Quantifying the Influence of the Crowded Cytoplasm on SmallMolecule DiffusionPeter M. Kekenes-Huskey,* Caitlin E. Scott, and Selcuk Atalay*

Department of Chemistry, University of Kentucky, Lexington, Kentucky 40506, United States

*S Supporting Information

ABSTRACT: Cytosolic crowding can influence the thermodynamics and kinetics of invivo chemical reactions. Most significantly, proteins and nucleic acid crowders reduce theaccessible volume fraction, ϕ, available to a diffusing substrate, thereby reducing itseffective diffusion rate, Deff, relative to its rate in bulk solution. However, Deff can be furtherhindered or even enhanced, when long-range crowder/diffuser interactions are significant.To probe these effects, we numerically estimated Deff values for small, charged molecules inrepresentative, cytosolic protein lattices up to 0.1 × 0.1 × 0.1 μm3 in volume via thehomogenized Smoluchowski electro-diffusion equation. We further validated ourpredictions against Deff estimates from ϕ-dependent analytical relationships, such as theMaxwell−Garnett (MG) bound, as well as explicit solutions of the time-dependent electro-diffusion equation. We find that in typical, moderately crowded cell cytoplasm (ϕ ≈ 0.8),Deff is primarily determined by ϕ; in other words, diverse protein shapes andheterogeneous distributions only modestly impact Deff. However, electrostatic interactionsbetween diffusers and crowders, particularly at low electrolyte ionic strengths, can substantially modulate Deff. These findings helpdelineate the extent that cytoplasmic crowders influence small molecule diffusion, which ultimately may shape the efficiency andtiming of intracellular signaling pathways. More generally, the quantitative agreement between computationally expensivesolutions of the time-dependent electro-diffusion equation and its comparatively cheaper homogenized form suggest that thelatter is a broadly effective model for diffusion in wide-ranging, crowded biological media.

■ INTRODUCTION

Intracellular biochemical reactions commonly rely on thediffusion of charged, small molecules1 between regions wheresubstrates are stored or synthesized to where they areultimately utilized. These intracellular reactions can be stronglyinfluenced by effective diffusion rates of their substrates,2 whichmay be depressed by up to an order of magnitude relative tounrestricted diffusion in bulk solution.3,4 In part, protein-,carbohydrate-, and nucleic acid−based “crowders” restrict theintracellular volume accessible to diffusing substrates andthereby reduce their rates of transport (reviewed here5).However, additional factors beyond volume exclusion caninfluence the mobility of molecular diffusers. These factorsinclude the distribution and activity of proteins or chargedsurfaces of organelles that selectively bind substrates, as well asthe substrate’s shape and charge.3,6−8 Improved quantitativepredictions that delineate the relative contributions of thesefactors on shaping biomolecule diffusion is a challenging, butnecessary, endeavor for understanding biochemical reactionsand signaling in vitro and in vivo.9−11

A wide range of experimental and simulation approacheshave provided considerable insight into the effective diffusionrates of small molecules involved in intracellular signaling (seereviews3,12). Methods such as fluorescence or raster imagecorrelation spectroscopy,13,14 and nuclear magnetic reso-nance15−17 have yielded refined estimates for diffusion ratesof small molecules such as calcium, magnesium, and AMP in

crowded intracellular media. A common theme emerging fromthese studies is that small molecule diffusion rates aresubstantially smaller than measurements in crowder-freesolutions and are frequently anisotropic, owing to theorganization of intracellular structures, such as actin filamentlattices. In part, these reductions in diffusion rates can beattributed to a reduced free volume fraction,3 although thenature of substrate/crowder interactions can substantiallystrongly modulate the effective rate.6 However, given theconsiderable variations in the sizes, molecular composition, andelectrostatic properties of cytoplasmic crowders, isolating thecontributions of each factor via experimental means ischallenging.9

In this regard, computational modeling is a strong comple-ment to experimental techniques for describing the molecularunderpinnings of experimentally observed diffusion rates andtheir influence on intracellular signaling.9,18 A variety ofcomputational approaches for modeling diffusion in crowdeddomains has been reported in the literature.18 The mostcommon of which are based on explicit, particle-basedrepresentations of the diffuser or both diffuser and crowders,lattice-based models that assume discrete, finite-length hops for

