Document [original]

Article

Particle Diffusivity and Free-Energy Profiles in

Hydrogels from Time-Resolved Penetration Data

Amanuel Wolde-Kidan,

Anna Herrmann,

Albert Prause,

Michael Gradzielski,

Rainer Haag,

Stephan Block,

and Roland R. Netz

Fachbereich Physik and

Institut f€

ur Chemie und Biochemie, Freie Universit€

at Berlin, Berlin, Germany; and

Institut f€

ur Chemie, Technische

Universit€

at Berlin, Berlin, Germany

ABSTRACT A combined experimental and theoretical method to simultaneously determine diffusivity and free-energy profiles

of particles that penetrate into inhomogeneous hydrogel systems is presented. As the only input, arbitrarily normalized concen-

tration profiles from fluorescence intensity data of labeled tracer particles for different penetration times are needed. The method

is applied to dextran molecules of varying size that penetrate into hydrogels of polyethylene-glycol chains with different lengths

that are covalently cross-linked by hyperbranched polyglycerol hubs. Extracted dextran bulk diffusivities agree well with fluores-

cence correlation spectroscopy data obtained separately. Empirical scaling laws for dextran diffusivities and free energies inside

the hydrogel are identified as a function of the dextran mass. An elastic free-volume model that includes dextran as well as poly-

ethylene-glycol linker flexibility quantitively describes the repulsive dextran-hydrogel interaction free energy, which is of steric

origin, and furthermore suggests that the hydrogel mesh-size distribution is rather broad and particle penetration is dominated

by large hydrogel pores. Particle penetration into hydrogels for steric particle-hydrogel interactions is thus suggested to be

governed by an elastic size-filtering mechanism that involves the tail of the hydrogel pore-size distribution.

INTRODUCTION

The penetration of particles into hydrogels is relevant for

technological applications (1,2), drug delivery (3), and bio-

logical systems such as biofilms (4), the extracellular ma-

trix (5), and mucus (6). Mucus, which is the most

common biological hydrogel, lines the epithelial tissues of

different organs, such as the respiratory, gastrointestinal,

and urogenital tracts. Mucus is mainly composed of mucins,

which are glycoproteins of varying length that absorb large

amounts of water and thereby lend mucus its hydrogel na-

ture, and additional components such as enzymes and ions

(7). Mucins are relevant in the cell signaling context and

presumably also play a role in the development of cancer

(8). But primarily, mucus is a penetration barrier against

pathogens, e.g., virions or bacteria, whereas it allows the

permeation of many nonpathogens, e.g., nutrients, that are

absorbed through the mucosa of the small intestine (9).

Studies have suggested that based on the type of mucus,

the combination of different mechanisms gives rise to the

protective barrier function (10,11), in addition to the advec-

tive transport of pathogens through mucus shedding

or clearance (12,13), which is not considered here. One

typically distinguishes steric size-filtering mechanisms

from interaction-filtering mechanisms (6,14); the latter

Submitted September 29, 2020, and accepted for publication December 23,

2020.

*Correspondence: rnetz@physik.fu-berlin.de

Amanuel Wolde-Kidan and Anna Herrmann contributed equally to this

work.

Editor: Jennifer Curtis.

SIGNIFICANCE The barrier function of mucus and other biological hydrogels against particles and pathogens depends

on their diffusivity and free-energy profiles. We introduce a method that allows for simultaneous extraction of these

quantities from non-normalized concentration profiles measured in penetration experiments. We apply our method to

fluorescently labeled dextran polymers diffusing into polyethylene-glycol-based hydrogels and explain the results by an

elastic free-volume model. We conclude that the penetration is governed by the large pores of the broad pore-size

distribution, which is most likely a general characteristic of hydrogels. Our method is generally applicable to various kinds of

labeled particles, including bacteria and virions, and can be used to help unravel the mechanisms behind mucus barrier

function.

Biophysical Journal 120, 463–475, February 2, 2021 463

https://doi.org/10.1016/j.bpj.2020.12.020

Ó2021 Biophysical Society.

This is an open access article under the CC BY license (http://

creativecommons.org/licenses/by/4.0/).

presumably play a major role in the defense of organisms

against pathogens because they allow for precise regulation

of the passage of wanted and unwanted particles and mole-

cules (15,16). Recent studies demonstrated that attractive

electrostatic interactions reduce the particle diffusivity in-

side hydrogels substantially and much more than repulsive

electrostatic interactions (17,18) and that salt concentration

and the distribution of charges and pore sizes are important

parameters that influence the permeation properties of

charged hydrogels (19,20).

Particle penetration into mucus and biofilms has been

studied by single-particle tracking techniques (21,22)as

well as by methods in which a diffusor ensemble is observed

(15,16,23,24). On short timescales, transient particle bind-

ing to the hydrogel (16–18) is important and leads to anom-

alous particle diffusion (25). On spatial length scales larger

than the hydrogel mesh size and on timescales larger than

typical binding escape times, particle diffusion is in a con-

tinuum description determined by the free-energy and diffu-

sivity profiles across an inhomogeneous hydrogel system. In

this framework, particle binding is effectively taken into ac-

count via a reduction of the diffusivity and a lowering of the

free energy. If the free-energy and diffusivity profiles are

known, particle penetration can be quantitatively predicted,

provided the particle concentration is low and the particles

do not modify the hydrogel properties in an irreversible

manner. In this context, it should be noted that both profiles

depend on the interactions between particle and hydrogel

and therefore are different for each distinct hydrogel-parti-

cle pair. Because of method restrictions, experiments pri-

marily focus on determining either the particle diffusivity

inside the hydrogel (6,10,21) or on the partitioning between

hydrogel and the bulk solution (26), from which the free en-

ergy inside the hydrogel (relative to the bulk solution) can

be determined. However, for prediction of the penetration

or permeation speed of particles into the hydrogel, both

the diffusivity and the free energy in the hydrogel are

needed.

In this work, we study synthetic hydrogels that consist of

polyethylene-glycol (PEG) linkers of different molecular

masses that are permanently cross-linked by hyperbranched

polyglycerol (hPG) hubs (2). Such synthetic hydrogels can

be regarded as simple models for mucus because they display

size-dependent particle permeabilities (14,27) similar to

mucus. As diffusing particles, we employ fluorescently

labeled dextran molecules of varying sizes. When using

confocal laser-scanning fluorescence microscopy to investi-

gate particle penetration into hydrogels, the sample can be

oriented such that the hydrogel-bulk interface is either paral-

lel (16) or perpendicular (28) to the optical axis, which makes

no significant difference from a scanning perspective. How-

ever, for laterally extended samples like cell cultures that

grow on a substrate, the parallel alignment causes the light

path to span substantially larger distances, making this setup

more prone to distortions in the imaging process. A perpen-

dicular alignment, as employed in this work and sketched in

Fig. 1, is therefore preferable for biological samples (28) and

is also compatible with future extensions of such penetration

assays to mucus-producing cell cultures.

