Kinetics of Nanoparticle-Membrane Adhesion Mediated by Multivalent

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Kinetics of Nanoparticle-Membrane Adhesion Mediated by Multivalent Interactions Roberta Lanfranco, Pritam Kumar Jana, Lucia Tunesi, Pietro Cicuta, Bortolo Matteo Mognetti, Lorenzo Di Michele, and Gilles Bruylants Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b02707 • Publication Date (Web): 14 Jan 2019 Downloaded from http://pubs.acs.org on January 14, 2019

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Langmuir

Kinetics of Nanoparticle-Membrane Adhesion Mediated by Multivalent Interactions Roberta Lanfranco,

∗,†,‡

¶

Pritam Kumar Jana,

¶

Bortolo Matteo Mognetti,

†Biological

Lucia Tunesi,

∗,†

Lorenzo Di Michele,

†

Pietro Cicuta,

and Gilles Bruylants

†

∗,‡

and Soft Systems, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom

‡Université

libre de Bruxelles (ULB), Engineering of Molecular NanoSystems, 50 av. F.D. Roosevelt, 1050 Brussels, Belgium

¶Université

Libre de Bruxelles (ULB), Interdisciplinary Center for Nonlinear Phenomena

and Complex Systems, Campus Plaine, CP 231, Blvd. du Triomphe, B-1050 Brussels, Belgium E-mail: [email protected]; [email protected]; [email protected]

Abstract

anchored receptors and of the bulk concentration of nanoparticles, and is observed to de-

Multivalent adhesive interactions mediated by a

crease substantially in regimes where the num-

large number of ligands and receptors underpin

ber of available receptors is limited compared

many biological processes, including cell adhe-

to the overall number of ligands. We attribute

sion and the uptake of particles, viruses, para-

such peculiar behaviour to the rapid sequestra-

sites, and nanomedical vectors. In materials sci-

tion of available receptors after initial nanopar-

ence, multivalent interactions between colloidal

ticle adsorption. The experimental trends and

particles have enabled unprecedented control

the proposed interpretation are supported by

over the phase behavior of self-assembled ma-

numerical simulations.

terials.

Theoretical and experimental studies

have pinpointed the relationship between equi-

Introduction

librium states and microscopic system parameters such as ligand-receptor binding strength and their density. In regimes of strong interac-

Adhesive interactions between nanoscale ob-

tions, however, kinetic factors are expected to

jects and lipid membranes are central for a num-

slow down equilibration, and lead to the emer-

ber of biological processes, including cell-cell

gence

communication mediated by extracellular vesi-

of

long-lived

out-of-equilibrium

states

13

viral infection,

46

cles,

self-assembly experiments and the adhesion of

therapeutic and diagnostic nanomedicine, nd-

particles to biological membranes. Here we ex-

ing reliable strategies to control the interaction

perimentally investigate the kinetics of adhe-

between biological membranes and nanoscale

sion of nanoparticles to biomimetic lipid mem-

probes is a key issue.

branes.

Multivalent interactions are produced

diated by specic ligands on the surface of the

by strongly interacting DNA constructs, play-

nano-objects that target specialised receptors

ing the role of both ligands and receptors. The

expressed on the cell membranes. The resulting

rate of nanoparticle adhesion is investigated as

multivalent interactions, mediated by a large

a function of the surface density of membrane-

number of interacting molecular agents, give

ACS Paragon Plus Environment 1

1012

and endocytosis.

79

that may signicantly inuence the outcome of

In

Adhesion is often me-

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Page 2 of 19

rise to complex phase behaviours, which emerge

adopted model system, Large Unilamellar lipid

from the interplay between enthalpic contribu-

Vesicles (LUVs) are targeted by gold nanopar-

tions to individual ligand/receptor interactions

ticles (NPs), and complementary DNA con-

and congurational/combinatorial entropic ef-

structs anchored on both substrates play the

fects.

The (bio)physics of multivalent inter-

role of ligands and receptors, where the latter

actions has been thoroughly characterised both

are able to freely diuse, as in many biologically

experimentally and theoretically, by means of

relevant situations.

analytical and numerical approaches.

1318

15,1822

15,4853

The programmabil-

In

ity and selectivity of Watson-Crick base pairing

the context of materials science, the acquired

ensures accurate control over the bond forma-

understanding enabled the design of colloidal or

tion, enabling a quantitative characterization of

nanoscale units, whose self-assembly behaviour

kinetic eects as a function of dierent param-

can be precisely prescribed.

eters characterizing the ligands, the receptors

2328

In targeted

drug delivery, multivalent interactions can be

and their complexes.

exploited to improve the binding selectivity of

Scattering (DLS), nanoparticle adsorption ki-

vectors, which can be designed to target cell

netics is characterized as a function of the sur-

membranes only if certain receptors are overex-

face density of membrane-anchored DNA recep-

pressed allowing one to discern between healthy

tors as well as the overall bulk concentration of

and diseased cells.

nanoparticles. We observe a substantial reduc-

Our

current

2932

understanding

of

the

Using Dynamic Light

equilib-

tion in adsorption rates in regimes where the

is not

number of available receptors is limited com-

matched by systematic studies of kinetic ef-

pared to the number of nanoparticles. The ob-

fects,

rium features of multivalent systems which

where

are

suciently

in

regimes

served trend is ascribed to the rapid and near-

interactions

become

irreversible sequestration of most of the initially

often

ligand-mediated

relevant

into

available receptors, following the adsorption of

long-lived metastable congurations that dif-

a relatively small number of nanoparticles. Re-

fer substantially from the equilibrium ground

ceptor depletion is a direct consequence of the

state.