Special Issue: J. Andrew McCammon Festschrift

Received: April 16, 2016Revised: June 18, 2016

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© XXXX American Chemical Society A DOI: 10.1021/acs.jpcb.6b03887J. Phys. Chem. B XXXX, XXX, XXX−XXX

the diffuser, or spatially continuous differential descriptions ofthe diffuser and its diffusion domain. These systems areprominently modeled as either deterministic or stochasticprocesses,18 or combinations thereof.19 Among these includeparticle-based approaches such as molecular dynamics20−22 andBrownian dynamics6,7,23−25 that explicitly represent thegeometry of the diffuser and its crowders. Their computationalcomplexity, however, generally limit their applications tonanometer spatial and microsecond temporal scales that areinsufficient for describing subcellular transport. On the otherhand, particle-based methods that represent diffusers as points,including Smoldyn26 and MCell27 scale well to longer timescales and to greater spatial extents appropriate for subcellularphenomena. Point representations, however, clearly sacrificedetails of crowder shape and crowder/diffuser interactions thatare known to influence transport and reaction ki-netics.18,24,28−31

As a compromise between these two extremes, partialdifferential equation (PDE)-based continuous diffusion modelsenable the preservation of atomistic details of crowder shapeand composition, as well as nonbound interactions (likeelectrostatic forces32) with an implicitly represented diffus-er.33−35 Such PDE approaches are most appropriate for acontinuum of diffusing species, such as ions and nucleotides,that are fairly concentrated, significantly smaller than typicalcellular crowders (100 kDa19), and do not substantiallyinfluence the local electrostatic potential of the medium. Inearlier works, we have applied such continuum models toquantify the influence of molecular crowding around enzymesand electrostatic interactions on biochemical reaction rates andefficiency.29,30,36,37 Nevertheless, the computational expensegrows exponentially with increasing number of explicitlyrepresented crowders, which renders these approachesimpractical for the nanometer length scales relevant tosubcellular transport phenomena.To reduce this computational burden, multiscale theories

such as “homogenization” can facilitate the extrapolation ofmolecular-scale information onto transport processes occurringover submicrometer and longer length scales.38−41 In essence,homogenization theory posits that a continuum transportprocess occurs on two decoupled spatial scales: a microscopicscale within which the system is in steady-state, and a multifoldlarger macroscopic scale. This leads to a modified PDE(described in the Methods section) that is solved on the

microscopic scale, which ultimately yields an effective diffusiontensor that is appropriate for transport on the macroscopicscale. We and others have shown that homogenized PDEmodels applied to ionic and nucleotide diffusion in structurallydetailed, intracellular environments are in excellent agreementwith experimentally measured trends in hindered diffusion ratesand anisotropy.8,42−48

To our knowledge, applications of homogenization theory tobiomolecular systems has been limited to a single or a smallnumber of crowder types, thus the challenge remains tounderstand transport in highly heterogeneous systems withvariable protein sizes, positions, and charge distributions.5 Inthis study, we therefore consider the diffusion of a small,anionic molecule (such as adenosine monophosphate (AMP))in an immobile, submicrometer scale lattice of commoncytoplasmic proteins (Figure 1), for which we precisely controlthe crowder distribution and charge density, as well as theelectrolyte concentration. Specifically, we used realistic three-dimensional geometries of the cytoplasm, as well as two-dimensional (2D) and three-dimensional (3D) geometriescomprising charged, spherical inclusions to unravel moleculardeterminants of small molecule diffusion rates. Our modelingresults for the realistic and 2D/3D simplified geometries areconsistent with prior studies,43,49,50 in that decreasing accessiblevolume progressively hinders diffusion, while long-rangeinteractions can further slow or accelerate diffusion (repulsiveversus attractive, respectively). Given that analogous trends areexhibited in both the 3D and 2D geometries, the majority ofour simulations are done using the latter geometries, namely forrigorously examining the effects of ionic strength, surfacecharge potential and heterogeneous crowder shapes anddistributions. Overall, our simulations offer quantitative insightinto the effective diffusion rates of small biomolecules inheterogeneous cytoplasmic fractions, thereby providing im-portant guidelines for electrokinetic transport in confined51−53

or crowded media.

■ METHODS

Structural Models of the Crowded Cytoplasm. Themodel cytoplasm considered in this study was based onBrownian dynamics (BD) trajectory snapshots of commoncytosolic proteins and nucleic acids in E. coli by McGuffee etal.24 (see Figure 2). We assume lattice proteins are immobilerelative to the rapidly diffusing small molecule, given the large

Figure 1. Configurations of domains used in this study. (a) Diffusion of charged particle (red spheres) in the absence of crowders (black spheres).Steady-state concentration gradient is indicated by higher concentration of diffusers (red background) at origin x = 0 and lower concentrations (bluebackground) at x = L. (b) Same as panel a, but with neutrally charged diffuser (black spheres) (c) Same as panel b, except crowders are assignedrepulsive electrostatic potentials.