We investigate the filtering function of hydrogels by theo-

retical analysis of time-resolved concentration profiles of

the labeled dextran molecules as they penetrate into the hy-

drogel. The employed numerical method allows for simulta-

neous extraction of free-energy and diffusivity profiles from

relative concentration profiles at different times and is a sig-

nificant extension of earlier methods (29–31) because it

does not require absolute concentration profiles but works

with relative, i.e., arbitrarily normalized, concentrations.

This is a crucial advantage because often fluorescence inten-

sity profiles are subject to significant perturbation due to,

e.g., laser light intensity fluctuations or fluorescence dye

bleaching over the course of the experiment, and using rela-

tive concentrations makes the often-difficult conversion of

measured intensity data into absolute particle concentra-

tions obsolete. The analysis framework we introduce here

can thus be used for a wide range of experimental setups

to simultaneously extract free-energy and diffusivity pro-

files from a variety of different biological systems. As a

check on the robustness of the method, the extracted dextran

bulk diffusivities are shown to agree well with fluorescence

correlation spectroscopy (FCS) data that are obtained sepa-

rately. The obtained particle free energies and diffusivities

inside the hydrogel are shown to obey empirical scaling

laws as a function of the dextran mass. The dextran free en-

ergy inside the hydrogel is described by a free-volume

model based on repulsive steric interactions between the

dextran molecules and the hydrogel linkers, which includes

dextran as well as hydrogel linker flexibility. This model

constitutes a modified size-filtering mechanism for repul-

sive particle-hydrogel interactions, according to which par-

ticle penetration into hydrogel pores is assisted by the elastic

widening of pores and the elastic shrinking of dextran mol-

ecules and matches the extracted particle free energies in the

hydrogel quantitatively. The model furthermore suggests

that the hydrogel mesh-size distribution is rather broad

and that particle penetration is dominated by the fraction

of large pores in the hydrogel.

MATERIALS AND METHODS

Hydrogel preparation

The hydrogel is formed by cross-linking end-functionalized polyethylene-

glycol-bicyclo[6.1.0]non-4-yne (PEG-BCN) linkers with hyperbranched

polyglycerol azide (hPG-N

) hubs via strain-promoted azide-alkyne cyclo-

addition. The two macromonomers PEG-BCN and hPG-N

are synthesized

as previously described (2,32). The ‘‘click’’ reaction of binding the PEG-

BCN linkers to the hPG-N

hubs works in water, at room temperature,

without the addition of a catalyst or external activation like heat or ultravi-

olet radiation and without the formation of byproducts. Two different sizes

of PEG-BCN linkers are employed, having a molecular weight of either

PEG

¼6orM

PEG

¼10 kDa (for details about the mass distributions,

Wolde-Kidan et al.

464 Biophysical Journal 120, 463–475, February 2, 2021

see Supporting Materials and Methods, Section S1), the hydrogels are de-

noted as hPG-G6 and hPG-G10, respectively. The number ratio of the

PEG-BCN linkers to the hPG-N

hubs (M

hPG

¼3 kDa, 20% azide) is

kept constant at 3:1 for both hPG-G6 and hPG-G10. This ratio can ideally

lead to a cubic lattice structure if each hPG-hub exactly binds to six PEG

linkers. The chemical structure of the hPG-N

hubs, however, allows on

average for eight binding sites, making the hydrogel presumably quite

disordered.

The two components of the hydrogel are stored as aqueous stock solu-

tions at concentrations of 8.5 wt% (6 kDa PEG-BCN), 8.4 wt% (10 kDa

PEG-BCN), and 5 wt% (hPG-N

). After very long storage times of the

stock solutions of about 1 year, the cross-linking click reaction of PEG

linkers and hPG hubs starts to become impaired, which is why storage

times are kept short. To minimize aging effects of the hydrogels, hydrogel

formation is always initiated shortly before the start of the experiments

by mixing the components according to Table 1. The resulting gel solution

is thoroughly vortexed before being placed as 1 mL drops on the glass

substrate. Both hydrogel solutions are adjusted to have the same mass

concentration. However, after drying and reswelling on the glass

substrate, volumes of the formed hydrogels are different and measured

as VhPGG6

tot ¼0.42 50.03 mLandVhPGG10

tot ¼0.31 50.04 mL

for hPG-G6 and hPG-G10, respectively (see Fig. S1 in Supporting

Materials and Methods, Section S2). This results in a final hydrogel

concentration of 9 wt% (90 mg/mL) for hPG-G6 and12wt%

(120 mg/mL) for hPG-G10.

Estimate of mean hydrogel mesh size

Assuming an idealized cubic hydrogel network structure, the mean mesh

size can be easily estimated. The length of a cubic unit cell l

0,ideal

follows

from the total gel volume V

tot

and the total number of hPG hubs ntot

hPG in

mol as

l0;ideal ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Vtot

ntot

hPGNA

s;(1)

where N

is the Avogadro constant. The total volumes for the rehydrated

gels are VhPGG6

tot ¼0.42 mL and VhPGG10

tot ¼0.31 mL as mentioned above.

The total number of hPG hubs is given as ntot

hPG ¼nhPG Vapp=Vsol

gel, with the

values from Table 1 for the respective gel and in which we account for the

fact that only V

app

¼1mL of the total gel solution Vsol

gel is applied onto the gel

substrate. This results in rough estimates for the mesh size of

lhPGG6

0;ideal ¼7.1 nm and lhPGG10

0;ideal ¼7.5 nm, which shows that even though

PEG linkers of significantly different masses were used, the mesh sizes

of the two gels differ only slightly. In deriving Eq. 1, one assumes an ideal

hydrogel pore connectivity that corresponds to a perfect cubic lattice. There

is no reason why the hydrogel should consist of a perfect cubic lattice; on

the contrary, entropy favors a disordered network topology. For cubic pores

with lower connectivity, Fig. 2 illustrates how the pore size l

can increase

for a fixed PEG end-to-end distance R

PEG

. Thus, except for the case of an

ideal cubic lattice, the pore size l

will be larger than the estimate of Eq. 1,

as indeed suggested by our elastic free-volume model.

Dextran preparation

Dextrans conjugated with the dye fluorescein isothiocyanate (FITC) are ob-

tained from Sigma-Aldrich as d4-FITC, d10-FITC, d20-FITC, d40-FITC,

and d70-FITC, the number stating the molecular weight in kDa of the com-

mercial product. To remove unbound FITC from the dextran solutions, all

batches are subjected to a desalting PD-10 column, which eliminates low-

molecular weight compounds such as free FITC dye. This step is done

according to the manufacturer’s recommendations, and the column is equil-

ibrated using phosphate-buffered saline (PBS). Afterwards, the molecular

weight distribution of all dextrans is determined by gel permeation chroma-

tography (see Supporting Materials and Methods, Section S1).

