comparative analysis of viscosity of complex liquids and...

Published: April 22, 2011

r 2011 American Chemical Society 2157 dx.doi.org/10.1021/nl2008218 |Nano Lett. 2011, 11, 2157–2163

LETTER

pubs.acs.org/NanoLett

Comparative Analysis of Viscosity of Complex Liquids and Cytoplasmof Mammalian Cells at the NanoscaleTomasz Kalwarczyk,† Natalia Zie-bacz,† Anna Bielejewska,† Ewa Zaboklicka,†,‡ Kaloian Koynov,§

Je-drzej Szyma�nski,^,† Agnieszka Wilk,|| Adam Patkowski,|| Jacek Gapi�nski,|| Hans-J€urgen Butt,§ andRobert Hozyst*,†,‡

†Department of Soft Condensed Matter, Institute of Physical Chemistry PAS, Kasprzaka 44/52 01-224 Warsaw, Poland,‡Cardinal Stefan Wyszy�nski University, WMP-SN�S, Dewajtis 5, 01-805 Warsaw, Poland,§Max-Planck-Institute for Polymer Research, D-55128 Mainz, Germany, and

)A.Mickiewicz University, Faculty of Physics, Umultowska 85, 61-614 Pozna�n, Poland

bS Supporting Information

The mobility of proteins and other probes is one of the mainregulating factors in processes taking place in living cells.

Small proteins (tens of kilodaltons with hydrodynamic radii rp∼1�5 nm) show diffusivities orders of magnitude faster than thosepredicted by the Stokes�Sutherland�Einstein (SSE) equationD = kT/6πηmacrorp on the basis of the macroscopic viscosityηmacro.

1,2 Also the measurements of the self-diffusion coefficientsof proteins and small molecules in E. coli,3 HeLa,4,5 and Swiss3T36 cells, could not be explained via the SSE equation and themacroscopic viscosity. Luby-Phelps et al.6 performed fluores-cence recovery after photobleaching (FRAP) measurements offluorescently labeled dextrans in Swiss 3T3 muscle cells. Theydescribed the data via the diffusion coefficient ratio (DCR),calculated as the ratio of the coefficient of self-diffusion of probesin the cytoplasm to the self-diffusion in water. They observed thatDCR decreased with increasing size of the probes up to the size of14 nm. For probes larger than 14 nmDCR had an approximatelyconstant values. FRAP measurements preformed on CHO cellsby the Verkman group suggested that translational mobility ofgreen fluorescence protein (GFP) was determined mostly by theconcentration of cellular obstacles.7 Lukacs5 studied the mobilityof DNA fragments (from 21 up to 6000 bp) in the cytoplasm ofHeLa cells. Fragments longer than 3000 bp presented unexpect-edly low mobility. On the contrary in a nucleus, diffusion of all

investigated DNA fragments (from 21 up to 6000 bp) wasstrongly hindered and did not change with size of diffusingentities. Dauty et al.4 compared self-diffusion coefficients (SDC)of fluorescently labeled DNA fragments diffusing in the solutionsof polymerized actin with SDC obtained in vivo in HeLa cells.They postulated that it was the cytoskeletal network of actinfilaments that hindered the diffusion of probes in the cytoplasm.Studies on viscoelastic properties of polymerized actin solutionswere reported in numerous works. Most of them, however,consider length scales of probes from 1 to 100 μm.8�12 Thesestudies showed that the viscoelastic properties of the actinnetwork are determined by the amount of bundles and cross-linkers connecting actin filaments.8,9,11 Additionally, varioustheoretical models mimicking the cytoskeleton behavior13 havebeen proposed. Also an opposite approach (biomaterials whichmimic polymers) has been established recently.14

The development of a theory relating the mobility of solutes ina cytoplasm to their hydrodynamic radii and to the cytoplasmicviscosity is important because mobilities of biomolecules influ-ence the transport, signaling, metabolism, and life cycle of living