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difference in diffusion constants (e.g., diffusion constant forAMP is at least 10-fold greater than the median for E. colicytoplasmic proteins54). Our numerical approach (see Support-ing Information) uses three-dimensional finite element meshesbased on these BD data, which were constructed by (1)exporting their solvent-accessible molecular surfaces fromVMD,55 (2) meshlab56 refinement to remove defects, (3)tetrahedralization via tetgen,57 (4) conversion to.xml format forcompatibility with the PDE solver FEniCS58 via DOLFIN-CONVERT. Details of this procedure are available in theSupporting Information (Volumetric Mesh Generation forCytoplasm Model section). The final result was a finite elementmesh of the representative volume element (RVE) representingthe aqueous phase between the crowders, of which an exampleis shown in Figure 4a.We additionally created lattices of cylindrical (2D) or

spherical (3D) primitives via GMSH57 to afford more control

over testing the influence of crowder shape, size, and surfacecharge on diffusion rates. The size distribution were chosen tobest approximate those of the original cytoplasm model, basedon our characterization of the crowder radius of gyration (Rg)(see Figure S2), for which the average Rg was approximately 30Å. The appropriate accessible volume fraction (accessible to thediffusing species), ϕ, was determined based on simulation datafrom McGuffee et al.24 that used 1008 proteins within an 808 ×808 × 808 Å3 box, or ϕ = 0.78. We note that our definition of ϕas the accessible volume fraction is consistent with59 and ourearlier publications,8,45 although in other contexts the volumefraction is based on the inaccessible fraction (e.g., 1 − ϕ).On the basis of a perfect simple cubic arrangement of Rg = 30

Å crowders with a center of mass distance, dcom, of 80.6 Åwould yield ϕ = 0.78, which is consistent with crowdingestimates in E. coli.4 Given the computational expense inevaluating large numbers of three-dimensional finite elementmeshes, we additionally defined two-dimensional referencesystems using dcom = 80.6 Å (based on surface distances forspherical crowders) and dcom = 113.0 Å for volume fractions ofϕ = 0.57 and ϕ = 0.78, respectively. We consider both volumefractions in order to examine the potential influences ofintercrowder distances and crowder volume fractions analogousto the 3D system. The intercrowder distance, H, is related tothe distance between centers of mass (e.g., dCOM = H + 2Rg);while dcom may be a more familiar metric, H is better suited forassessing the range of electrostatic interactions betweencharged protein surfaces. Where indicated in the text, we

additionally randomize the protein size and position inaccordance with Figure S2, based on a Monte Carlo protocol(see Supporting Information, the Monte Carlo Protocol forGenerating Crowder Ensembles section). We additionallyassumed crowders have boundary potentials of ψ0 = ±19.2[mV], based on ζ potential measurements ranging from 0.5 to0.75 kBT for bovine serum albumin in 150 mM potassiumchloride (KCl) and approximately neutral pH.60 Theseparameters are summarized in Table S1.

Diffusion Models. Our numerical approach is based on atime-dependent diffusion model governing the diffuserconcentration c(x, t) on a domain Ωm,

∂∂

= −∇· ∇ Ωct

D c on m (1)

where D is the small molecule diffusion constant. To reflect theinfluence of an electrostatic potential of mean force acting onthe diffuser, we formulate the concentration function as aSmoluchowski equation such that

∂∂

= −∇· ∇ Ω

= −∇· ̃∇ ̃ Ω

β ψ β ψ−ct

De e c

D c

( ) on (2)

on (3)

q qm

m

where q is the diffuser charge and ψ is the electrostaticpotential. β ≔ 1/kBT where kB is the Boltzmann constant, andT is the temperature. The second line reflects the Slotboomform using D̃(y) ≔ D(y) e−βqψ(y) and c(̃y) ≔ eβqψ(y)c(y). Theevaluation of the electrostatic potential arising from chargedcrowders is described in the Supporting Information, Poisson−Boltzmann Model for the Electrostatic Potential.Direct numerical simulation of eq 2 can yield effective