Penetration assay of FITC-labeled dextrans

After preparation of the hydrogel solutions and purification of the dextrans

(see above), penetration assays are performed with five different dextran so-

lutions and two different gels. For these assays, coverslips (Menzel #1;

ABFIGURE 1 (A) Schematic drawing of the experi-

mental setup. Concentration profiles of fluorescently

labeled dextran molecules (green) are measured as

they penetrate from the bulk solution (blue) into

the hydrogel (black). The origin of the zaxis is posi-

tioned such that experimentally measured profiles

range from z¼0toz¼z

bot

. The hydrogel-bulk so-

lution interface is located at z¼z

int

. In the range

from z¼z

top

to z¼0, only numerically deter-

mined concentration profiles are available. (B)

Exemplary experimental concentration profiles for

two different penetration times for M

dex

¼4 kDa

dextran diffusing into the hPG-G10 hydrogel are

given; positions of the hydrogel-bulk solution inter-

face z

int

and the hydrogel-glass bottom interface z

bot

are indicated. To see this figure in color, go online.

TABLE 1 Composition of the Hydrogels Used in this Study

PEG

Vsol

PEG

hPG

Vsol

hPG

VH2OVsol

gel V

app

hPG-G6 142 nmol 10 mL 47 nmol 2.8 mL 13.0 mL 25.8 mL1mL38mg 7.3 nmol

hPG-G10 84 nmol 10 mL 28 nmol 1.7 mL 12.7 mL 24.4 mL1mL38mg 4.6 nmol

Here, Vsol

PEG and Vsol

hPG denote the volumes of the stock solutions, VH2Ois the volume of purified water added to the resulting gel solutions, and n

PEG

and n

hPG

denote the amount of PEG linkers and hPG hubs in the gel solutions. From the total resulting volume of the gel solutions Vsol

gel, only V

app

¼1mL was placed as

a gel spot on the glass substrate, leading to the combined applied amount n

app

and the combined applied mass m

app

of PEG linkers and hPG hubs.

Solution is of 8.5 wt% for 6 kDa PEG and 8.4 wt% for 10 kDa PEG.

hPG solution is of 5 wt%.

Diffusivity and Free Energy of Hydrogels

Biophysical Journal 120, 463–475, February 2, 2021 465

VWR, Darmstadt, Germany) with a diameter of 25 mm and a thickness of

0.13–0.16 mm are thoroughly washed with water and absolute ethanol and

subsequently dried under a stream of nitrogen. For every experiment, 1 mL

of the respective hydrogel solution is placed on the center of the coverslip.

The substrates with the applied gel spots are kept in a humid environment

overnight, allowing hydrogel formation to be completed before the hydro-

gel spots are left to dry for 30 min at ambient conditions. Permeation exper-

iments are performed within 1 day after hydrogel formation. To start a

permeation experiment, a home-made polydimethylsiloxane stamp (1 

1 cm) prepared with a cylindrical cavity in the middle (5 mm diameter)

is placed on the coverslip, so that the dried hydrogel is located in the middle

of the stamp’s cavity. The polydimethylsiloxane surrounding the dried hy-

drogel allows for the addition of solutions such as buffer or dextran. Before

the measurement, 30 mL of PBS buffer are added to reswell the hydrogel for

30 min, which typically creates hydrogel volumes of semispheroid shape

with a base radius of 1050 mm and heights of 150 mm for hPG-G10

and 210 mm for hPG-G6 (see Supporting Materials and Methods, Section

S2). Afterwards, the coverslip is mounted on a Leica SP8 confocal laser-

scanning microscope (CLSM; Leica, Wetzlar, Germany) and imaged using

a20objective (0.75 HC PL APO water immersion objective with correc-

tion ring). In a first step, the hydrogel is visually identified by imaging the

sample with a 488 nm laser and collecting the transmitted light using the

transmission photomultiplier tube of the CLSM, allowing us to place the

optical axis of the CLSM in the center of the hydrogel and to place the focal

plane 30 mm below the glass-hydrogel interface. After aligning the sample

like this, the PBS buffer is removed from the cavity and replaced by 35 mL

of the FITC-dextran solution (0.07 mg/mL for all dextrans). This fixes the

total length from the bottom of the glass dish at z¼z

bot

to the air-water

interface at z¼z

top

, where z¼0 corresponds to the end of the measurement

region (see Fig. 1 A). The total length of the solution is thus z

tot

¼z

top

bot

¼1780 mm. The individual contributions to z

tot

vary because of

different gel thicknesses changing the extent of the measured region,

ranging from z¼0toz¼z

bot

(cf. also Fig. 1 A).

About 10 s after the application of the dextran solution, the spatial distri-

bution of the FITC-based fluorescence intensity is measured using a z-stack

that starts 30 mm below and ends 410 mm above the glass-hydrogel interface

(with 10 mm increments). The recorded intensities are afterwards truncated

to probe the spatial FITC distribution within the hydrogel starting from the

glass bottom (located at z

bot

) and extending 100 mm into the bulk solution,

away from the gel-water interface located at z¼z

int

(cf. Fig. 1 A). In these

measurements, the sample is excited at l¼488 nm, and the emission is re-

corded between 500 and 550 nm using a photomultiplier tube. For the

dex

¼4 kDa to the M

dex

¼40 kDa dextrans, one z-stack is recorded every

Dt¼10 s, yielding time-resolved FITC distributions after the penetration of

the dextran molecules into the hydrogel network over time. For the M

dex

70 kDa dextrans, a period of Dt¼30 s is used instead to account for the

much smaller diffusion coefficient of the larger dextran molecules. The em-

ployed temporal resolutions can be easily estimated to be larger than time-

scales on which effects of anomalous diffusion are present; for diffusion

over length scales larger than the mesh size of the hydrogel, normal diffu-

sion is expected. An upper bound for the corresponding crossover timescale

can be estimated as t¼l2

0/D

gel

, where l

¼24 nm is an upper estimate for

the hydrogel mesh size and D

gel

¼0.15 mm

/s is the smallest obtained diffu-

sion constant in the hydrogel (see below for explicit results). The resulting

value of tz0.2 ms, beyond which normal diffusion is expected, is several

orders of magnitude lower than the experimental temporal resolution. Thus,

anomalous diffusion cannot be observed in the experimental data, and the

normal diffusion equation that is used to model the time-dependent exper-

imental concentration profiles should be valid.

For all dextran types, measurements are performed at least three times

with total measurement times of 30 min, with the exception of the

dex

¼70 kDa dextrans. Here, only one measurement is performed for

each gel, but with a longer recording time of 1h.

FCS of FITC-labeled dextrans

Reference diffusion coefficients for the FITC-labeled dextran molecules in

the bulk solution are obtained using FCS. The measurements are performed

on a Leica TCS SP5 II CLSM with an FCS setup from PicoQuant (Berlin,

Germany). The CLSM is equipped with an HCX PL APO 63/1.20 W

CORR CS water immersion objective. Samples are put on high-precision

cover glasses (18 18 mm, 170 55mm thick) and excited with the

488 nm Argon laser line. The fluorescent light is passed through a 50/50

beam splitter with a lower wavelength cutoff of l¼515 nm. Both channels

are detected separately with a single photon avalanche diode. Afterwards, a

pseudo-cross correlation is performed between both channels to eliminate

the influence of detector afterpulsing. Before a measurement, the optical

setup is calibrated with the water-soluble Alexa-Fluor 488 dye. The corre-

lated signal is fitted with two components and accounting for triplet states.

