm/ >> 1/q on

SimulationAnalytic

10 -2 10 -1 10 0 10 1

m q on/

0

0.2

0.4

0.6

0.8

Def

f/D0

N=1N=2

N=5N=10

N = 5

n = 2

N=1

A

B

FIG. 2. Diffusion slow down controlled by the ratio ofinertial to binding timescales, mqon/Γ. (A) Effective dif-fusion for a N=1-legged particle with respect to the inertia tobinding ratio, from direct stochastic simulations and analyticEq. (5), with friction coefficients given by Eq. (6). (B) Effec-tive diffusion with respect to the number of legsN , in the largeor small mass limit, from direct stochastic simulations and an-alytic Eq. (5) (solving numerically the linear system for the{Γn}) with mqon/Γ = 0.01 (red) and mqon/Γ = 100 (black).(top inset) Effective diffusion relative to inertia to bindingratio, for different N . (bottom inset) Schematic of a mobileparticle with N legs including n bound legs to the surface.Other simulation parameters are γ = 0.1Γ, qoff = 0.8k/Γ, andqon = k/Γ. Error bars correspond to one standard deviationfor 100 independent simulations.

Reciprocally, if binding timescales are long comparedto inertial relaxation (τon ∼ τoff � τm) we expect in-ertia to play a negligible role: the particle has the timeto accelerate and reach an overdamped, limit motion be-fore any further change of state occurs. In this case thefriction coefficients are

Γm=00

Γ= 1,

Γm=01

Γ= 1 +

γeff

Γ. (8)

Eq. (8) was obtained in Ref. [28], starting directly withoverdamped motion of the particle.

We therefore find that the onset of inertial effects isgoverned by the ratio of timescales, τon ∼ τoff comparedto τm. A posteriori, it is natural that this onset is con-trolled by timescales, yet it was not obvious which of thediversity of timescales at play in ligand-receptor systemswould matter. For example, the timescale for relaxation

4

of leg fluctuations k/γ does not control the occurrence ofinertial slow-down.

However, k/γ does control the magnitude of the iner-tial slow-down, via γeff . The relative slow down betweenthe underdamped and the overdamped regime is indeed

Dm=∞eff

Dm=0eff

=1 + γeff

Γ

1 + γeffΓ + p0p1

γ2eff

Γ2

. (9)

If the leg is very stiff (k � Γqoff , implying γeff � Γ), thendiffusion can be significantly slowed for massive particles,Dm=∞

eff � Dm=0eff . In fact, stiff legs greatly reduce motion

in the bound state, Γ1 � Γ. Since a particle with largemass does not have the time to accelerate while its leg isunbound, we also have increased friction in the unboundstate Γm=∞

0 � Γ and the particle’s overall mobility is de-creased, by up to orders of magnitude (as seen in Fig. 1).For an overdamped particle, on the contrary, even witha stiff leg, the particle can still move when it is unbound,as it has time to accelerate (Γm=0

0 = Γ), and its diffusioncoefficient remains finite.

Let us now consider a particle with many legs involvedin the binding process, say N ≈ 100 − 1000, as in someDNA-coated colloids at low temperatures [38, 39]. Thelinear system satisfied by {Γn} can be simplified when theaverage number of bonds is large, N = qon

qon+qoffN � 1,

to

Γeff(m) 'N�1

Γ + Nγeff . (10)

Now the diffusion coefficient does not depend on the massof the particle. Stochastic simulations, as well as numer-ical solutions of the linear system satisfied by the {Γn},confirm this result: the diffusion coefficient at large num-ber of legs N converges to a value independent of mass(Fig. 2B). Interestingly, the transition to the slowed-down diffusion regime occurs when τm/τon ∝ N (Sup-plementary 2.3.3).

Why do inertial effects vanish with a large number oflegs? For a particle with large mass, the friction coef-ficients for each state are equal (as in Eq. (7) for the1-legged case), because the particle does not have timeto accelerate between changes, and hence is only sensi-tive to the average configuration: Γm=∞

0 = Γm=∞1 = ... =

Γm=∞N = Γ+Nγeff (Supplementary 2.3.2). The difference

is that now an average of N legs exerts extra recall forces.For a particle with very small mass, friction coefficientsfor each state are different, but their sum in Eq. (5) isdominated by the most likely state, which is N when thisaverage is large, leading to Γm=0

eff ≈ Γm=0N

= Γ + Nγeff .Numerous legs can thus be thought of as self-averaging,and hence transitions between states do not matter asmuch as when there are a small number of bound legs.

We now explore the possible emergence of such iner-tial effects in biological or biomimetic systems. Withinour theory, the inertial relaxation time τm = m/Γ isquite fast: for a R ∼ 1 µm particle near a surface, Γ ∼2×6πηR ∼ 10−8 kg/s with water viscosity η ∼ 10−3 Pa.s

and m ∼ ρ 43πR

3 ∼ 10−15 kg with density ρ ∼ 103 kg/m3,such that τm = m/Γ ∼ 100 ns, accumulating to slightlylonger timescales if we account for the momentum re-laxation of the fluid [2]. In any case, to observe inertialeffects, the typical binding times (τon = q−1

on , τoff = q−1off )

have to be faster. As typical adhesive systems have atleast qoff . 10qon, we focus mainly on qon in the follow-ing, as it will be the limiting binding rate affecting Deff .We report the orders of magnitude for the momentumrelaxation time τm = m/Γ and binding times τon = q−1

on

for a variety of multivalent ligand-receptor systems – seeFig. 3 (and details in Supplementary 3.2). To keep thediscussion simple we limit ourselves to a small numberof average bound legs N ' 1, which is inherent or canbe achieved (e.g. with temperature control [39]) in allsystems explored.