Received: March 11, 2011Revised: April 16, 2011

ABSTRACT: We present a scaling formula for size-dependentviscosity coefficients for proteins, polymers, and fluorescent dyesdiffusing in complex liquids. The formula was used to analyze themobilities of probes of different sizes in HeLa and Swiss 3T3mammalian cells. This analysis unveils in the cytoplasm two lengthscales: (i) the correlation length ξ (approximately 5 nm inHeLa and7 nm in Swiss 3T3 cells) and (ii) the limiting length scale thatmarks the crossover between nano- and macroscale viscosity(approximately 86 nm in HeLa and 30 nm in Swiss 3T3 cells).During motion, probes smaller than ξ experienced matrix viscosity:ηmatrix ≈ 2.0 mPa 3 s for HeLa and 0.88 mPa 3 s for Swiss 3T3 cells. Probes much larger than the limiting length scale experiencedmacroscopic viscosity, ηmacro ≈ 4.4 � 10�2 and 2.4 � 10�2 Pa 3 s for HeLa and Swiss 3T3 cells, respectively. Our results arepersistent for the lengths scales from 0.14 nm to a few hundred nanometers.

KEYWORDS: Viscosity, diffusion, cytoplasm, complex liquids, scaling

2158 dx.doi.org/10.1021/nl2008218 |Nano Lett. 2011, 11, 2157–2163

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cells.15 On the other hand studies on diffusion of solutes indifferent systems such as polymer solutions have also beendone.16�29 Schachman et al.16 showed that small and largeprobes sedimenting in a solution of DNA differ dramatically insedimentation coefficients. Currently most researchers17�30

accept an empirical ansatz for the relation between the viscosityexperienced by the nanoscopic probes and their hydrodynamicradius rp

η ¼ η0 expðKrμp cνÞ ð1ÞHere η0 is the viscosity of the solvent (i.e., water),K is a constant,and c denotes the concentration of the polymer. μ and ν areexponents whose values vary from system to system.29 In ourprevious paper,31 we have shown that eq 1 may be replaced by amore general expression describing viscosity.

η

η0¼ exp

dξ

� �a !

for d , Rg ð2Þ

η

η0¼ exp

Rg

ξ

� �a !

for d . Rg ð3Þ

Here ξ is the correlation length interpreted as the averagedistance between the point of entanglement in the polymernetwork, Rg is the radius of gyration of the polymer coil, and d isthe diameter of the probe. The physical meaning of the exponenta (a constant of order 1) is still under discussion, and as wasshown by Odijk29 it may vary from system to system. ξ is afunction of the concentration of the polymer and of Rg

32

ξ ¼ bRgcc�� β

ð4Þ

Here b ∼ 1, c* is the overlap concentration defined for flexiblepolymers as

c� ¼ Mw

4=3πR3gNA

ð5Þ

and Rg is a gyration radius of the polymer. The value of the βexponent depends on type of the solvent. In a good solventβ = �0.75, while in θ solvent β = �1.33 Equations 2 and 3 arevalid only in two separate regimes: for d , Rg or for d . Rg.Equations 2 and 3 provide a uniform description of viscosityexperienced by nanoprobes (eq 2) and macroscopic viscosityexperienced by the macroscopic probe (eq 3). Additionally onecan expect a crossover length scale above which the probeexperiences the macroscopic viscosity of the solution. Equation1 transforms into the form of eq 2 with the constant K =(2bRgc*)

�a, μ = a and ν = βa.Here we generalize the description of viscosity further to all

sizes of probes including probes of radius comparable to theradius of cosolute (polymer or micelle). We also show that theviscosity experienced by the probe of radius rp depends on thehydrodynamic radius of polymer coils, Rh, and not as in eq 3 ontheir radius of gyration. d in eq 2 should be interpreted ashydrodynamic radius of probes rp and not as their diameter. Wedevelop a scaling law of viscosity of flexible polymer solutions.We also reanalyze the data of the viscosity of solutions of rigidmicelles34 according to developed scaling law. Finally we use ourmodel to analyze published diffusion coefficients of solutes

diffusing in the cytoplasm of two different types of mammaliancells (HeLa and Swiss 3T3).4,6,35�38

We have shown previously31,34 that the Stokes�Sutherland�Einstein relation can describe diffusion at the nano-scale (D = kT(6πη(rp)rp)) if one uses the scale-dependentviscosity, η(rp). We use this relation to calculate the viscosityexperienced by the probe frommeasurements of its self-diffusioncoefficient

η

η0¼ D0

Dð6Þ

where η is the viscosity experienced by the probe, η0 is theviscosity of the solvent, D is the self-diffusion coefficient of theprobe in the studied solution, and D0 is the diffusion constant ofthe same probe in a pure solvent.