The first component is fixed to a freely diffusing FITC dye molecule for

which only the fraction is a fit parameter. The second component is set to

a log-normal distributed species. The component fractions and means of

distribution are fitted, and the width of distribution is taken from previously

performed gel permeation chromatography measurements (for details about

the fitting procedure, see Supporting Materials and Methods, Section S3).

The fitted diffusion times are used to calculate the diffusion coefficients

and hydrodynamic radii using the Stokes-Einstein relation.

Numerical model and discretization

Extending a previously introduced method (29–31), spatially resolved

diffusivity and free-energy profiles are estimated from experimentally

measured concentration profiles. Numerical profiles are computed by dis-

cretizing the entire experimental setup from the glass bottom of the sub-

strate to the air-water interface (z

bot

to z

top

in Fig. 1 A). In the regime

in which concentration profiles are measured (z¼0toz¼z

bot

), the

experimental resolution is used as the numerical discretization width

Dz¼10 mm. For the range without experimental data (z¼0toz¼z

top

in total, six bins are employed. Two of those bins are spaced with Dz¼

10 mm; for the other four bins, discretization spacings between Dz¼300

and 400 mm are used, depending on the z-length measured in the respective

experiment z

bot

. The z-dimension of the total system is the same for all ex-

periments and given as z

tot

¼z

top

þz

bot

¼1780 mm. The experimentally

measured region always extends from the glass bottom through the gel

and at least 100 mm into the bulk solution, away from the hydrogel-bulk

interface, which leads to values of z

bot

z300 mm, depending on the exact

thickness of the hydrogel in the respective measurement.

FIGURE 2 A cubic pore with lower connectivity to the right, containing

two PEG linkers per edge instead of one, leads to an effectively larger unit-

cell length l

at the same PEG end-to-end distance R

PEG

. Only for a perfect

cubic lattice to the left is the estimate of Eq. 1 valid and l

¼l

0,ideal

¼R

PEG

Wolde-Kidan et al.

466 Biophysical Journal 120, 463–475, February 2, 2021

The numerical optimization problem is given by the cost function, which

is defined as

s2D;F;~

f:¼1

NMP

j¼1P

i¼1cnum

itjfjcexp

itj2;

(2)

with Nthe total number of experimental profiles, Mthe total number of

experimental data points per concentration profile and s

(D,F,~

f) being

the mean squared deviation between the experimental and numerical pro-

files. The diffusivity profile D¼D(z), the free-energy landscape F¼

F(z), and the vector containing all scaling factors (see below for details)

f¼(f

,.,f

) are all optimized to find the minimal value of s

This nonlinear regression is performed using the trust region method imple-

mented in Python’s scipy package (33).

The numerical profiles

cnumtj¼cnum

1tj;.;cnum

itj;.;cnum

MtjT

are computed from the diffusivity and free-energy profiles as

cnumtj¼eWtj~

cinit;(3)

where the rate matrix W(D,F) is defined as

Wi;k¼DiþDk

2Dz2eFiFk

2kBT;with k¼i51

as explained previously (29). Numerical profiles at time t

depend on the

initial profile ~

cinit at t¼0, which is determined as explained below.

The numerically computed profiles are fitted to the rescaled experi-

mental profiles ~

cexpðtjÞat time t

>0. The scaling factors ~

fare obtained

simultaneously from the fitting procedure and correct drifts in

the experimentally measured fluorescence intensity profiles (see

Supporting Materials and Methods, Section S4). As a check, the numer-

ical model is compared to the analytical solution for a model with piece-

wise constant values of the diffusivity and free energy in the respective

regions. Results from the numerical model agree perfectly with those

from the analytical solution (see Supporting Materials and Methods,

Section S5).

Construction of the initial concentration profile

The initial profile~

cinit, used for the computation of all later profiles accord-

ing to Eq. 3, needs to cover the entire computational domain and is gener-

ated by extending the first experimentally measured profile~

cexp (t¼0) into

the bulk regime (from z¼0toz¼z

top

, cf. Fig. 1 A). We define t¼0 as the

time of the first measurement, which is performed 10 s after application of

the dextran solution onto the gel-loaded substrate. For the spatial extension

of the profile, a constant initial concentration is assumed in the bulk, the

value of which is taken as the experimentally measured value furthest

into the bulk c

:¼cexp

1(t¼0) at z¼0. This leads to the following expres-

sion used for the initial profile

cinit

i:¼c0;if ztop%zi%0

cexp

iðt¼0Þ;if 0<z

i%zbot

;(4)

which by construction is continuous at z¼0. The initial profiles used

for the fit procedure are shown in Fig. 3,Band Fas black lines. To obtain

concentration profiles in physical units, we set the first measured value

furthest into the bulk equal to the applied dextran concentration c

70 mg/L.

Free-energy and diffusivity profiles

Because the experimental system consists of two regions, namely the hy-

drogel and the bulk solution, and to reduce the number of parameters of

the numerical model to avoid overfitting, we employ sigmoidal profiles

for the diffusivity D(z) and free energy F(z), which transition continuously

from the value in the bulk solution to their values in the hydrogel. This

sigmoidal shape is modeled using the following expressions:

DðzÞ¼Dsol þDgel

2þDsol Dgel

2erfzzint

ﬃﬃﬃ

pdint ;

FðzÞ¼DFgel

2þDFgel

2erfzzint

ﬃﬃﬃ

pdint ;(5)

where erf(z):¼1/ ﬃﬃﬃ

pRz

zez02dz0is the error function. The fit parameters

int

and d

int

determine the transition position and width, respectively, and

are the same for the free-energy and diffusivity profiles. Because only

free-energy differences carry physical meaning, the free energy in the

bulk solution is set to zero so that F

sol

¼0. The values of the diffusivity

and free energy in the hydrogel and in the bulk solution are thus determined

by fitting the five parameters of Eq. 5, namely D

gel

,DF

gel

sol

int

, and d

int

to the experimentally measured concentration profiles.

Confidence intervals for the obtained parameters of D

sol

gel

, and DF

gel

are estimated by determining the parameter values that change sby not

more than 50% (for details, see Supporting Materials and Methods, Section

S6). The error bars shown in Fig. 5 are then obtained by averaging the con-

fidence intervals over all measurements.

RESULTS AND DISCUSSION

Fluorescence intensity profiles of FITC-labeled dextran

molecules penetrating into PEG-based hydrogels are

analyzed using the procedure explained in the Materials

and Methods. The analysis is based on numerical solutions

of the one-dimensional generalized diffusion equation (35)

vcðz;tÞ

vt¼v

vzDðzÞebFðzÞv

vzcðz;tÞebFðzÞ;(6)

where c(z,t) is the concentration at time tand depth z(see

Fig. 1), D(z) and F(z) are the spatially resolved diffusivity

and free-energy profiles that the dextran molecules experi-

ence, and b¼1/k

Tis the inverse thermal energy. Whereas

the diffusivity D(z) describes the mobility of dextran mole-

cules at position z, the free-energy profile F(z) uniquely de-

termines the equilibrium partitioning of dextran molecules.