Interactions of such strength can oc-

mobility of membrane-anchored DNA linkers,

cur in biological recognition schemes, where

and is expected to occur also in biological sys-

ligand-receptor binding free energies can ex-

tem, where adhesion is mediated by proteins

ceed

strong,

33

25 kB T .

tems,

34

kinetic

In

driving

the

multivalent has

been

colloidal

sys-

that can diuse in the membrane. Experimen-

characterised

tal evidence is backed up by state-of-the-art

and exploited to design self-protected inter-

simulations that accurately account for the

action schemes or sequential self-assembly pro-

eect of molecular reaction rates on the ad-

tocols.

sorption dynamics of the nanoparticles.

3539

arrest

system

Yet, with the exception of a lim-

ited number of studies

4045

including the recent

Our ndings demonstrate that, in the presence

investigation of the adhesion kinetics of DNA-

of strong ligand-receptor interactions, factors

functionalized

dendrimers

such as receptor mobility and concentration

systematic quantitative

should be taken into account when designing

poly(amidoamine)

to solid surfaces,

46,47

nanoscale probes for membrane targeting.

investigations have not been reported on the kinetics of multivalent nanoparticle-surface interactions.

Particularly signicant and unex-

plored is the role of kinetic eects in the pres-

Materials and Experimental

ence of mobile likers, in these directions will

Methods

be relevant to the understanding and design of the aforementioned biological and nanomedical processes.

DNA

In this paper, we present a quantitative inves-

ligands

and

receptors

tigation of the adsorption kinetics of nanopar-

Single stranded DNA was purchased from IDT

ticles to biomimetic lipid membranes.

(Integrated DNA Technologies, Leuven, Bel-

In the


Page 3 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir Strands mimicking ligands (5' -TGC

extrusion, using an Avanti mini-extruder. First,

GTG TGT GCG TTT TTT TTTT-Thiol-3') 0 feature a thiol modication on their 3 end, en-

a dry lipid lm is obtained by evaporation of a

Receptor strands (5'

The lipid lm is then rehydrated with experi-

-56FAM-TCG CAC ACA CGC TTTT TTT

mental buer (10 mM phosphate buer pH 7.4

TTT-Chol-3') are modied with both a uores0 0 cein (5 ) and a cholesterol molecule (3 ) (Fig. 1).

+ 0.5M NaCl).

The latter is connected to the DNA via a Tri-

incubating for 1 hour.

Ethylene Glycol (TEG) spacer,

and enables

extruded 31 times through a poly-carbonate

Ligand and Re-

membrane featuring track-etched pores with a

ceptor strands feature mutually complementary

diameter of 200 nm (Whatman). A high num-

sticky ends of 11 nucleotides, driving their hy-

ber of extrusion steps is necessary to obtain a

bridization and NP-membrane adhesion. Poly-

homogeneous size distribution of the liposomes.

T domains are included between the sticky over-

As prescribed by the manufacturer of the extru-

hangs and the grafting moieties to improve con-

sion kit (Avanti Polar Lipids), an odd number

gurational freedom.

of steps is required so that any particulate con-

gium).

abling grafting to NPs.

grafting onto lipid bilayers.

chloroform solution under vacuum for 1 hour.

Rehydration is facilitated by

vortexing the sample for 1 min and further The solution is then

taminant larger than the pore size is blocked by the membrane and removed from the sample. The formed liposomes have a hydrodynamic diameter, measured with DLS (Malvern Zetasizer NanoZS, Malvern, UK), of either 160±31 nm or 189±37 nm (SI, section 1.1).

The overall

lipid concentration (2 mM) and the hydrodynamic diameter were used to estimate the bulk concentration of the liposomes.

Vesicles were

functionalized with dierent concentrations of Receptor strands, to obtain surface densities in a biologically relevant range

54,55

(SI, sec-

tion 1.2). Functionalization was performed by adding DNA Receptors to the LUV solution and agitating on an Eppendorf thermomixer for 16 hours at 750 rpm. The cholesterol moiety spontaneously inserts into the lipid bilayer, providing a mobile but stable anchoring.

5658

Figure 1: Hybridization between DNA ligands

To demonstrate the grafting of all DNA strands

and receptors drives the adhesion of a NP to a

to the LUVs, samples were analysed by Ultra

lipid membrane.

Centrifugation (Optima LE-80K from Beck-

The sequences of ligand and

receptor DNA strands are reported. They fea-

man) in a sucrose gradient (SI, section 1.3).

ture, respectively, a thiol and a TEG-cholesterol moiety to anchor them to NPs and LUVs.

NPs

synthesis,

functionalization,

and characterization The

Liposome preparation, functional-

materials

used

to

synthesize

NPs:

Potassium gold (III) tetrachloride (KAuCl4 ,

ization, and characterization

CAS

number

Large Unilamellar Vesicles (LUVs) were pre-

CAS

number

pared

1-palmitoyl-2-oleoyl-glycero-3-

(Na3 C6 H5 O7 , CAS number 51 804), and Tris-

phosphocholine (POPC) lipids (Avanti Polar

HCl buer (1 M, pH 7, CAS number T6455),

Lipids, CAS number 26853-31-6) by membrane

were

from

dithiothreitol

3483-12-3),

purchased


450235),

from

trisodium

Sigma-Aldrich,

(DTT, citrate

while

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Page 4 of 19

phosphate buer and NaCl used for particle

NaCl + 10 mM phosphate buer at concentra-

functionalization were purchased from Merck.

tions reported in the paragraph List of Sam-

Gold nanoparticles (NPs) were produced via a

ples, tipically in the range of 0.1 - 0.2 nM,

modied Turkevich method

and immediately

was characterized via DLS. The scattering au-

dialyzed against a 0.1 mM citrate solution. The

tocorrelation function (ACF) was used to quan-

nanoparticles were characterized by TEM, UV-

tify the diusion parameters of free NPs, dis-

vis absorption, and DLS measurements, as re-

cussed in the Results and Discussion section.

ported in SI, section 2.