According to eq 6 we determined the viscosity of aqueoussolutions of poly(ethylene glycol) (PEG) and of solutions ofpolystyrene (PS) in acetophenone. Properties of these polymersare listed in Table 1. Hydrodynamic (Rh) and gyration (Rg) radiiof PEG were calculated according to the relations39 Rg =0.0215Mn

0.58 and Rh = 0.0145Mn0.57 where Mn is the number

average molecular weight of the polymer obtained from gelpermeation chromatography. Hydrodynamic and gyration radiiof PS were determined according to the procedure describedelsewhere.40 In addition, we studied the viscosity of solutions ofrigid micelles of surfactant (hexa(ethylene glycol) monododecylether, C12E6).

34 For concentrations of C12E6 lower than 35% andat room temperature the micelles are L ≈ 23 nm in lengthcorresponding to an equivalent hydrodynamic radius Rh ≈5.3 nm.41 Diffusion coefficients of the probes were measuredwith fluorescence correlation spectroscopy technique (for detailssee Supporting Information). We used Rhodamine B (rp =0.58 nm) and lysozyme (rp = 1.9 nm) labeled with TAMRA(carboxytetramethylrhodamine) as probes for the solutions ofPEG . In the solutions of PS 220000 in the acetophenone we usedN-(2,6-diisopropylphenyl)-9-(p-styryl)perylene-3,4-dicarboximide(PMI) (rp = 0.53 nm) and PMI-labeled polystyrene (PMI-PS34000 and PMI-PS 255000, cf. Table 1). The viscosity ofsurfactant solutions (for experimental details see SupportingInformation) was probed with water (rp = 0.14 nm), TAMRA(rp = 0.85 nm), and TAMRA labeled proteins such as lysozyme(rp = 1.9 nm), chymotrypsinogen (rp = 2.6 nm), ovalbumin (rp =3.4 nm), bovine serum albumin (rp = 4.2 nm), and apoferitin

Table 1. Characteristics of Poly(ethylene glycol) and Poly-styrene. a

symbol Mn or Mw (g/mol) Rg (nm) Rh (nm)

PEG 400 325 0.6 0.4

PEG 6 000 3461 2.4 1.5

PEG 20000 10944 4.7 2.9

PEG 35000 15040 5.7 3.5

PEG 600000 276862 30.8 18.3

PEO 8000000 7019190 201 115.8

PS 34000 34000 5.8 4.4

PS 220000 220000 14.9 11.5

PS 255000 255000 16.3 12.5aNumber averaged molecular weight Mn of PEG, weight averagedmolecular weights Mw of PS, gyration (Rg), and hydrodynamic(Rh) radii.


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(rp = 6.9 nm). Additionally we used quantum dots (rp = 12.5 nm)and fluorescent silica spheres (rp =35, 57, and 95 nm, re-spectively). Probes used in experiments were chosen accordingto solubility in the studied system, different for water and foracetophenone solutions.

Viscosity of the complex liquid depends on the concentra-tion of objects which create the fluid (polymer coils or