The numerical solution of Eq. 6 provides a complete model

of the penetration process into the hydrogel and at the same

time allows for extraction of the diffusivity and free-energy

profiles by comparison with experimentally measured con-

centration profiles. A direct conversion of measured fluores-

cence intensities into absolute concentrations is often

difficult because of drifts of various kinds. The method

developed here circumvents this problem and allows for

in-depth analysis of arbitrarily normalized concentration

profiles, as explained in Numerical Model and Discretiza-

tion. Complete profiles of free energies and diffusivities,

Diffusivity and Free Energy of Hydrogels

Biophysical Journal 120, 463–475, February 2, 2021 467

both in the bulk and in the PEG hydrogel, are obtained, and

the results for different hydrogels and dextran molecules of

varying sizes will be analyzed in the following.

Comparison between experimental and modeled

concentration profiles

Fig. 3,Aand Eshows exemplary concentration profiles for

dextran molecules with molecular masses of M

dex

¼4 kDa

and M

dex

¼40 kDa penetrating into the hPG-G10 hydrogel

(see Hydrogel Preparation). Measurements are performed

over a total time span of 30 min, and concentration pro-

files are recorded every 10 s, leading to a total of 180 con-

centration profiles as input for the numerical extraction of

the diffusivity and free-energy profiles. The first measured

concentration profile at t¼0 min represents the start of

the experiment, 10 s after the dextran solution was applied

onto the gel (see Penetration Assay of FITC-Labeled Dex-

trans). The numerically determined concentration profiles

(lines) reproduce the experimental data (data points) very

accurately, as seen in Fig. 3,Aand E. The deviation is esti-

mated from the normalized sum of residuals, s(according to

Eq. 2), which is below 2 mg/L for both measurements. A

stationary concentration profile is obtained in the theoretical

model only after 4 h of penetration for the smaller 4 kDa

dextran (see Fig. 3 B); for the larger dextran molecule, the

stationary profile is reached only after an entire day (see

Fig. 3 F). These times significantly exceed the duration of

the experiments.

The extracted diffusivity and free-energy profiles in

Fig. 3,C,D,G, and Hreveal the selective hydrogel perme-

ability for dextran molecules of varying size. The free-en-

ergy difference in the hydrogel is positive DF

gel

>0 for

both dextran sizes, indicating that dextran is repelled from

the hydrogel. The dextran partition coefficient K

gel

between

the hydrogel and the bulk solution is related to the change in

the free energy DF

gel

Kgel ¼ebDFgel :(7)

According to Eq. 7, the obtained free-energy differences

gel

¼0.6 k

Tand DF

gel

¼1.9 k

Tcorrespond to partition

coefficients of about K

gel

z1/2 and K

gel

z1/7 for the

smaller and the larger dextran molecules, respectively,

which illustrates a significant exclusion in particular for

the larger dextran. Compared with the partition coefficients,

the diffusion constants in the hydrogel decrease only

slightly as a function of the dextran mass. This suggests

FIGURE 3 Exemplary time-dependent dextran concentration profiles from experimental measurements (circles) and numerical modeling (solid lines) for

the hPG-G10 hydrogel. Results for the smallest dextran with M

dex

¼4 kDa in (A)–(D) are compared with results for M

dex

¼40 kDa in (E)–(H). (Aand E)

Experimental and modeled concentration profiles agree very accurately; note that concentration profiles are shifted vertically for better visibility. The initial

bulk concentration of dextran is c

¼70 mg/L. (Band F) Modeled concentration profiles are presented for a wide range of penetration times. The initial

profile c

!init (black line) is based on experimental data (see Construction of the Initial Concentration Profile). (Cand G) Extracted diffusivity profiles are

given, showing that the diffusivity in the hydrogel is only slightly reduced compared to the bulk solution. (Dand H) Extracted free-energy profiles are shown.

Significant exclusion of dextran from the hydrogel is observed, with a stronger effect for the larger dextran. To see this figure in color, go online.

Wolde-Kidan et al.

468 Biophysical Journal 120, 463–475, February 2, 2021

that the dextran molecules are only modestly hindered in

their motion, a conclusion that will be rationalized by our

elastic free-volume model further below.

Fig. 4 shows the temporal evolution of the average

dextran concentration cin three different regions, namely

inside the gel for z

int

<z<z

bot

, in the near solution for

0<z<z

int

, and in the far solution for z

top

<z<0 for

the same data shown in Fig. 3. The lines show the predic-

tions based on the extracted diffusivity and free-energy pro-

files and the circles the experimental data, which are not

available in the far solution range. The average concentra-

tion in the gel (black) increases monotonically and saturates

after about 1 h for both dextran sizes. Note that the station-

ary final concentration in the hydrogel is considerably less

for the larger dextran with M

dex

¼40 kDa. In contrast, the

average concentration in the far solution saturates more

slowly and shows a slight nonmonotonicity for both dextran

masses (blue). This nonmonotonicity is more pronounced in

the near solution (red) and is caused by the fact that dextran

molecules diffuse quickly into the hydrogel from the near

solution in the beginning of the experiment, whereas the

replenishment from the bulk solution takes a certain time,

as also seen in the concentration profiles in Fig. 3,Band

F. Very good agreement between experiments and modeling

results is observed.

Influence of dextran size on hydrogel penetration

The same analysis is performed for dextran molecules of

molecular masses ranging from M

dex

¼4 kDa to M

dex

70 kDa that penetrate into PEG hydrogels with two different

linker lengths, namely hPG-G6 with a PEG linker size of

PEG

¼6 kDa and hPG-G10 with M

PEG

¼10 kDa. Fig. 5

shows the extracted diffusivities and free energies, which

result from averages over at least three experiments for

each system, except for M

dex

¼70 kDa dextran, for which

only one experiment was performed.

Fig. 5 Ashows the bulk diffusivities D

sol

extracted from

measured concentration profiles as colored symbols; in prin-

ciple, there should be no difference between results for hPG-

G6 and hPG-G10. A power-law relation between the dextran

mass and the diffusivity according to D

sol

fMn

dex is shown as

straight lines for n¼1(broken line) and for n¼1/2 (dotted

line). An exponent of n¼1/2 agrees nicely with our FCS data

(solid black triangles; see FCS of FITC-Labeled Dextrans)as

well as with literature fluorescence recovery after photo-

bleaching (FRAP) measurements (34)(open black triangles).

The value n¼1/2 follows from combining the generally

applicable Stokes-Einstein relation D

sol

¼k

T/6ph

(36)

with the scaling of the dextran hydrodynamic radius accord-

ing to r

fMn

dex (37,38) by assuming that the bulk solution is

a theta solvent for dextran polymers (39,40)(seeSupporting

Materials and Methods, Section S7 for details). The exponent

n¼1/2 is only expected for linear polymers, whereas dextran

is in fact a branched polymer. The good agreement of FCS

and FRAP data with the power law for n¼1/2 suggests

that the degree of branching is low (41) or that branching

effectively compensates self-avoidance effects. The dextran

hydrodynamic radii estimated from the FCS measurements

compare well with the values reported by the supplier (see

Table 2). The data for D

sol

obtained from the time-dependent

dextran concentration profiles show rather large uncer-

tainties, which is due to the fact that the concentration pro-

files are rather insensitive to the bulk diffusivities; they are

within error bars consistent with our FCS results but do not

allow extraction of the power-law scaling with any reason-

able confidence.