Then, LUVs were added to reach the sought

59

An average diameter

of 16.5±1.3 nm was determined by processing

NPs:LUVs bulk concentration ratio

TEM images with ImageJ. NPs were then func-

and

tionalized with DNA Ligands using thiol chem-

pipette.

istry.

cally 10 s after mixing, and 60 ACFs were ac-

60

The grafting density was estimated by

the

sample

was

rapidly

RNP:LUV ,

mixed

with

a

DLS data acquisition started typi-

measuring the DNA concentration of the su-

quired at 5 s intervals.

pernatant after a 20-minute centrifugation step

tween subsequent measurements was then in-

at 18,000 g. The concentration of DNA in the

creased to 30 seconds, 3 minutes, and 15 min-

supernatants after the rst washing step was

utes to monitor adhesion over the following

measured as 2.24

µM,

concentration was 3.23

while the initial DNA

µM,

14 h. All measurements were performed at 25°C

demonstrating the

occurrence of DNA grafting.

The time interval be-

and in 0.5 M NaCl ionic strength.

Higher-

Afterwards, the

than-physiological ionic strength of 0.5 M NaCl

same washing procedure was repeated 5 times

was used in this study to obtain suciently

to remove all the non-grafted DNA. We ob-

high ligand-receptor binding strength to allow

tained an average of 300 Ligand strands per

the emergence of signicant kinetic eects as-

NP. The presence of a DNA layer on NPs was

sociated to the near-irreversibility of the NP

conrmed by both DLS and UV-vis, as reported

adhesion to the target vesicles.

in SI, section 2.

of a higher-than-physiological ionic strength is

61

The eect

largely limited to the DNA-hybridization free energy, and therefore our results are readily ap-

Characterizing NP-LUV adhesion

plicable to any system with comparably strong

and dissociation temperature.

ligand-receptor interactions, regardless of the

The adhesion of NPs to LUVs was conrmed by

ionic strength. Similar ionic conditions are rou-

means of UV-vis absorption spectroscopy (Schi-

tinely used in DNA-nanosystems to make DNA

madzu UV-3600 equipped with a Peltier T con-

duplexes more stables.

6265

troller) monitoring the shift in the wavelength of the NP extinction peak induced by the re-

List of samples

ciprocal proximity between LUV-bound NPs. The dissociation temperature of the NP-LUV

A number of samples were tested with dier-

complexes was also determined and found to

ent surface concentration of receptors on the

increase with increasing surface density of the

LUVs,

receptors

ρR−LUV .

ρR−LUV ,

vant range,

Details are provided in SI,

54,55

concentration

section 3.

spanning a biologically releand dierent NP:LUV bulk

ratios,

RNP:LUV .

All

conditions are listed in Table 1. ples

Characterizing NP-LUV adhesion

with

a

nanoparticles

with Dynamic Light Scattering

-10

RNP:LUV =7

and

concentration

M, while RNP:LUV =290 ρN P =2.2×10-10 M. 10

Experiments to monitor the adhesion kinetics of NPs to LUVs were carried out using Dynamic Light Scattering (DLS). Initially, a sample of DNA-functionalized NPs in 0.5 M


70,

ρN P

tested

For samthe

bulk

is

1.1×

corresponds to

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Langmuir

Table 1:

List of tested samples:

RNP:LUV

Simulation Algorithm

in-

dicates the NP:LUV bulk concentration ratio,

ρR−LUV

We model NPs as hard spheres freely dius-

the average surface concentration

NR−LUV

the aver-

ing in a parallelepipedal simulation box (Fig. 2).

age total number of receptors on each LUV,

Nanoparticles interact with the basal surface

of receptors on the LUVs,

A,

equal to

number of Receptors available for each NP in

that of a LUV. The surface carries

NR−LUV

the system.

The number of ligands on each

receptors, a number compatible with the one

nanoparticle is xed and equal to 300, see SI,

used in experiments. As often done when mod-

section 1.2 for details on the estimation of these

elling DNA-functionalized surfaces ,

values.

glect receptorreceptor,

NR−NP = NR−LUV /RNP:LUV

and

(nm-2 )

of the simulation box having area

the average

16,18

we ne-

receptorligand,

ligandligand steric interactions.

and

We assume

Sample

RNP:LUV

1

7

3.9 10-3

370

53

2

7

8.2 10-3

770

110

-3

1190

170

pable of free diusion within a layer of thick-

-3

ness

3

ρR−LUV

7

12.7 10 17.5 10

NR−LUV

NR−NP

4

7

1640

234

5

70

3.9 10-3

370

5

6

70

8.2 10-3

770

11

7

70

12.7 10-3

1190

17

8

70

17.5 10-3

1640

23

370

1

that the sticky ends tethered to the surface of the LUV are uniformly distributed and ca-

` = 3 nm

surrounding the surface of the

LUV (see Fig. 2).

`

has been estimated using

typical endtoend distances of ssDNA.

66

Sim-

ilarly, nucleic acids on the NPs are simulated as

10

290

770

3

11

290

12.7 10-3

NL−NP sticky-ends uniformly distributed within a shell of thickness ` surrounding the parti-

1190

4

12

290

17.5 10-3

1640

6

cle (see Fig.