micelles). Viscosity experienced by the probe diffusing in thecomplex liquid is affected not only by the concentration butalso by the size of the diffusant itself. In order to visualizedifferences in the viscosities experienced by the differentprobes, in Figure 1 we plot the viscosities as a function of thepolymer or of the surfactant concentration for different ratiosof hydrodynamic radii rp/Rh. We measured the viscosity in twosolutions for different ratios of rp/Rh: one composed of PEG400 (Rh = 0.4 nm) and of PEG 20000 (Rh = 2.9 nm) withlysozyme (rp = 1.9 nm) as the probe (Figure 1a). When the rp <Rh (rp/Rh < 1) the probe diffused with nanoscopic viscositylower than the macroscopic viscosity of the solution. The sameprobe in polymer solution composed of the smaller polymercoils (i.e., rp > Rh and rp/Rh > 1) experienced the macroscopicviscosity. In the PS 220000 solutions (Rh = 11.5 nm), probessuch as PMI (rp = 0.53 nm) or fluorescently labeled polystyr-ene, PMI-PS 34000 (with PMI as fluorophore), experiencedmuch lower viscosity than the macroscopic one (Figure 1b).On the other hand rp of the PMI-PS 255000 was larger than thehydrodynamic radius of the matrix obstacles. Surprisingly,PMI-PS 255000 in PS 220000 solutions (rp > Rh and rp/Rh ≈ 1) experienced a viscosity lower than the macroscopicviscosity of the solution. This result showed that the crossoverbetween the two regimes (nano- and macroscopic viscosity)could not be described with a sharp threshold determined bythe radii of the probe and of the obstacles (i.e., nanoscopicviscosity when rp/Rh < 1 and macroscopic viscosity when rp/Rh > 1). Figure 1c shows the data obtained for C12E6solutions34 for three different probes (lysozyme rp = 1.9 nm,apoferitin rp = 6.9 nm, and nanoparticles rp = 57 nm). Thediffusion coefficient of the nanoparticles (rp . Rh, rp/Rh =10.75) was related to the macroscopic viscosity. When rp < Rh(lysozyme) or rp G Rh (apoferitin), the probe experiencednanoscopic viscosity, much lower than the macroscopic visc-osity. The analysis of the data shown in Figure 1 leads to theintroduction of a new empirical equation, which describes theeffective viscosity of complex liquids as a function of thehydrodynamic radius of the probe

η ¼ η0 expReff

ξ

� �a" #

ð7Þ

Figure 1. Viscosities of PEG (a), PS (b) and C12E6 (c) solutions as afunction of concentration measured with different nanoprobes. In panela, we present macroscopic viscosity of low molecular weight oligomerPEG 400 (O, rp/Rh = ¥, Rh = 0.4 nm), nanoscopic viscosity measuredwith a nanoprobe (lysozyme) in the PEG 400 matrix (9, rp/Rh = 4.75,rp = 1.9 nm), macroscopic viscosity of PEG 20000 (b, rp/Rh = ¥, Rh =2.9 nm) and one measured by diffusion of lysozyme in PEG 20000 (4,rp/Rh = 0.66, rp = 1.9 nm). In panel b we show viscosity measured withdifferent probes for solutions of PS 220000 in acetophenone. As probeswe use: PMI (4, rp/Rh = 0.09, rp = 0.53 nm), PMI-PS 34000 (b, rp/Rh =0.4, rp = 4.45 nm), and PMI-PS 255000 (0, rp/Rh = 1.09, rp = 12.5 nm).Macroscopic viscosity of PS 220000 matrix is represented by9 (rp/Rh =¥, Rh = 11.5 nm). Panel c shows viscosity of aqueous solutions of rigidmicelles C12E6. Macroscopic viscosity (9, rp/Rh = ¥, Rh = 5.3 nm) andviscosities measured with different probes: fluorescent particles (O, rp/Rh = 10.75, Rh = 57 nm), apoferitin (4, rp/Rh = 1.3, rp = 6.9 nm), andlysozyme (1, rp/Rh = 0.36, rp = 1.9 nm).

Table 2. Hydrodynamic Radii of Obstacles (Rh), Probes (rp),rp/Rh, and the Effective Hydrodynamic Radii Reff (eq 8) forDifferent Systems Shown in Figure 1

matrix/probe Rh (nm) rp (nm) rp/Rh Reff (nm)

PEG 20000, macroscopic viscosity 2.9 ¥ ¥ 2.9

PEG 20000/lysozyme 2.9 1.9 0.66 1.6

PEG 400, macroscopic viscosity 0.4 ¥ ¥ 0.4

PEG 400/lysozyme 0.4 1.9 4.75 0.4

PS 220000, macroscopic viscosity 11.5 ¥ ¥ 11.5

PS 220000/PMI 11.5 0.5 0.04 0.5

PS 220000/PMI-PS 34000 11.5 4.5 0.4 4.2

PS 220000/PMI-PS 255000 11.5 12.5 1.09 8.5

C12E6, macroscopic viscosity 5.3 ¥ ¥ 5.3

C12E6/lysozyme 5.3 1.9 0.36 1.8

C12E6/apoferitin 5.3 6.9 1.3 4.2

C12E6/particles 5.3 57 10.75 5.3


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Here Reff is an effective hydrodynamic radius related tohydrodynamic radii of the probe rp and of the polymer/micelle,Rh