Values for the diffusion constant in the hydrogel D

gel

are

compared with power laws with exponents n¼1/2 and n¼1

in Fig. 5 B. The difference of the diffusion constants be-

tween the two different hydrogels is within the error bars,

FIGURE 4 Comparison of experimental results (circles) and modeling

results based on the extracted diffusivity and free-energy profiles (lines)

for the mean dextran concentration cover time in three different regions,

the far solution (z

top

<z<0), the near solution (0 <z<z

int

), and the

gel (z

int

<z<z

bot

); see Fig. 1. The systems are the same as shown in

Fig. 3. A nonmonotonic dextran concentration is measured over time in

the near and far solution regions. The fact that cin the gel does not vanish

for t/0 reflects that the first measurement at t¼0 is done 10 s after the

application of the dextran solution onto the gel. The initially employed bulk

dextran concentration is c

¼70 mg/L. To see this figure in color, go online.

Diffusivity and Free Energy of Hydrogels

Biophysical Journal 120, 463–475, February 2, 2021 469

which reflects the fact that the estimated mean hydrogel

mesh sizes, using a very simplistic hydrogel network model

with a perfect cubic structure, are lhPGG6

0;ideal ¼7.1 nm and

lhPGG10

0;ideal ¼7. 5nm (see Estimate of Mean Hydrogel Mesh

Size) and thus quite similar to each other. It is to be noted

that for M

dex

%20 kDa, the estimated mesh sizes are larger

than twice the dextran hydrodynamic radii from Table 2,

which would not suggest any dramatic confinement effect

on the diffusion constant (42). Interestingly, for the data

for which M

dex

T20 kDa, the hydrogel with the larger

linker length (hPG-G10), which has a slightly higher

mesh size, is seen to reduce the diffusion constant slightly

more, which at first sight is counterintuitive. This finding

can be rationalized by the fact that the hPG-G10 gel has a

higher mass density compared to the hPG-G6 gel (see Hy-

drogel Preparation), and thus, the effective pore size is pre-

sumably substantially smaller. This is schematically

illustrated in the inset in Fig. 5 B. A diffusivity scaling

with an exponent n¼1, which describes the data for

hPG-G10 slightly better, could be rationalized by screened

hydrodynamic interactions or by reptation-like diffusion

(43). In fact, a crossover in the scaling of the diffusivity

with increasing hydrogel density from n¼1/2 to n¼1

has been described before for dextran penetrating into

hydroxypropyl cellulose (38). However, because of the large

error bars, extraction of the diffusivity scaling with respect

to dextran mass in the two gels is not uniquely possible.

This is mostly due to the fact that the diffusivities change

rather mildly with varying dextran mass. This is why we

do not attempt to model the scaling of the extracted diffusiv-

ities, as was done elsewhere before (18,19,44), but rather

focus on the mechanism behind the extracted free-energy

differences in the following.

Fig. 5 Cshows the extracted values of DF

gel

for the two

hydrogels as a function of the dextran mass. In all measure-

ments, we find DF

gel

>0, which suggests exclusion of the

dextran molecules from the hydrogel. Also, the value of

gel

increases with the dextran mass. Because dextran,

as well as the PEG-hPG based hydrogels, is uncharged

(45), this exclusion must be due to steric repulsion, possibly

enhanced by hydration repulsion (46,47).

Elastic free-volume model for dextran penetration

in hydrogels

For the larger dextran molecules, the hydrogel with the

smaller PEG linkers, hPG-G6, displays a slightly stronger

exclusion. The power-law relation between the hydrogel

free energy and dextran mass according to DF

gel

fMa

dex

with an exponent of a¼1/2 describes the data well for larger

dextran masses M

dex

T20 kDa, as shown by the dotted black

line in Fig. 5 C. This power-law behavior is in fact compat-

ible with a simplistic elastic free-volume model for the pene-

tration of dextran molecules into hydrogels, which yields the

solid lines and will be derived in the following.

The model geometry is sketched in Fig. 6 Aand consists

of a single dextran molecule of radius r(green sphere) in-

side a cubic unit cell of the PEG-based hydrogel (gray cyl-

inders), similar to previous coarse-grained hydrogel models

(18–20). The presence of the hPG hubs connecting the

PEG linkers is neglected in the following. The dextran

TABLE 2 Dextran Radii

dex

FCS

4 kDa 1.4 nm 1.5 nm

10 kDa 2.3 nm 2.7 nm

20 kDa 3.3 nm 3.2 nm

40 kDa 4.5 nm 4.3 nm

70 kDa 6.0 nm 6.4 nm

Hydrodynamic radius r

as reported by the supplier, in comparison to esti-

mated hydrodynamic radius r

FCS

based on our FCS measurements using the

Stokes-Einstein relation and the viscosity of water as h

¼0.8 10

3

Pa s.

ABC

FIGURE 5 Results for the diffusivity and free energy obtained from the experimental measurements as a function of dextran mass. (A) Fitted diffusivities

in the bulk solution (squares and circles) agree within the error with FCS data measured in this work (solid black triangles) and with FRAP measurements

from literature (34)(open black triangles). (B) Fitted diffusivities in the hydrogel are reduced relative to the bulk values and are compared to different power

laws. (C) Dextran molecules are excluded from the hydrogel and DF

gel

>0 for all dextran masses. For larger dextran molecules, DF

gel

increases as a square

root with the dextran mass. The results from the free-volume model of Eq. 12 (continuous lines) agree nicely with the measurements. Error bars have been

estimated as explained in Supporting Materials and Methods, Section S6. The inset in (B) presents a schematic depiction of the two different gels. Even

though the hPG-G10 gel is composed of larger linkers, the mass density is larger than in the hPG-G6 gel, which results in an effectively smaller pore

size. To see this figure in color, go online.

Wolde-Kidan et al.

470 Biophysical Journal 120, 463–475, February 2, 2021

experiences a reduction of its free volume compared with

the bulk solution because of steric interactions with the

PEG linkers. In the simple model geometry, the PEG linkers

are located at the edges of the cubic unit cell and are

modeled as impenetrable cylinders of radius aand length

l. Conformational fluctuations of the PEG linkers are not

treated explicitly in this model; instead, the linker length l

and radius aare to be understood as average values over

different confirmations of the linker chains. The excluded

volume V

for dextran in the cubic unit cell consists of a

quarter of each of the 12 cylinders at the edges. The acces-

sible or free volume in the hydrogel V

free

depends on the

sum of sphere radius rand cylinder radius aand is given by

Vfree ¼Vunit Vex

¼l312

4pðrþaÞ2lþ2Vcyl:(8)

Here, V

unit

¼l

is the volume of the unit cell and V

cyt

ð16 =3Þ(rþa)

is the volume of two intersecting cylinders

(48), which is subtracted from the excluded volume to avoid

over counting of the unit-cell corners. The entropic contri-

bution to the total free energy is given by

DFvol ¼kBTlnVfree

Vunit

¼kBTln13prþa

l2

þ32

3rþa

l3:

(9)

Because dextran and the PEG linkers are elastic poly-

mers, they are both flexible and can deform. For small defor-

mations, the polymers behave like Gaussian chains (39,40).