9

290

-3

3.9 10

-3

8.2 10

2).

Given that in our mod-

elling all ligands can bind,

NL−NP

has been es-

timated by calculating the maximum number of ligands that could bind the surface using a geometrical construction (see SI, section 4).

Nanoparticle

To improve computational eciency, also the ligands are regarded as freely diusive, rather than anchored to a point on the NP surface. This approximation has a negligible eect on the results of this work. We apply the simula-

LUV modeled as a flat surface

tion algorithm presented in reference

67

to study

DNA-directed self-assembly of LUVs, adapted to the present system.

Improving on conven-

tional algorithms, our method synchronizes the

0 Figure 2:

n

binding kinetics of DNA with the diusion ki-

25

netics of the nanoparticles,

enabling one to

study the eects of nite interaction rates on

Modelling ligand-receptor mediated

interactions between NPs and membranes. Lig-

the adsorption kinetics.

and and receptor stickyends are uniformly

does not account for the deformability of the

distributed, respectively, in the blue and red

LUVs.

stained regions surrounding the NP and the

tainly pivotal in processes such as endocyto-

membrane surface. Inset: simulation snapshot

sis

showing NPs anchored at the surface or freely

tween bound particles,

diusing in bulk. The colormap represents the

little role in the present study to the extent

number of formed NP-membrane bonds.

that reactions between dierent ligand-receptor

6873

Note that our model

While membrane deformability is ceror membranemediated interaction be-

74,75

it is expected to play

pairs can be considered as independent events. We have veried this hypothesis in a recent publication

19

using a more detailed model account-


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Page 6 of 19

of the paper. Indeed, here we describe how re-

λi ({r}) are geometric factors (see SI, section 4), ∆G0 is the hybridization free energy of 61 0 is free sticky ends in solution, β = 1/kB T , kon the on rate of the sticky ends, and ρ0 is the stan-

ceptors, once bound, no longer participate in

dard concentration equals to 1 mole per liter. In

the later stages of the adsorption.

the following we report on rates in simulation 0,∗ 0 2 −1 units: kon = kon ρ0 ` D . In physical units, −1 0 0,∗ = 3.1 × 106 ρ−1 = 1 corresponds to kon kon 0 s .

ing for membrane deformability. We also notice

where

that an eventual wrapping of the NPs by the lipid bilayer would not change the conclusions

This sce-

nario is not aected by the NPs being wrapped or not.

→

Nanoparticle positions (ri ) are updated (ri

ri + ∆ri )

Using the rates of Eq. 3 we calculate the probi i abilities pon / poff of forming / breaking a bond between nanoparticle i and the membrane, de-

using a Brownian dynamics scheme

∆ri = ri (t + ∆t) − ri (t) =

ned as (1)

√ D ∆t + D∆tN (0, 1) , fi kB T where

D

fi

aion

is the total force acting on NP

i,

Boltzmann constant and the index dividual NPs.

i

box.

is the

∼ exp[−¯ τ /atot ]). We increment a reaction clock (τreac ) by τ ¯ (τreac = τreac + τ¯) and, if τreac < ∆t, we update the vals s ues of {ni , ni , n }, the anities and probabili-

kB

for it to happen (Prob(¯ τ)

a short-range force fij due to entropic compres-

fi

Using Eq. 4, we sample one of all possi-

labels in-

Fig. 2), nanoparticles repel each other through

18

ties (Eqs. 5, 4), and re another reaction until

(Eq. 1) is then given

j fij , where fR,i is the force between the surface and NP i

τreac > ∆t.

fR,i = kB T [nsi Γ1 ({r}) − ns Γ2 ({r}) − ni Γ3 ({r})] .

at time

by

f = fR,i +

(5)

ble reactions, along with the average time (τ ¯)

In the (very diluted) bulk (see

sion of grafted DNA.

aioff

total number of nanoparticles in the simulation

is a vector of inde-

pendent normal-distributed numbers,

(4)

aion and aioff are the corresponding aniPNp i i ties and atot = i=1 (aon +aoff ) , where Np is the

in diluted conditions (estimated using Stokes7 2 −1 Einstein equation as 2.8 × 10 nm s ), ∆t is

N

aioff atot i = nsi koff ,

pioff =

where

is the nanopaticle bulk diusion coecient

the integration step,

aion , atot i = ni ns kon

pion =

Importantly, the last reaction is s s never used to update the values of {ni , ni , n }. Using the outcomes of the Gillespie algorithm

P

∆t,

we calculate forces using Eq. 2 and

(2) update nanoparticle positions using Eq.1. At which point, we reiterate the algorithm startIn Eq. 2,

nsi

ing from the calculation of the new on /o rates

is the number of ligandreceptor

bonds between nanoparticle i and the mems brane, n is the number of free receptors, and

(Eq. 3).

ni the number of free ligands on particle i. Γ1 , Γ2 and Γ3 are geometric factors and have been

To simulate the low dilution regimes used in experiments, we use grandcanonical moves in

derived in SI, section 4. s s We estimate ni , n , and ni using the Gille76 spie method. At a given conguration {r}, i i we calculate all the on /o rates (kon /koff ) of

which we insert/delete nanoparticles at the top

forming/breaking a bond between nanoparticle

more of them are adsorbed by the membrane us-

i and the membrane, with the assumption i i kon = koff = 0 if nanoparticle i is not in

con-

ing µ = µ0 + log (1 − nadh /RNP:LUV ), where µ0 , nadh , and RNP:LUV are, respectively, a reference

tact with the interface. In the SI, section 4, we

chemical potential, the number of adsorbed par-

prove that

ticles, and the stoichiometric ratio between NPs

layer of the simulation box. To simulate a xed NP:LUV concentration ratio, we gradually decrease the chemical potential

that

39

i 0 kon = kon λi ({r})

i 0 koff = kon ρ0 exp[β∆G0 ],

µ

of the NPs as

and LUVs. Grandcanonical moves are neces(3)

sary to design aordable simulations but intro-


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Langmuir

Figure 3:

Quantifying adhesion kinetics with DLS. A) Autocorrelation functions acquired with

DLS for a system with dierent times yellow

t=490 s,

t

RNP:LUV = 70

and

ρR−LUV =17.5 ×

at the mixing LUVs and NPs (dark blue

orange

t=2220 s,

red

t=7200 s,

pink

10

-3

-2

nm . The ACF was collected after

t=0 s,

t=18000 s

light blue

t=120 s,

green

t=57600 s.)

and purple

t=290 s,

B) Fraction

fadh , is obtained by tting the data from panel A with Eq. 6, extracting the two Afree (t) and Aadh (t), and using Eq. 7. The color code is as in panel A. The fraction adhering nanoparticles, fadh , for all the perfomed experiments are reported in SI, section 9, Fig

of bound NPs, amplitudes of

S9.1. Notice that only a subset of the collected ACFs are reported in panel A.

duce a bias in the diusion time required by

to longer decay times of the scattering auto-

nanoparticles to reach the membrane, making

correlation function (ACF), as demonstrated in

it signicantly faster than in reality.

Fig. 3A, which can thus be monitored to gain information on nanoparticle adhesion. For any measurement, we can identify two distinct NP

Results and Discussion

populations: free nanoparticles and nanoparticles bound to the LUVs. Over time, the popula-

We use DLS to investigate kinetic aspects of

tion of free NPs progressively shrinks, while the

the interaction between NPs and LUVs deco-

bound population grows.

rated by DNA ligands and receptors (Fig. 1)

sured ACF can be decomposed into the sum

as a function of receptor surface density on the LUVs, ratio,

of two exponentials, one describing the free

ρR−LUV , and NP:LUV stoichiometric

RNP:LUV ,

in Table 1.

Therefore the mea-

NPs population and the second associated to

spanning the ranges described

membrane-bound NPs

In DLS experiments, the signal

α 2τ/τ free )

is dominated by the strongly scattering NPs,

ACF(t, τ ) = Afree (t)e−(

−2τ/τcomp

+Aadh (t)e

while owing to the similarity between the re-

(6)

fractive index of the vesicles and that of the

where

surrounding medium, the scattering contribu-

of the experiment, and

tion of the LUVs is comparatively negligible.

time over which each scattering-intensity cor-

(see SI, section 5).

relation is calculated. The timescales

When an individual NP

adheres to a liposome, whose size is approxi-

τ comp

t

is the time elapsed from the beginning

τ

indicates the delay

τ free

and

are related to the diusion coecient of

mately one order of magnitude larger, the dy-

the free nanoparticles and the NP-LUV com-

namic diusion coecient of the nanoparticle is

plexes, respectively, while

drastically reduced. After NPs are exposed to

the polydispersity of the NP population.

LUVs, this eect produces a progressive shift

that using a stretched exponential to model the


α < 1

follows from

77

Note

,

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 19

Figure 4: Adhesion kinetics depends on receptor density and nanoparticle concentration. A) Adhe-

τadh for three dierent NP:LUV concentration ratios RNP:LUV as a function of the receptor surface density ρR−LUV . B) The same data as in panel A, shown as a function of the average number of receptors per NP in the system NR−NP . The vertical dashed line marks the maximum number max of bonds a nanoparticle can form due to geometrical limitations Nbonds = 25 (see SI, section 7.1). sim sim C) Simulated adhesion times τadh , using τadh = −∆t/log(1 − f (∆t)), where f (∆t) is the fraction of nanoparticles adsorbed at time ∆t, with ∆t small enough to allow tting data with an exponential sion times

function.

The predictions of panel A are bigger than the values of panel B because the average

time between consecutive NP-LUV encounter is smaller in simulations. Nevertheless, the similarity between the two gures conrms that the reaction kinetics is limiting adsorption.

contribution of the complexes did not improve

7, 70,

the ts, and the stretching factor was always

exponential trend hints at a hindered adsorp-

found to be comparable to 1. In Eq. 6, the am-

tion dynamics in which the rate of NP ad-

plitudes

Afree (t)

and

Aadh (t)

are directly pro-

and 290, respectively.

The stretched-

hesion decreases over time.

Similar trends

portional to the number of free and adhering

have been ascribed to a number of processes

nanoparticles, which allows us to extract the

including depletion of adsorbing agents,

fraction of bound nanoparticles as

steric hindrance and irreversible adsorption.

Aadh (t) . fadh (t) = [Aadh (t) + Afree (t)]

The progressive decrease of

RNP:LUV

(7)

parameters

and kept constant.

describe intrinsic char-

The decay time

τ comp

de-

the

evolution over the course of an experiment is (8)

analysed in a dedicated section below.

adhesion ss timescales, and the plateau value fadh represents the steady state fraction of adsorbed nanoparticles.