Reff�2 ¼ Rh

�2 þ rp�2 ð8Þ

In eq 7, a is an exponent of the order of 1 and has samemeaningas in eqs 2 and 3. In the limit of rp. Rh, eq 7 becomes eq 3 withRg replaced by Rh. In the opposite limit (rp, Rh) eq 7 becomeseq 2 with d replaced by rp. The novelty of eq 7 is in thecombination of hydrodynamic properties of the probe (rp) andcosolute (Rh) with the structural property of the network (ξ).Equations 2 and 3 reflect only structural properties of the

matrix liquid (gyration radius Rg and ξ). In addition, eqs 2 and3 exhibit a discontinuity since one boundary refers to the radiusof gyration and the second to the diameter of the probe. Incontrast eq 7 provides a smooth transition between the nano-and macroscopic scale. In Table 2 we present effective radii Refffor different probes presented in Figure 1. Now we canformulate the crossover criterion, i.e., a criterion which statesat what length scale the probes start to experience the macro-scopic viscosity. The probe experiences macroscopic viscositywhen Reff ≈ Rh. For practical reasons when rp > 4Rh theviscosity experienced by the probe is very close (within1�5%) to the macroscopic viscosity according to eq 8.

In Figure 2 we present viscosity values scaled with the factorReff/ξ (in PEG, Figure 2a; in PS, Figure 2b). In all plots the datafollowed the same master curve defined by eq 7, same for themacroscopic probes (falling ball viscometer) and for the nano-probes (proteins or dyes). The scaling law describes the viscosityexperienced by the proteins (compact structures similar to hardspheres) and labeled polymer coils (entangling with polymercoils � matrix obstacles) whose motion in concentrated solu-tions of polymers was used to be described with the reptationmodel.42We tested the universality of our scaling law by applyingit to solutions of elongated rigid micelles as shown in Figure 2c.In the studied system, above critical micelle concentration thesurfactants aggregate forming elongated, ellipsoidal micelles oflength L.41 Further increase of concentration causes the micellesto overlap. In this system the correlation length ξ (eq 4) isdefined by the distance between two touching points of micelles.It is a similar definition to the one used in polymer solutionswhere ξ is defined as the distance between entangle points ofpolymer coils. In micellar systems we defined correlation lengthas in eq 4. Namely ξ = bL(c/c*)β where L is the length of themicelles. The overlap concentration in the solution of rigidmicelles (with length much larger than their diameter) is givenby c* = 3AnMw/(NAL

3), where An is an average number ofsurfactants in a micelle (aggregation number) and Mw is themolecular weight of the surfactant. The value of β = �1 forsolution of micelles guarantees that ξ does not depend onmicellar length (similarly as ξ in polymer solutions does notdepend on the molecular weight of the polymers).32 Theviscosity data scaled with Reff/ξ for the micellar solution(Figure 2c) follows the same curve given by eq 7 as in the caseof the polymer solutions.

Each of the systems described above had only one type ofcrowding agent (objects which crowd the environment, micellesor polymers in our case) that formed the complex liquid.Viscosity in these systems depended on two length scales:correlation length ξ and an effective hydrodynamic radius Reff.More generally, complex liquids (including cytoplasm in cells)may comprise many types of crowding agents that differ in sizeand shape. In cells apart from the cytoskeleton and actinfilaments one can find lots of cytoplasmic proteins and othermacromolecules and organelles which crowd the environment.43