diffusion parameters, as we describe in Sec. Estimatingeffective diffusion rate from continuum diffusion simula-tions; in practice, however, this entails evaluating the PDE overa considerable number of time intervals, which can beprohibitively expensive. Therefore, we propose solving a‘homogenized’ steady-state problem that directly yields effectivediffusion constants, albeit at a computational cost that can be atleast an order of magnitude smaller. The two-scale homoge-nization approach39,61 applied to eq 2 assumes that Ωm has twolength scales of interest, x and y ≔ x/ϵ where the constant ϵ isassumed to be small compared to the dimensions of Ωm, andthat Ωm is periodic in y where y corresponds to a nanometerlength-scale, whereas x corresponds to submicrometer orlonger. The y-scale periodic unit, otherwise known as theRVE, is denoted by Ω, within which the accessible region isdefined as Ωϵ. This relationship is illustrated in Figure 2b,where the RVE Ω (colored) is a subset of the macroscopicdomain Ωm (gray). We further assume that D is y-periodic, suchthat D(x, y) can be written as Dϵ(y).Before continuing, we briefly discuss the implications of this

assumption. Namely, assuming “microscopic” periodicity allowsus to represent the influence of interacting crowders ondiffusion implicitly through an effective diffusion coefficient,Deff. In practice, time-dependent and spatially dependentsimulations of intracellular signaling in cell-shaped geometriescould then proceed using the Deff in eq 2, for which the Deffterm effectively “coarse-grains” the impact of crowders onsubcellular diffusion.Two-scale homogenization proceeds by rewriting eq 2 using

the expansion of c in powers of ϵ (i.e., c = ∑i ciϵi) and the

revised gradient operator (i.e., ∇ = ∂x + ϵ−1∂y). In previous

Figure 2. (a) 808 Å × 808 Å × 808 Å snapshot of crowded bacterialcytoplasm simulation from McGuffee et al.24 (b) Prototypicalrepresentative volume element (RVE) used for homogenizationrepresented in color.

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works,8,45 we demonstrated that the resulting equation yieldsthe homogeneous y-scale steady state problem: find χ such that

δχ

δχ

χ

∇· +∂∂

= Ω

+∂∂

· ̂ = Γ =∂Ω

∂∂

· ̂ = Γ Γ = ∂Ω ∂Ω

ϵϵ

ϵϵ ϵ

ϵϵ ϵ

⎛

⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟⎞

⎠⎟⎟

Dy

Dy

n

Dy

n

0, on

0, on :

0, on \ : \

ij jkk

j

ij jkk

j

ijk

j (4)

where Γϵ corresponds to a molecular boundary and Γ\Γϵ istypically the RVE boundary, while Dϵ is the small moleculediffusion rate within the RVE. Equation 4 is similar to eq 2, butincludes the Kronecker delta, δjk, which essentially couples themicroscopic and macroscopic coordinate systems. Givensolutions for χ, the effective diffusion tensor (applicable to eq2 on the macroscopic domain) is determined via

∫ δχ

=|Ω|

+∂∂Ω

ϵ

ϵ

⎛⎝⎜⎜

⎞⎠⎟⎟D D y

yy

1( ) dij ij jk

k

j (5)

where Ω is the RVE volume. In this study, we reportnormalized self-diffusion constants along the x-direction; for

example, Deff≡ ϵD

D y( )xx

xx, except where otherwise noted. The reader

may consult the references8,42,61,62 for additional details of thederivation. The PDEs defined in this study were numericallyevaluated via the finite element method using FEniCS,58 forwhich a piece-wise linear basis and the default direct linearsolver were used. Details of the numerical procedure areidentical to protocols outlined in refs 8 and 45. All code writtenin support of this publication is publicly available at https://bitbucket.org/pkhlab/pkh-lab-analyses. Simulation input filesand generated data are available upon request.

■ RESULTS AND DISCUSSIONDiffusion of a Neutral Diffuser in the Crowded

Cytoplasm. Model Validation for Two- and Three-Dimen-sional Domains. The homogenized Smoluchowski equation(HSE) enables the prediction of effective diffusion parametersbased on the crowder configurations in nano- to micro-scaledomains. We first demonstrate the accuracy of ourimplementation for a neutral diffuser in the two- and three-