VirusesMolecular motorWhite blood cellsNuclear pore complexDNA coated colloidsEscherichia Coli

slow bindingdynamics

very light particles

onset of inertial effects

nanoparticles

microparticles

P selectin

L selectin

VirusesMolecular motorWhite blood cells

Nuclear pore complexDNA coated colloidsEscherichia Coli

Sars CoV Influenza A

1 2

FIG. 3. Sorting biophysical systems. Ashby chart com-paring the typical inertia versus binding timescales for varioussystems. Full disks represent the range of values found in theliterature for each system. Systems are color coded accord-ing to their category in the legend. When multiple systemsbelong to a category, details are indicated next to the circles.Large dashed circles represent global categories of systems.

Numerous multivalent ligand-receptor systems – suchas spike proteins on the Influenza A virus [40, 41], molec-ular motors transporting cargos [42] or pili adhesion ofEscherischia Coli [43–45] – have binding kinetics thatare simply too slow to observe intertial effects, typicallyqon . 100 s−1. Such slow binding prevents inertial slowdown (blue dashed circle in Fig. 3). These systems areindeed overdamped.

Other systems do have fast binding kinetics (qon &104 s−1), but not fast enough to incur inertial effects onthe lighter systems they are connected to. These oftencorrespond to smaller particles, and since m/Γ ∝ R2,this decreases the maximum binding timescale requiredto observe inertial effects. This is the case of Sars CoV 1and 2 viruses that are barely R = 50 nm in radius [46, 47],

5

of DNA-coated nanocolloids that have particle radii inthe R = 10 nm range [48] and of protein transportersin the nuclear pore complex [49] (orange dashed circlein Fig. 3). Some systems are both light and with slowbinding kinetics, as is the case of e.g. Influenza A, andhence are also overdamped.

Inertial effects may therefore occur for a specific com-bination of large particles and fast binding kinetics. Wehave isolated two systems for which such a combinationmay be obtained: (a) DNA-coated colloids near the melt-ing point, where the colloids are micron-sized (R ∼ 1µm)and (b) white blood cells with adhesion mediated by theL-selectin protein only – see Table. S1. For both thesesystems, typical existing experimental designs possess aninertial time to binding time ratio mqon/Γ ' 10−3−10−1,which is already within, or close to, the range where wepredict inertial slow down – see Fig. 2.

DNA-coated colloids offer a promising route to probesuch inertial slow down as they may be finely tuned toincrease the inertial relaxation time, for example by us-ing coating processes on gold particles [38] (to increasem), larger particles [39] (to increase the ratio m/Γ ∝ R2)or by modifying the viscosity of the solution (to alter Γ).The average number of bound legsN for DNA-coated col-loids can also be tuned over a wide range (N = 1−1000),for example by using different coating densities on eithersurface, or by changing temperature [39]. With typi-cal experimental parameters [28, 39, 50], Eq. (9) pre-dicts a slow down of Dm=∞

eff /Dm=0eff ' 92%, which could

be amenable to experimental measurements (see Supple-mentary 3.1). Note that targeted experimental designs(such as stiffer legs) could produce a greater decrease.Therefore, we expect that inertial slow down of the long-time diffusion coefficient could be probed with existingexperimental techniques on DNA-coated colloids.

In summary, we have found that inertia can modifythe diffusion coefficient of particles with ligand-receptorcontacts, inducing a diffusion slow-down with increasedinertia. Within our analytic framework, the onset of thisslow-down occurs when the binding time scale τon = q−1

on

is faster than the timescale for the inertial relaxation,which is τm = m/Γ in our model, a lower bound on the

actual timescale since momentum decays algebraically inmost fluid systems [2, 5]. The magnitude of the effect iscontrolled by the stiffness of the ligands and the averagenumber of bound legs, with stiff ligands and fewer boundlegs promoting increased inertial effects. We predict thediffusion slow down could be probed experimentally byfine-tuning existing DNA-coated colloids. Such experi-ments may also give further insight into the coupled dy-namics between the solvent and the ligand-receptor bind-ing dynamics, and identify other inertial effects, beyonddiffusion slow down. For example, in dense active sus-pensions, particles need time to pick up speed after col-lisions, which modifies both their long time diffusion [6]and phase separation behavior [12]. Improvements to ourmodel could include fluid memory kernels to investigatethe decay of velocity autocorrelation functions [2] or in-homogeneities in the lateral direction to probe inertialeffects on the subdiffusion regime [38]. For DNA-coatedcolloids, one could envision that such inertia-modifieddynamics could also impact collective properties suchas crystallographic alignment into self-assembled struc-tures [33, 51]. Understanding the dynamics of such com-plex micronscale particles is a key step to pave the waytowards controlled design at the microscale, e.g. to im-prove synthesis of materials with advanced optical prop-erties [52, 53].

ACKNOWLEDGMENTS

The authors acknowledge fruitful discussions withAleksandar Donev and Brennan Sprinkle. S.M. acknowl-edges funding from the MolecularControl project, Euro-pean Union’s Horizon 2020 research and innovation pro-gramme under the Marie Sk lodowska-Curie grant awardnumber 839225. All authors were supported in part bythe MRSEC Program of the National Science Foundationunder Award Number DMR-1420073. M.H.-C. was par-tially supported by the US Department of Energy underAward No. DE-SC0012296, and acknowledges supportfrom the Alfred P. Sloan Foundation.

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