α

scribes the diusion of NP-LUV complexes and

γ t ss fadh (t) = fadh 1 − exp − , τadh quanties

and

and can therefore be independently determined

increases with time as a

stretched exponential

τadh

τ free

acteristics of NPs that do not change over time,

vated and discussed in SI, section 6. Figure 3B

where

we observed indicates that the specic

nanoparticles increases. Note that in Eq. 6 the

This assumption is moti-

fadh

with increasing

comes more severe as the number of available

and membrane-bound NPs have the same scat-

demonstrates how

80

factor limiting adsorption in our system be-

Eq. 7 is valid under the assumption that free tering eciency.

γ

78,79

typical

Reaction-limited

γ

in the range

of 0.9-1, 0.6-0.7 and 0.4-0.5 for

RNP:LUV =

ad-

hesion and receptor depletion

Optimal tting is obtained by

restricting the stretching factor

nanoparticle

Figure

4A

demonstrates

the adhesion time


τadh

the

dependence

of

on the NP:LUV rela-

Page 9 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Figure 5:

Simulations highlight the eect of kinetic rates on particle adhesion.

A) Number of 0,∗ adhering nanoparticles nadh as a function of time for dierent reaction rates parametrized by kon 0,∗ (see sections on Simulation Algorithm). When kon < ∞ the system fails to reach the equilibrium 0,∗ state corresponding to the steady value of the kon = ∞ trajectory. Receptors surface density was −3 −2 set to ρR−LUV =4.2×10 nm and the NP:LUV concentration ratio to RNP:LUV = 70. B) Average number of bonds

nbonds

formed by each adhering nanoparticle for the simulated trajectories of panel

A.

RNP:LUV

and the sur-

netic trends can be proposed by comparing the

face density of DNA receptors on the LUVs

timescales of the individual processes leading to

ρR−LUV .

observed to

NP adhesion. Adsorption is initiated by a colli-

If the lat-

sion between a NP and a LUV, and the average

tive concentration ratio For xed

ρR−LUV , τadh is RNP:LUV .

increase with increasing

τadh decreases monotonically with increasing ρR−LUV . The samples with the lowest RNP:LUV , where τadh remains con-

time

stant, constitute an exception.

These trends

ing each collision, the NP-LUV spend a much

can

studying

of

shorter time within their interaction range, τcoll ≈ `2 /D = 0.3 µs. While a NP is in contact

receptors per NP, obtained by dividing the av-

with a LUV, the time required for the forma-

ter is kept xed,

be

better

dependence of erage number on each LUV

rationalized

τadh

by

NR−LUV of receptors by RNP:LUV (Table 1).

and 450 ms (see SI, section 8.1). However, dur-

τbond . 200 ms

anchored

tion of a ligand-receptor bond is

The data

(see Eq.3 and SI, section 8.2), indicating that

in Fig. 4B show indeed a smooth trend, and clearly highlight two distinct regimes:

τadh

several thousands of unsuccessful collisions (natt

is

NR−NP , while increasNR−NP decreases below a

low and constant for high ing monotonically as

between two of these events, at the

beginning of the experiment, ranges between 80

the

NR−NP

on the number

τdiff

≈ τbond /τcoll )

natt

are on average required for

a NP to dock on a LUV. Assuming a collision −1 rate of the order of ∼ τdiff , τdiff < 0.5 s, the typical timescale at which particles start adsorbing 3 can then be estimated using natt · τdiff ≈ 10 4 10 s which is compatible with Fig. 4A. After

threshold. The switching point coincides with

NR−NP ∼ 20 − 50, which matches well the maximum number of ligand-receptor bonds that a single NP can form due to geometrical limitamax tions, estimated as Nbonds ∼ 25 (see SI, section 7.1).

the rst bond is established, preventing the LUV and NP from diusing apart, more connections are quickly formed as mobile receptors are converging towards the adhesion zone. Re-

A qualitative explanation of the observed ki-

ceptor accumulation occurs on the timescale


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 19

Figure 6: Receptor availability inuences steady state conguration. A) Steady state fraction of ss adhering NPs fadh as estimated from experiments (full symbols) and simulations (empty symbols) as a function of RNP:LUV and ρR−LUV . B) Experimental estimate of the steady-state average number ss of bonds per adhering NP nbonds .

τbond natt · τdiff ,

so that before another NP

trends in Fig. 4A supporting the interpretation

has a chance to bind with the LUV, the rst

that the reaction kinetics is the factor limit-

has already maximised the number of bonds, max capturing Nbonds receptors. If NR−NP exceeds max , the process continues until all nanoparNbonds max ticles have formed Nbonds connections, leaving

ing NP adsorption in experiments.

free receptors on the LUVs. In turn, for lower

that, as compared to experiments, drastically

NR−NP ,

reduces the time between successive NP-LUV

in simulations the typical adhesion timescales are much faster than the experimental ones. This is due to the simulation scheme employed

the system enters a regime where the

τdiff , while properly reproducing reτbond , and the time taken by NPs

initially binding NPs quickly deplete receptors,

encounters,

drastically reducing their availability.

action times,

tor scarcity causes

τbond

Note that

Recep-

to increase, reducing

to diuse away from the interacting region,

the chances of a successful docking and sub-

τcoll .

stantially slowing down adhesion.