dimensional crowder domains in Figure 3a,b by (1)approximating the effective diffusion constant using semi-analytical bounds and (2) explicitly solving the time-dependentdiffusion eq (eq 1) in the presence of crowders to obtain aneffective diffusion constant. Figure 3c depicts the normalizedeffective diffusion coefficients of a neutral diffuser predicted viaHSE (solid symbols) for two- and three-dimensional crowderdomains with accessible volume fractions ranging from ϕ =0.5−1.0. For ϕ →1.0, Deff approaches the (normalized) bulkrate at small occluded volume fractions. As the free volumefraction decreases toward ϕ = 0.5 with increasing crowderdensity, Deff monotonically decreases to roughly 30% and 40%of its bulk value for spherical and cylindrical inclusions,respectively. We further note that Deff values predicted forcylindrical inclusions tend to be slightly smaller (up toapproximately 25%) than those for spherical crowders;therefore our simulations using two-dimensional geometrieswill very modestly overestimate the effects of crowders on smallmolecular diffusion in analogous three-dimensional domains.Nevertheless, near typical intracellular volume fractions (ϕ ≈0.78), Deff values for both spherical and cylindrical inclusionsare comparable at roughly approximately 60−70% of the bulkdiffusion rates. These HSE predictions are consistent withestimates based on two semianalytical bounds for uniformlydistributed inclusions, the Hashin−Shtrikman (HS) upperbound63 for spherical crowders (solid line),

ϕϕ

=−

D2

(3 )HS(6)

and the MG bound for cylindrical particles (dashed line),64

ϕϕ

=−

D(2 )MG

(7)

where ϕ is the accessible volume fraction. Lastly, we findcomparable Deff predictions based on explicit time-dependentsimulations of the continuum diffusion equation (opensymbols, see Supporting Information section Deff fromContinuum Diffusion Simulations for more information anderror analyses). Given that Deff estimates for two- and three-dimensional lattices show similar dependencies on accessiblevolume fraction and are comparable in magnitude, subsequentanalyses will primarily use 2D geometries for reasons ofcomputational efficiency, except where otherwise indicated.

Influence of Protein Shape and Distribution on EffectiveDiffusion Rates for Neutral Diffusers. After validating our HSEmodel against primitive, uniformly distributed crowders, wenext determined Deff based on three-dimensional snapshots

Figure 3. Two-dimensional (a) and three-dimensional (b) simulations domains used for validating continuum and homogenized Smoluchowskiequation (HSE) models of hindered diffusion. (c) Comparison of 2D (red) and 3D (black) effective diffusion coefficient for HSE (solid symbol) andcontinuum model (open symbol) as a function of accessible volume fraction with appropriate analytical bounds for noninteracting crowders(Maxwell-Garnett (MG) for cylinders and Hashin-Shtrikman (HS) for spheres) are shown with dashed and solid lines, respectively.

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taken from published BD simulation data.24 We note that theHSE model permits a computationally inexpensive evaluationof Deff for a complex, crowder-laden cytoplasmic domain(several minutes on 16 processors), and to our knowledge, isthe first application of homogenization to a structurallydetailed, angstrom-resolution cytosol fraction. In Figure 4a,

we present the three-dimensional solution for χ (see eq 4);generally speaking, the reddish and bluish tones representregions where diffusion is more strongly hindered relative tothe grayish regions. In Figure 4b we report the effectivediffusion constants along the x, y, and z axes for RVE sizes of100−300 Å. The diffusion tensors are nearly isotropic, withranges from 0.40 and 0.55; such a result is not surprising, giventhat the cytoplasmic proteins are roughly spherical in shape andhave no preferential, anisotropic arrangement. Furthermore,our predictions are consistent with estimates indicating thattranslational diffusion rates are within an order of magnitude ofbulk diffusion rates for diffuser sizes < 500 kDa3,4 and closelymatch the Deff = 0.51 predicted by the HS bound. The Deffpredictions from these data have no obvious dependence onthe RVE size, which we attribute to small sample sizes (onecase for each cell length).To better understand the variance of effective diffusivity with

respect to crowder positions and sizes, we characterized the Rg(see Figure S2) of crowders reflected in the McGuffeecytoplasm data. The skewed distribution had an average Rg ofapproximately 30 Å, with prominent peaks at 15 and 30 Å,although some radii were as much as 80 Å in size. Given thesensitivity of the effective diffusion tensor to the free volumefraction and the difficulty in constructing atomistic-resolutionthree-dimensional meshes of complex topologies,65 wegenerated randomized, 2D crowder distributions (11 per celllength ranging from 100 to 2000 Å). Specifically, eachrandomized distribution was chosen such that their radii ofgyration well-approximated the statistics derived from the BDdata shown in Figure S2. Their randomized positions weregenerated using a Monte Carlo approach (see SupportingInformation section Monte Carlo Protocol for GeneratingCrowder Ensembles) that ensured crowders were nonoverlap-ping and within the RVE boundaries. We note that theaccessible volume fraction of the original three-dimensionalgeometry could be modulated by shrinking or enlarging thesolvent probe radius for generation of the Connolly solvent-accessible surface.66 However, there is considerable burden ofmanually revising commonly occurring meshing defects65 such