In vitro studies show that the self-diffusion of small (few to tensof nanometers) probes in cells is hindered by the cytoskeletonnetwork.4 On the other hand studies of diffusion of proteins inprotein solutions44 suggest that viscosity of such solutions maybe only slightly higher than the viscosity of water. Therefore incells we could expect that diffusion of probes will depend onviscosity which is a product of both viscosities (viscosity of theprotein solution and of the viscosity of cytoskeleton network). Inthis case we expect that the effective viscosity of the system

Figure 2. Scaled viscosities (same as in Figure 1) of PEG, PS, and C12E6solutions divided by the viscosity of water at 25 �C (relative viscosity). Inpanel a we show scaling plot for the logarithm of relative viscositymeasured for different probes in aqueous solutions of PEG. In panel b weshow scaling plot obtained for PS 220000 solutions in acetophenone. Inboth panels (a and b), all data follow the same curve given by eq 7 withparameters a = 0.62 ( 0.02 and b = 0.24 ( 0.02 (cf. ξ eq 4) for PEGsolutions and a = 0.71( 0.1 and b = 0.43( 0.02 for solutions of the PS.In panel c we show scaled relative viscosities of aqueous solutions ofC12E6 consisting of rigidmicelles. All data follow the same curve given byeq 7 with parameters a = 0.87 ( 0.02 and b = 1.18 ( 0.04.


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should depend on the properties of all types of crowding agents.In particular we expect that the protein solution exhibits viscosityηmatrix and that the actin filament network is responsible for largechange of viscosity according to the exponential factor exp[(Reff/ξ)a] (similarly as for complex liquids). We assume that theviscosity of the composite solution of proteins and actin filamentscan be described by the following scaling law

η ¼ Aη0 exp½ðReff=ξÞa� ð9Þ

where ηmatrix = Aη0, Reff is defined by eq 8, and A is a constant ofthe order of 1.

We reanalyze the data of diffusion in a cytoplasm of HeLacells for various probes: DNA,4 water,35 EGFP,36 and nano-diamonds.37 We also reanalyzed diffusion data of fluorescentdye38 and dextrans6 in Swiss 3T3 mammalian cells. Variablessuch as hydrodynamic radius of the probes (rp) were partiallyavailable in the literature. However, for the analysis of diffusion ofDNA, the hydrodynamic radius of DNA was determined sepa-rately. Robertson et al.45 studied diffusion of linear DNA frag-ments in buffer solutions. From those data we determined anempirical equation for the hydrodynamic radius of DNA frag-ment as a function of its molecular weight.

rp ¼ 0:024M0:57w ðnmÞ ð10Þ

rp obtained from eq 10 is the hydrodynamic radius of the sphereequivalent to the DNA fragment. Hydrodynamic radii andviscosities experienced by all probes together with the cell typeare listed in Table 3. For fitting of viscosity of the cytoplasm data,we combined eqs 8 and 9 to obtain the following form of thescaling law

lnη

η0

� �¼ lnðAÞ þ ξ2

Rh2 þ

ξ2

rp2

!�a=2

ð11Þ

Here ξ, A, a, and Rh (equivalent hydrodynamic radius ofobstacles) were fitting parameters. In Figure 3 we show theviscosity experienced by the probes listed in Table 3. We fit thedata obtained for living cells with eq 11. We found that in HeLacells the viscosity of the matrix ηmatrix =Aη0 = (1.3( 0.3)ηwater =0.9 mPa 3 s. When the radius of the probe exceeds the correlationlength ξwe expect a sudden increase in the values of the viscositymeasured with the probe. For HeLa cells we found ξ = 5( 4 nm.Furthermore for ξ < rp < Rh viscosity values increased with theradius of the probe. When rp > Rh we observed a plateau in thevalue of viscosity plotted against rp. For rp . Rh the values ofviscosity correspond to the macroscopic viscosity. We assumedthat Rh corresponds to the mean value of hydrodynamic radii ofobstacles. In HeLa cell cytoplasm Rh ≈ 86 nm. The value ofexponent a for this type of cell (obtained from fitting of eq 11) isequal to a = 0.49 ( 0.22. Analogous parameters obtained fromfits for Swiss 3T3 cells were equal ηmatrix = Aη0 = (2.9 (0.6)ηwater ≈ 2.0 mPa 3 s, ξ = 7 ( 2 nm, Rh ≈ 30 nm and a =0.62 ( 0.36. Under the assumption that the radius of the actinfilament in HeLa cells is equal r = 7 nm,46 the length of cellularobstacles can be calculated using the relation47 Rh = L/(2s �0.19 � 8.24/s þ 12/s2) where s = ln(L/r) and L is the length ofthe filament. We estimate the obstacles’ length L ≈ 660 nm inHeLa and ≈125 nm in Swiss 3T3 cells. The estimated filamentlength for cells seem reasonable when compared to the results of