In our ex-

used above to justify the timescale of Fig. 4A,

periments, receptor depletion is expected to be

explains the similarity between simulation and

eectively irreversible due to the strong anity

experimental proles that only dier by a time

between DNA ligands and receptors resulting i 6 in a bond breakup rate koff ∼ 10 exp (β∆G0 ) s−1 (see Eq. 3). The initial transient, therefore,

given by the ratio between the experimental

drives the system out of equilibrium. At high

tion rates on adsorption kinetics, in Fig. 5A we

RNP:LUV = 290

visualize the time evolution of the number of

steric hindrance can further limit nanoparti-

adsorbed nanoparticles, calculated using dier-

cle adsorption at later stages given that at full NP (SI, section 7.2).

ent reaction rates. The latter are controlled by 0,∗ changing kon while keeping the ligand-receptor binding free energy, and therefore the ther-

These qualitative arguments can be veried by

modynamics of the system, constant.

comparing experimental data with simulated

sequently, the ligand-receptor o rate is also i 0,∗ rescaled as koff ∝ kon (Eq. 3). 0,∗ i For kon , koff = ∞, the system experiences no

NP:LUV concentration ratio

packing a LUV can host a maximum of

and simulated

τdiff .

To better clarify the role played by the reac-

∼ 150

trends. Figure 4C reports adhesion timescales sim τadh , estimated by tting the early stages of adsorption as detailed in the caption of Fig. 4.

This argument, along with the reasoning

Con-

kinetic hindrance from ligand-receptor reac-

Simulation results reproduce experimental

tions, and can therefore reach thermodynamic


Page 11 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

equilibrium.

In these conditions, mobile re-

tions, but the latter appear to underestimate ss in regimes of low NR−NP . We ascribe fadh this deviation to the fact that, in simulations, ss the average number of bonds Nbonds formed by each adhering NP at steady state is always max maximised, and equal to Nbonds = 25 (Fig. 5).

ceptors are observed to redistribute instantaneously among available NPs, progressively decreasing the average number of bonds per NP to stabilize a greater number of adsorbed nanoparticles, as displayed in Fig. 5B. In turn, Fig.

5A shows that for nite reaction rates

This is not always the case in experiments, ss as shown in Fig. 6B. Nbonds is estimated experimentally by assuming that i) all receptors

the asymptotic number of adsorbed nanoparticles is much smaller than the one observed 0,∗ 0,∗ with kon = ∞, implying that for kon < ∞ the system fails to reach equilibrium. Indeed, 0,∗ even for the largest (nite) kon value we could i −9 test, koff becomes negligibly small (≈ 10 −1 s ) owing to the strong ligand-receptor an-

are bound, and

ii)

receptors are evenly parti-

depletion eect hypothesized above, where all

tioned among adsorbed NPs. We observe that ss the experimental Nbonds is aways smaller than, max or comparable to Nbonds . More specically ss Nbonds increases as a function of ρR−LUV for max for the highRNP:LUV = 70, approaching Nbonds est tested receptor density, while remaining rel-

adhering NPs quickly and irreversibly max-

atively small and constant for

imise the number of formed bonds, which in max simulations is set to Nbonds = 25 (Fig. 5B). To enter a regime where adsorption is not lim-

Note that condition

ited by DNA denaturation, one would need 0,∗ kon & exp [−β∆G0 ]. Unfortunately, such val0,∗ ues of kon are not computationally aordable. 0,∗ For kon = ∞ bond formation occurs instantly

Fig. 6. There are a number of eects that can max ss result in Nbonds < Nbonds , not accounted for in simulations. First, in Fig. 6B we assume that

upon NP-LUV collision, implying that initial

adsorbed NPs, while it should not be excluded

aggregation is diusion limited and occurs over

that those adhering at later stages may form

ity. Bond irreversibility results in the receptor-

typical timescales

RNP:LUV = 7,

receptors are evenly distributed among all the

Figure 5 shows that, at 5 0,∗ early times, the curve calculated for kon = 10 follows the same trend, indicating that in this

τbonds

i) cannot be veried for max NR−NP > Nbonds , thus the

estimates for these samples are not included in

τdiff .

regime the bond formation time

where

RNP:LUV = 290.

less bonds than earlier nanoparticles. This nonideal behavior could emerge, for instance, as a consequence of nonselective (e.g.

steric) in-

is still

teractions between ligands and receptors, that

signicantly smaller than the collision time 0,∗ τcoll . The same is no longer true if kon ≤ 10−1 , for which the onset of NP adhesion is signif-

could signicantly slow down bond formation and receptor sequestration by increasing

τbond ,

permanent NP-LUV bond upon collision is pro-

allowing more nanoparticles to bind the LUV. ss max Additionally, Nbonds < Nbonds may imply a degree of bond reversibility enabling recep-

gressively reduced.

tor redistribution.

icantly delayed as the likelihood of forming a

This could be explained

by an underestimation of the o rates, which in crowded environments may deviate sensibly from the theoretical predictions obtained

Steady state

in diluted conditions (Eq.

From the time evolution of the fraction of

fadh (t)

once again that for

LUV-adhering

3).

81

We point out

RNP:LUV = 290

there is

NPs we can extract the ss asymptotic value fadh , corresponding to fraction of adhering nanoparticles at steady state ss (Eq. 8). In Fig. 6A, the experimental fadh is show as a function of RNP:LUV and ρR−LUV

not enough room on the LUVs to accomodate

(full symbols), and compared to simulations re-

asymptotic fractions experimentally observed

sults (empty symbols). We observe qualitative

for

all available nanoparticles, regardless on receptor availability.

The geometrical limit to NP

adsorption per LUV is to

fadh ∼ 0.5.


corresponding

This value is similar to the

RNP:LUV = 290

agreement between experiments and simula-

∼ 150,

in Fig. 6A.

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 19

Figure 7: Diusion time of NP-LUV complexes suggest LUV clustering. Decay time

τcomp , describ-

ing the diusivity of NP-LUV complexes shown as a function of the receptor surface density on LUVs

ρR−LUV

for dierent NP:LUV concentration ratios

RNP:LUV = 7,

70 and 290.