The elastic deformation free energy for a cubic unit cell con-

sisting of 12 equally deformed PEG linkers can be written as

(for a detailed derivation, see Supporting Materials and

Methods, Section S8)

DFPEG ¼12

2kBT0

@l

l02

14hl

l0i2

2þhl

l0i21

:(10)

Here, l/l

is the relative stretching of the PEG linkers,

where l

denotes the edge length of the unit cell in the

absence of dextran molecules. The elastic deformation en-

ergy of dextran is obtained in the same fashion and reads

DFdex ¼3

2kBTr

r02

þhr0

ri2

2;(11)

where rdenotes the deformed dextran radius and the unper-

turbed dextran radius is denoted by r

and is taken from

Table 2. The complete free energy follows as

DFgelðr;lÞ¼DFvolðr;lÞþDFPEGðlÞþDFdexðrÞ:(12)

The equilibrium free energy is given by the minimal value

of this free-energy expression, obtained for the optimally

stretched unit-cell length l* and the optimal dextran radius

r*, which are determined numerically. The values of the

unit-cell length l

and the PEG linker thickness aare adjusted

by fits to the experimental data. The model results are shown

in Fig. 6 Bin terms of the partition coefficient as solid lines

and compared with the experiments (circles and squares)asa

function of the length ratio r

. The inset shows the obtained

equilibrium values for l*andr* for the hPG-G6 gel. A

considerable stretching of PEG linkers and compression of

dextran are observed, which shows that elasticity effects of

both PEG linkers and dextran molecules are important and

cannot be neglected when estimating the free volume.

The fit to the experimental data yields lhPGG6

0¼16.7 nm,

lhPGG10

0¼23.7 nm, a

hPG-G6

¼3.4 nm, and a

hPG-G10

5.4 nm. The fit values of acertainly represent an effective

AB C

FIGURE 6 Elastic free-volume model for the partitioning of a particle in a hydrogel. (A) Schematic sketch of the cubic unit-cell model for the hydrogel is

given, made up of connected linkers of length land a finite radius of a. The diffusing particle is modeled as a sphere of radius r. Both the particle and the

linkers are elastic and can stretch or contract. (B) Partition coefficient K

gel

extracted from the experimentally measured dextran concentration profiles (sym-

bols) is shown in comparison with the elastic free-volume model predictions according to Eq. 12 (solid lines). The results of the nonelastic model according to

Eq. 9 are shown as dashed lines. Error bars have been estimated as explained in Supporting Materials and Methods, Section S6. The inset shows the equi-

librium values of l* and r* obtained for the hPG-G6 gel. (C) Illustration of a disordered pore in the hydrogel that has a mesh size l

and consists of more than

four linkers is given (see also Fig. 2). To see this figure in color, go online.

Diffusivity and Free Energy of Hydrogels

Biophysical Journal 120, 463–475, February 2, 2021 471

PEG linker radius and include the layer of tightly bound hy-

dration water. They are indeed, close to the respective equi-

librium PEG radii R

PEG

¼b

N3=5

PEG=ﬃﬃﬃ

p, given as

RhPGG6

PEG ¼4.4 nm and RhPGG10

PEG ¼5.99 nm, where b

0.4 nm denotes the Flory monomer length (49) and N

PEG

is the respective number of PEG monomers. In fact, the

free-volume model yields estimates of the number of hydra-

tion waters per PEG monomer that scatter around 8, in

rough agreement with literature values (see Fig. S8;Sup-

porting Materials and Methods, Section S9).

The fit values for the unit-cell length l

are significantly

larger than the mean mesh size estimated based on Eq. 1,

which for a perfectly ordered cubic lattice predicts

lhPGG6

0;ideal ¼7.1 nm and lhPGG10

0;ideal ¼7.5 nm, but still consider-

ably shorter than the PEG contour lengths L¼bPEG

PEG

which are L

hPG-G6

¼48.5 nm and L

hPG-G10

¼80.9 nm, where

bPEG

0¼0.356 nm is the PEG monomer length (49). Although

the large unit-cell lengths obtained from the fit to the elastic

free-volume model could reflect a substantial stretching of

individual PEG polymers, there is no a priori reason why

the linkers should be stretched to such a considerable fraction

of their contour length. We therefore rationalize this surpris-

ing result in terms of a broad distribution of pore sizes that

exhibit different topologies. To illustrate this, a random

pore is schematically shown in Fig. 6 C. Based on the 3:1

number ratio of linkers/cross-linkers in the hydrogel formu-

lation (cf. Hydrogel Preparation and Fig. 2), a perfectly cubic

lattice could form, in which each hub is connected to six

different linkers. Such an ideal cubic connectivity is, of

course, entropically highly unfavorable, and the connectivity

distribution of hubs, i.e., the distribution of the number of

linkers that connect to one hub, will be rather broad and

the network topology disordered, in which case the PEG

end-to-end distance R

PEG

will be significantly smaller than

the pore size l

(cf. also Estimate of Mean Hydrogel Mesh

Size). Whereas in a cubic lattice, each cubic facet consists

of four hubs and four linkers, the pores present in the actual

hydrogel will show a broad distribution of the number of

participating linkers. For illustration, the pore shown in

Fig. 6 Cconsists of eight linkers. Clearly, dextran molecules

will tend to be located in larger pores to maximize their free

volume, and therefore, the fit parameters of our model will be

dominated by the tail of the pore-size distribution, which ex-

plains the large fit values for l

. This finding also allows us to

rationalize the larger extracted free energy in the hydrogel in

the case of the hPG-G6 gel, even though the hPG-G10 gel

mass density is higher (cf. Fig. 5 C). The tail of the pore-

size distribution of the hPG-G10 gel presumably contains

larger pores that can stretch even further to minimize the un-

favorable dextran-PEG interactions. Clearly, the precise to-

pology and compositional distribution of pores cannot be

predicted by our analysis; our results should thus be merely

interpreted as an indication of the presence of large pores

and a disordered network topology.

An approximate nonelastic version of the free-volume

model is obtained by neglecting the polymer deformation

term and just keeping the excluded volume term, Eq. 9,

which becomes accurate in the limit of l

>> r

,where

r*zr

and l*zl

. These approximate results are shown

as broken lines in Fig. 6 Band describe the experimental

data only for small values of r

. When additionally

approximating the logarithm in Eq. 9, the obtained expres-

sion for the free energy is similar to results derived for a

random-fiber network (50). Our free-volume model is

valid only for short-ranged steric and hydration repulsive

interactions between diffusor and linkers; if long-ranged

and, in particular, attractive interactions are present—for

example, electrostatic interactions for low salt concentra-

tions—the model would need to be adjusted accordingly.