as overlapping and self-intersecting vertices. Moreover, ourMonte Carlo approach permits more exhaustive sampling ofcrowder locations and spatial configurations than would bepossible from the original Brownian dynamics trajectory datamade available by McGuffee et al.24 In Figure 5, we report the

mean Deff values as a function of RVE size for configurationswith randomized positions and radii (RandPos/RandRad) aswell as randomized positions and constant (Rg = 30 Å) radii(RandPos/ConstRad). We found that Deff ≈ 0.65 across RVElengths, regardless of whether randomized or fixed radii wereused. As expected for larger sample sizes, the standard error inthe mean Deff decreased with increasing RVE size. These datatherefore indicate that for neutral diffusers, (1) reliable Deffestimates may be obtained using relatively small RVE lengths(300 nm) provided sufficiently large sample sizes and (2) Deffestimates are apparently insensitive to crowder position andsize variations at cytosolic volume fractions (ψ ≈ 0.78). In otherwords, fine-resolution details of cytosolic globular proteins’three-dimensional shape are likely unnecessary for predictingtheir influence on diffusion rates of small molecules. Therefore,we expect that the deviations in Deff as a function of directionand cell size relative to the HS bound in Figure 4 were due tosmall sample sizes, as opposed to variations in shape. We clearlyanticipate exceptions to this trend if the protein positions orshapes are strongly anisotropic, such as for myosin/actinfilaments in contractile cells,67 especially if the proteins stronglyinteract with the diffuser (examined in the next section).

Diffusion of a Charged Molecule in the CrowdedCytoplasm. Effects of Protein Charge on Electric Potential.The electrostatic potential arising from surface-exposed polaramino acids is known to influence protein−protein andprotein−ligand association thermodynamics and kinetics.68−70

Therefore, it can be expected that the electrostatic interactionsof a small, charged diffuser with densely packed, immobileproteins comprising the cytoplasm would influence theireffective diffusion rate, although the contributions of (1)typical protein surface potentials and (2) protein distribution inlarge submicrometer cytoplasmic fractions has not been well-explored. Before investigating these contributions, we firstpredict the electrostatic potential from distributions of charged2D crowders by numerical solution of the Poisson−Boltzmann(PB) equation. Further details on the determination of theelectrostatic potential and assignment of crowder surfacepotential can be found in the Supporting Information sectionEstimating Effective Diffusion Rate from Continuum DiffusionSimulations.

Figure 4. (a) Three-dimensional solution of homogenized diffusionequation for an uncharged diffuser (see eq 4) based on the angstrom-resolution cytoplasm data in Figure 2. (b) HSE-predicted effectivediffusion constants along the x (red), y (yellow), and z (black)directions as a function of RVE size (100, 200, and 300 Å) and thequantitative comparison of them with the analytical Hashin−Shtrikman (HS) upper bound (blue).

Figure 5. Predicted effective diffusion rates, Deff, for a neutral diffuserin crowded, two-dimensional RVEs with volume fraction ϕ = 0.78, as afunction of domain size (cell length). Geometries are derived from thecrowder distributions in Figure 2.

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In Figure 6, we predict the electric potentials arising from auniform distribution (ϕ = 0.78) of crowders with −19.2 mVsurface potentials subject to ionic strengths ranging from 5 to500 mM. The 2D surface plot electric potential distributionsare shown, as well as the spatial distribution of electric potentialbetween two neighboring particles in one dimension (seeFigure 6b). The ionic strength is reported as the dimensionlessquantity, κH, for which κ is the inverse Debye length, and H isthe edge-to-edge average distance between proteins.The magnitude of the electrostatic potential is clearly