The dashed

lines represent the characteristic diusion time of a 180 nm diameter liposome. The grey areas mark the range of adhesion times

LUVs

aggregation

τadh

measured for the dierent

driven

by

NP

ρR−LUV

(Fig. 4).

eectively passivate the LUV surface against NP bridging.

bridging

Note that LUV aggregation is

not expected to aect our ability to quantify Further information on the evolution of the sys-

the fraction of bound NPs, used to monitor ad-

tem can be gathered by monitoring the time

sorption kinetics: all NPs, whether adhering to

dependence of the diusion timescale

τcomp

of

an individual LUV or bridging two of them,

the NP-LUV complexes (equation 7). As shown

would still be considered as bound.

τcomp exhibits dierent trends deon ρR−LUV (shown in dierent colors)

in Fig. 7, pending

RNP:LUV (shown in dierent panels). cases τcomp remains roughly constant for and

In turn,

NP-mediated LUV bridging may have an eect on aggregation kinetics, as NPs can no longer

In all

be considered as "inactivated" after adhering

times

to a LUV, and would still be able to seques-

shorter than the typical adhesion timescale τadh ,

trate more receptors by forming LUV-NP-LUV

marked as a grey-shaded band spanning the ad-

bridges. However, in view of the faster diusion

ρR−LUV .

kinetics of individual NPs compared to that of

hesion timescales observed for dierent For

RNP:LUV = 7,

an increase of

τadh

above

NP-LUV complexes, the adhesion of free NPs to

the value expected for an individual LUV is

LUVs is expected to dominate over the forma-

observed at later times, suggesting the possi-

tion of aggregates, as long as free NPs remain

bility of NPs mediated LUV-LUV aggregation.

available.

The latter can occur because, after all NP are adsorbed, for

RNP:LUV = 7

the system still

Conclusions

features free receptors that can bind NPs on other LUVs, causing aggregation. Consistently, the same phenomenon is observed for samples

In summary, through a combination of dedi-

RNP:LUV = 70 and large receptor densities,

cated experiments and coarse-grained simula-

which as demonstrated in Fig. 6A also reach

tions, we investigated the kinetics of adhesion

a regime where NPs are depleted but recep-

of gold nanoparticles to articial lipid vesicles

tors are still available. As expected, in samples

mediated by near-irreversible ligand-receptor

with

with

RNP:LUV = 290,

bonds.

LUVs remain monodis-

DNA strands grafted onto the surface

perse at all times, as the large excess of NPs

of the nanoparticles and the liposomes play

is always sucient to deplete all receptors and

the role of both ligands and receptors.

This

model system is chosen to mimic interactions


Page 13 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

between nanoscale vectors and the surface of

modeling community. In particular, they high-

biological cells, a key process in several biolog-

light the importance of explicitly considering

ical and medical scenarios, including cell inva-

reaction rates when designing numerical simu-

sion by viruses and parasites and cell targeting

lations.

by means of synthetic nanomedical vectors. The rate of nanoparticle-vesicle adhesion is

Acknowledgement

studied over a broad range of biologically and technologically relevant conditions,

and GB acknowledge support from the Wiener-

by vary-

Anspach foundation. LDM acknowledges sup-

ing the surface density of available membrane-

port from the Leverhulme Trust and the Isaac

anchored receptors and the bulk concentration

Newton Trust through an Early Career Fel-

of nanoparticles. In both experiments and sim-

lowship (ECF-2015-494),

ulations, we nd that adhesion rates are heavily

ship (UF160152).

in excess compared to available receptors. Ob-

CAPITALS (EP/J017566/1). PKJ and BMM

the timescales of individual kinetic processes

are supported by the Fonds de la Recherche

at play, including the frequency and duration

Scientique de Belgique - FNRS under grant ◦ n MIS F.4534.17. Computational resources

of nanoparticle-vesicle collisions and the typical time required to form a ligand-receptor

have been provided by the Consortium des

We determine that the observed slow

Équipements de Calcul Intensif (CECI), funded

kinetics arises from receptor depletion, as difmembrane-anchored

receptors

by the Fonds de la Recherche Scientique de ◦ Belgique - FNRS under grant n 2.5020.11. RL

quickly

segregate within the nanoparticle-vesicle adhe-

acknowledge Simon Croegaert for the produc-

sion region upon initial docking of individual

tion of AuNPs and TEM images, Jonathan

nanoparticles. Nanoparticles adhering at early

Goole from Pharmaceutics and Biopharmaceu-

aggregation stages therefore sequestrate a sig-

tics group (Université libre de Bruxelles) to give

nicant proportion of the available receptors,

her access to Malvern Zetasizer NanoZS and

reducing the rate of bond formation for late-

Guy Vandenbussche from Structure and Func-

coming nanoparticles and slowing down their adhesion.

tion of Biological Membranes group (Université

The near-irreversibility of ligand-

libre de Bruxelles) for letting her use Ultra Cen-

receptor connections hinder bond redistribution,

PC and LDM acknowledge

support from the EPSRC Programme Grant

served trends are rationalized by comparing

fusive

and the Royal So-

ciety through a University Research Fellow-

suppressed in regimes where nanoparticles are

bond.

RL, BMM, PC, LDM

trifuge.

leading to kinetically trapped congu-

rations in which the number of membrane-

Supporting Information Avail-

adsorbed nanoparticles is not maximised. Our observations have direct application in the

able

design of medical nanovectors targeting cell membrane receptors: for a xed receptor sur-

The Supporting Information is available free of

face density, the adsorption rate of nanoscale

charge.

probes can be maximised by improving bond reversibility or by reducing the number of con-

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