Table 3. An Equivalent Hydrodynamic Radii and Experi-enced Viscosities for Different Probes in Cytoplasm of Stu-died Cells

probe cell type rp (nm) η/η0 reference

H2O HeLa 0.14 1.5 ref 35

EGFP HeLa 2 1.35 ref 36

DNA HeLa 5.44 4.8 ref 4 and eq 10




nanodiamonds HeLa 80 32.5 ref 37


BCECF Swiss 3T3 0.3 3.7 ref 38

dextran Swiss 3T3 2.2 5.1 ref 6





dextran Swiss 3T3 7 8.0 ref 6




Figure 3. Viscosity of cell cytoplasm divided by the viscosity of water at37 �C, as a function of the probe radius in HeLa (a) and Swiss 3T3 (b)cells. In panel a we show data of viscosity measured with different probessuch as water,35 EGFP,36 DNA fragments4 and nanodiamonds.37 Inpanel b we present data of viscosity measured with two different probes:small fluorescent particle BCECF38 and dextrans of different molecularweights.6 Viscosities of both cytoplasms were fitted with equation(eq 11). In HeLa cells ηmatrix was equal (1.3 ( 0.3)ηwater. Correlationlength ξ = 5 ( 4 nm, Rh ≈ 86 nm, and a = 0.49 ( 0.22. For Swiss 3T3cells we obtained ηmatrix = (2.9( 0.6)ηwater, ξ = 7( 2 nm, Rh≈ 30 nm,and a = 0.62 ( 0.37.


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Dauty et al.4 who measured the filaments to range from 100 to500 nm. In addition our analysis yields macroscopic viscositiesfor both cell types. For HeLa cells ηmacro ≈ 64ηwater ≈ 4.4 �10�2 Pa 3 s since for Swiss 3T3 cells ηmacro ≈ 34ηwater ≈ 2.4 �10�2 Pa 3 s. These values are in agreement with the literature dataof the cytoplasmic viscosity in an amoeba Dictyostelium discoi-deum cells (5� 10�2 Pa 3 s).

48 According to the criterion that wasproposed for complex liquids the probe will experience macro-scopic viscosity when rp g 4Rh. One can estimate minimalhydrodynamic radius of the probe which would experience themacroscopic viscosity of the cytoplasm of HeLa and Swiss 3T3cells. For HeLa cells hydrodynamic radius of such probe shouldexceeds 350 nm while for Swiss 3T3 cells rp G 120 nm.

The data shown in Figure 3 reflect a universal trend of viscositydependence on hydrodynamic radius of the probe. The dataobtained in the cytoplasm of mammalian cells are different fromthose in polymer or in surfactant solutions in one respect: in thecytoplasm both ξ and Rh are fixed. In the polymer or in thesurfactant solution we vary ξ by changing the concentration ofthe polymer coils or surfactant micelles. Rh is controlled bychanges of the molecular weight of the polymer. We representdata for complex liquids (polymers and micelles) in the mostgeneral form shown in Figure 2. The scaling law (eq 7) is basedon the ratio Reff/ξ. Therefore correct determination of thescaling form may be achieved by use one of two different ways.One of them is to change the hydrodynamic radius of the probeunder ξ = constant. This is the method used for mammalian cells,because by definition, ξ is fixed. Another way is to change themolecular weight of the polymer, hydrodynamic radius of theprobe, and the concentration of the polymer or of the micelles.This method was used for polymer and surfactant solutions.