maximal at the protein surface (ψ0 = −19.2 [mV]) andminimal midway between neighboring proteins. As would beexpected for strong electrolyte solutions, the decay of thepotential as a function of distance from the crowder increaseswith increasing ionic strength. At κH = 10.69 corresponding toa concentration of 500 mM, the electric potential rapidly decaysto zero within 4.3 Å of the crowders, thus the influence ofelectrostatic interactions are expected to be negligible withinmost of the accessible diffusion volume. However, as κHdecreases, significant overlap of potentials from neighboringcrowders is observed throughout the entire domain. Tounderstand the impact of this potential on the distribution ofcharged diffusers, we assume local equilibrium inside the RVE,which is commonly done to ensure validity of the constitutivemodel at the microscale despite nonequilibrium conditions atthe macroscale.40 In this regime, the Boltzmann relationc(x,zieψ) = c0 exp[−βzieψ] may be used to describe the diffuserconcentration, c, in the diffuse layer (DL) relative to the bulksolution (c0). Hence, electronegative DLs are predicted toexclude negatively charged diffusers, which is analogous toincreasing the crowders’ effective radii; in complement,attractive forces would localize positively charged diffuserstoward the charger, thereby reducing the crowders’ effectivesize. This interpretation holds for the weakly attractiveelectrostatic interaction energies assumed in this study (lessthan −1 to −2 kBT) and low diffuser concentrations relative tothe bulk electrolyte; beyond this regime, the diffuser would be

expected to compete with the bulk electrolyte and potentiallydecrease the Debye length.To better approximate the diverse electrostatic properties of

proteins comprising the cytoplasm, we also present electricpotentials corresponding to an electro-neutral system (nearlyequal numbers of positively and negatively charged crowders)for alternating (Figure S4) and randomly distributed crowders(Figure S5). For alternating charged crowders, ψ = 0 midwaybetween crowders (e.g., d = H/2). Despite the inversesymmetry of the electric potential across the midpoint,however, an asymmetric distribution of diffusers is expected,as c(r = 0, zieψ = 0.75kT) ≈ 0.47 and c(r = 0,zieψ = 0.75kT) ≈2.12 for diffusers with zi = 1 and zi = −1, respectively. Forrandomly distributed charged crowders (Figure S5), we observelocalized electropositive (red) and electronegative (blue)regions that we expect will give rise to strongly heterogeneousdiffuser distributions.

Effective Diffusion Rates As a Function of ChargedCrowder Distribution. To investigate the influence of chargedcrowders on small molecule diffusion, we evaluated the HSEsubject to the PB solutions from the previous section. Similar tothe neutral diffuser case in the previous section, we predictedDeff values for charged cylindrical and spherical crowders;however, since we were unaware of effective diffusion raterelationships that account for both electrostatic interactionstrength and volume fractions, we validated our HSEpredictions against the time-dependent Smoluchowski model(eq 2). Namely, in Figure 7 we compute the normalizedeffective diffusion coefficient for charged diffusers (z = −1, 0,+1) using a lattice of cylindrical crowders with varying volumefractions (ϕ = 0.5−1.0). For these cases, we assumed that thecrowders have uniform, negatively charged surface potentials of−19.2 mV and are immersed in a background 150 mM KClsolution (λD = 7.8 Å).Similar to our results assuming a neutral diffuser, the

normalized Deff decreases with reduced accessible volumefraction regardless of charge, although the rates of decrease arecharge-dependent. While the effective diffusion rates of the

Figure 6. Electric potential in (a) two dimensions and (b) one dimension between negatively charged crowders for various background KClconcentrations (5, 20, 150, and 500 mM, corresponding to a κH of 1.06, 2.14, 5.86, and 10.69, respectively), given a crowder edge spacing (H) of 5.2nm and ϵ = −0.75 kBT.

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neutral diffusers (z = 0, black squares in Figure 7) follow theMG analytical bound (red crosses), negatively charged diffusersexhibit slower diffusion (blue dots). In contrast, for attractiveinteractions (red triangles), the normalized diffusion coef-ficients exceed those of the uncharged and negatively chargeddiffusers. Moreover, the HSE results (hollow symbols) and thetime-dependent continuum models (line-symbols) are inquantitative agreement, which validates our use of the HSE inmodeling electrokinetic phenomena, albeit at a fraction of thecomputational expense. We attribute hindered diffusion underrepulsive interactions to the diffuser’s exclusion from thecrowders’ DLs, as illustrated in Figure 6. In contrast, thediffusional acceleration by weakly attractive interactionsobserved here and in related experiments7,23,29,30,43,71−73 hasbeen rationalized as the “smoothing out” of the chemicalpotential energy surface roughened by crowder-inducedexcluded-volume regions.7