Derivation of particle permeabilities through

hydrogel barriers

Permeation through biological barriers is quantified by the

permeability coefficient P, which is defined as (51)

Pðz1;z2Þ¼ J

cðz1Þcðz2Þ;(13)

where c(z

) and c(z

) are the particle concentrations at the

two sides z

and z

of the barrier and Jdenotes the particle

flux through the barrier. Based on the diffusion equation

(Eq. 6), the inverse permeability can be written as (for a

detailed derivation, see Supporting Materials and Methods,

Section S10)

Pðz1;z2Þ¼Zz2

ebFðzÞ

DðzÞdz:(14)

For a step-like barrier, one obtains

P¼ebDFgel

Dgel

L:(15)

Here, DF

gel

and D

gel

are the particle free energy relative

to the solution and the diffusivity inside the hydrogel, and

Ldenotes the width of the hydrogel barrier.

Fig. 7 Ashows normalized permeability coefficients PL

for a single step-like barrier according to Eq. 15,whichare

independent of the thickness of the barrier L, as a function

of the gel free energy and the gel diffusivity. The values ex-

tracted from the experimental data for different dextran

molecules in the two gels from Fig. 5 are indicated by

data points. Obviously, the highest permeability is

observed for a low free-energy barrier and a high particle

diffusivity, as is the case for the smallest dextran molecules

(lower right corner in Fig. 7 A). On the other hand, perme-

ation is hindered by either a high free-energy barrier or a

low diffusivity in the hydrogel, both of which are observed

Wolde-Kidan et al.

472 Biophysical Journal 120, 463–475, February 2, 2021

for dextran molecules with larger molecular weights.

Because of counterbalancing effects of stronger exclusion

from the hPG-G6 gel and increased immobilization in the

case of hPG-G10, both hydrogels display comparable

permeability coefficients for the chosen dextran molecular

masses.

CONCLUSIONS

The method introduced in this work allows for the simulta-

neous extraction of diffusivity and free-energy profiles of

particles that permeate into spatially inhomogeneous hydro-

gel systems; we demonstrate the method using concentra-

tion profile measurements of fluorescently labeled dextran

molecules permeating into PEG-hPG-based hydrogels.

The advantage over alternative methods is that both diffu-

sivity and free-energy profiles are obtained from a single

experimental setup. This is important because only the com-

bination of diffusivity and free-energy profiles completely

determines the diffusion of particles.

The extracted diffusivities and free energies are analyzed in

terms of empirical scaling laws as a function of the dextran

mass, and a modified elastic free-volume model is developed

that quantitatively accounts for the particle free energy in

the hydrogel. Although the free volume accessible to a

diffusor inside a hydrogel has been previously shown to deter-

mine diffusion properties in biological systems, such as

crowded cellular membranes (52), our modified free-volume

model additionally includes the elasticity of linkers and of

the diffusing molecules and thereby quantitatively accounts

for the free energies we extracted from the experimental

data of dextran diffusing in PEG-based hydrogels. This dem-

onstrates that elastic deformations of both the diffusor and

the hydrogel network are important, in line with previous

computational (53–55) and experimental studies (56). Our

model furthermore unveils significant topological disorder

of the hydrogel pores and suggests that the dextran molecules

preferentially partition into exceptionally large pores, which

are locally even more enlarged because of PEG strand

elasticity.

Diffusional barriers in biological systems often show

a layered structure, as previously demonstrated for skin

(29–31) and also known to be true for mucous membranes,

which are found, for instance, in the gastrointestinal tract,

schematically indicated in Fig. 7 B. For a layered system,

Eq. 14 shows that the individual piecewise constant perme-

ability coefficients P

add up inversely as

Ptot ¼X

Pi¼X

ebDFi

Li¼X

DiKi

;(16)

where the sum goes over all layers, represented by their

respective diffusion constants D

, free-energy values DF

or partition coefficients K

, and thicknesses L

. Here, P

tot

de-

notes the total permeability, which is dominated by the

smallest permeability in the inverse sum.

Fig. 7 Bschematically illustrates permeation through a

layered system which represents the mammalian stomach

(57). The outermost layer of mucus is only loosely bound

and characterized by the permeability P

; it is followed

by a layer of more tightly bound mucus, characterized by

, and adheres onto the first layer of epithelial cells, char-

acterized by P

. The total thickness of this diffusional bar-

rier is about a millimeter, with the two mucus layers

spanning a few hundred micrometers only (58). Measure-

ments in rat gastrointestinal mucosa suggest typical values

FIGURE 7 (A) Normalized permeability coefficient PL through a single

box-like hydrogel barrier of width Las a function of the hydrogel free en-

ergy DF

gel

and the hydrogel diffusivity D

gel

from Eq. 15. High permeability

is observed for low free-energy barriers and high diffusivities in the hydro-

gel. The symbols denote the experimental data from Fig. 5. Because of

opposing trends in the free-energy barrier and the diffusivity, both hydro-

gels display comparable permeability coefficients. (B) Schematic layered

structure of a mucous membrane, as found in the stomach, is given. Exam-

ples for different diffusors are shown, including nutrients such as glucose

and pathogens such as virions or bacteria. The diffusors have to penetrate

different layers of varying permeabilities to enter the tissue below the mu-

cous membranes, the total permeability of a layered structure follows from

Eq. 16. To see this figure in color, go online.

Diffusivity and Free Energy of Hydrogels

Biophysical Journal 120, 463–475, February 2, 2021 473

of L

¼109 mm, L

¼80 mm, and L

(59), which are

close to the range of gel thicknesses studied in this work.

The total permeability is determined by the free energies

and the mobilities inside all layers. Nutrients, for instance,

can easily penetrate through the epithelia of the gastrointes-

tinal tract, displaying large permeabilities in the different

layers. Pathogens, on the other hand, are in healthy environ-

ments kept from reaching the epithelium because of low

permeability in the tightly bound mucus layer (P

<< P

)

(57). From Eq. 16, it is apparent that the lowest permeability

in such a layered system dominates the total permeability,

leading to an effective barrier function that for different parti-

cles can be caused by different parts of the layered barrier

structure.

The method introduced in this work determines free-en-

ergy and diffusivity profiles from experimental data and

thereby can be used to predict effective permeabilities of

different kinds of molecules, particles, or even organisms

into various layered systems, including systems that contain

hydrogels and mucus. A multilayered structure, as shown in

Fig. 7 B, can be produced by cultivating mucous-producing

cells in vitro and can be studied using the framework intro-

duced in this work. This would allow the detailed analysis of

permeabilities of different diffusors through an in vitro rep-

resentation of an actual biological barrier. We believe

that the technical advances described in this work will

help to shed light on the underlying mechanisms of the func-

tion of general biological barriers including mucous

membranes.

SUPPORTING MATERIAL

Supporting Material can be found online at https://doi.org/10.1016/j.bpj.

2020.12.020.

AUTHOR CONTRIBUTIONS

A.H., S.B., and R.H. designed and performed the CLSM experiments for

the determination of the dextran concentration profiles. A.P. and M.G. de-

signed and performed the FCS experiments. A.W.-K. and R.R.N. designed

the models, analyzed the experimental data, and wrote the manuscript.

ACKNOWLEDGMENTS

The authors acknowledge funding by the Deutsche Forschungsgemein-

schaft via grant SFB 1449.

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