For big probes (rp. Rh) changes in the viscosity experiencedby the probe are negligible when two probes of different sizes arecompared. However in order to obtain precise results in biolo-gical systems, measurements for all length scales (rp < ξ, ξ < rp <Rh, rp.Rh) are needed. This is not the case for synthetic systemswhere values of Reff/ξ can be controlled by all variables (rp, ξ, andRh). Furthermore measurements at the nanoscale for complexliquids might be not necessary in order to determine the scalingformula. In particular one canmeasure the macroscopic viscosity,and whole needed range of Reff/ξ values can be covered bychanges of the concentration only.

Our work is complementary to the work of Arcizet et al.,48

who proposed to probe the cytoplasm at the micrometer scaleand distinguished between active and passive transport.49 In ourwork we probed hydrodynamic radii below 100 nm. Onecan propose an experiment to probe all length scales (from nanoto macro) and to determine, for investigated cell strain, scalingof viscosity across all length scales. Such experiment shouldbe based on single molecule measurements (e.g., FCS) to probenanoscopic length scale and on single particle tracking48 toprobe microscopic scale. Additionally submicrometer lengthscale (>100 nm) can be probed with the technique describedby Wirtz.50 In all techniques however, at which hydrodynamicradius of the probe is smaller than a few hundred nanometers,results should be analyzed with scale dependent viscosity takeninto account.

In our model we did not consider viscoelastic effects. Wesuppose that when the probe size exceeds a certain value,elasticity of the cytoplasm may affect the probe’s motion. Theinterior of a cell is a gel, and below a certain length scale, motionis controlled by purely viscous effects.27 We also did not account

for possible effects of electrostatic charge that may also influencethe diffusion of microscopic probes.51 We performed our mea-surements on synthetic systems for nonionic polymers andsurfactants which eliminate the problem of electrostatic interac-tions between the probe and the matrix. Additionally in livingcells charge should also not influence the diffusion of the probes,because the Debye’s screening length for living cells is short (lessthan 1 nm),52 significantly below the value of ξ.

In summary we showed that viscosity depends strongly on thehydrodynamic radius of the probe (the scale of flow). Thereforeall measurements of viscosity, diffusion, or electrophoreticalmobility (via dynamic light scattering, capillary electrophoresis,chromatography, and many others) in complex liquids should betreated very carefully when the probes are nanoscopic. The SSEequation can be applied to complex liquids at all length scales,with a proper account for the viscosity dependence on Reff (eq 8).Additionally viscosity experienced by the probes of a given rp in asolution comprising obstacles of Rh = rp (polymer in polymers ormicelle in micellar solution) is lower than the macroscopicviscosity of the solution. Furthermore if the scaling form of theviscosity is known in a given system, one can use the SSEequation with the scale dependent viscosity to predict the self-diffusion coefficient for any probe of known hydrodynamicradius. As far as we know, this is the first model of viscosity thattakes into account and unites both the relationship between theprobe and the obstacles (Reff, eq 8) as well as the structuralproperties of the matrix (ξ(Rg,c) or ξ(L,c)), introduced byLangevin and Rondelez.53 In addition, our results show thatusing simple self-diffusion measurements performed in livingcells for a series of probes, it is possible to estimate such variablesas the matrix and the macroscopic viscosity. Our investigations ofviscosity of cytoplasm of living cells should be followed by furtherstudies to extend our understanding of the internal structureof cells.

’ASSOCIATED CONTENT

bS Supporting Information. Materials and methods anddescription of the experimental details. This material is availablefree of charge via the Internet at http://pubs.acs.org/.

’AUTHOR INFORMATION

Corresponding Author*E-mail: [email protected].

Present Addresses^EMBL Heidelberg

’ACKNOWLEDGMENT

The work was supported by the Project operated within theFoundation for Polish Science Team Programme cofinanced bythe EU “European Regional Development Fund’’ TEAM/2008-2/2 and from the budget of the Ministry of Science and HigherEducation 2007-2010. The research was partially supported bythe European Union within European Regional DevelopmentFund, through grant Innovative Economy (POIG.02.02.00-00-025/09). The research was also supported by a research grantfrom the Human Frontier Science Program Organization. Weacknowledge fruitful discussion with Piotr Garstecki. N.Z.


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acknowledges a Ph.D. scholarship from the President of thePolish Academy of Sciences.

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