We additionally examined the dependence of charged-mediated Deff values on the electrostatic interaction strength.In Figure 8a, Deff values are reported from solutions based onthe time-dependent diffusion equation, in which ψ isdetermined from the nonlinear (solid line) and linearized(dashed-line) PB equation, and based on the HSE (squaressymbols) using the linearized PB equation. We assumed for

these calculations a physiologically reasonable volume fraction(ϕ = 0.78) and a bulk electrolyte with λD = 7.8 Å. Thepredicted Deff values from these three approaches are inquantitative agreement, with the most hindered diffusion rates(0.54) reported at ψ = −25.6 mV, corresponding to anelectrostatic interaction energy of ϵ ≡ qψ = 1kBT, given z = −1for q ≡ ze; on the other hand, the fastest rates (0.81) werefound for ϵ → −1kBT. In the absence of electrostaticinteractions, ϵ = 0kBT, and we recovered the MG bound(Deff = 0.64).It is interesting to note that log(Deff) is nearly linear with

respect to the electrostatic interaction energy as shown in

Figure 8b for βγ= ϵψ′ψ=

log DD

( )

0. We found the greatest agree-

ment at low ionic strengths (see dashed-dot line for CKCl = 1mM), for which the spatial decay in the electrostatic potentialenergy, ϵ(r), and diffuser probability density, CAMP(r), weremostly linear (see Figure S7). Given this correspondence formodest electrostatic interaction energies, these data suggest asimple correction to the MG bound of the form.

ϕϕ

=−

βγϵD e2eff

(8)

where γ may be determined by a linear fit to the data in Figure8b. This expression is unlikely to hold for strongly attractiveinteractions (−qψ0 ≫ (2−3)kBT), which would result inadsorption of the diffuser and a corresponding reduction indiffusion rates.7 This topic is further discussed in theLimitations section.In Figure 8a we compared the HSE model (red square) with

continuum simulations (linearized and nonlinearized PBequations) for different surface potentials when λ = 7.85 Å,which corresponds to a 150 mM background electrolyteconcentration and ϕ = 0.78. Strong agreement for thepredictions from the HSE model and the time-dependentmodels are observed. We additionally present in Figure 8aeffective diffusion rates that were calculated using the HSEmodel for electro-neutral (nearly equal numbers of positively-and negatively charged crowders) crowder distributions asrepresentations of a typical crowded cytoplasm. For thesecalculations, we assumed alternating positive/negative crowders(diamond) as well as randomly charged (circle) or randomlycharged and randomly distributed crowders (triangle symbols).For alternating positive and negatively charged crowders, Deff

Figure 7. Predicted normalized diffusion coefficients based onnegatively charged crowders with volume fraction ϕ, similar to theconfiguration in Figure 6. Coefficients are computed using the HSE(open symbols), continuum (closed symbol-dashed line), andMaxwell-Garnett (MG) bound (cross-symbol). Diffuser charges of z= −1 (circle), z = 0 (square), and z = +1 (triangle) are given, assumingCKCl = 150 mM and ψ0 = −19.2 mV (0.75 kBT for q = 1) for thecrowders. The black dotted-line corresponds to the center of massdistance between crowders. We present an analogous plot for sphericalinclusions in Figure S6.

Figure 8. (Left) Normalized effective diffusion coefficient of the diffuser as a function of electrostatic potential energy, ϵ = qψ[kBT], where we haveassumed crowder surface potentials of |ψ| < 25.6 [mV] and z = −1 for q ≡ ze. Results based on the linearized and nonlinear PB equations arecompared with HSE when λ = 7.85 Å, which corresponds to 150 mM salt concentration and ϕ = 0.78. (Right) Logarithm of Deff from left figure isshown, in addition to a linear fit to Deff values predicted at 1 mM ionic strength, based on eq 8.

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G

appears to modestly increase (approximately 5%) for increasing|ϵ|. This may arise due to the stronger influence of attractiveinteractions on diffusion relative to repulsive interactions of thesame amplitude. This effect, however, is largely lost when thecrowder positions or distribution of positive versus negativecrowders are randomized; for these cases, the Deff coefficientsare nearly uniform for interaction strengths of |ϵ| ≤ 1kBT andcomparable to predictions for an uncharged diffuser.To understand the dependence of Deff on the bulk ionic

strength for uniformly charged crowders illustrated in theprevious section, we present HSE simulations for crowders inelectrolyte solutions ranging from dilute to concentrated. Forweak electrostatic interactions occurring predominantly for thinDLs (i.e., κH ≫ 10), Deff approaches the neutral diffuser limit(star symbol). As κH → 0, the electrostatic interactionsbetween crowders and the diffusion medium are shielded to alesser extent; as a result, the hindrance of diffusion due torepulsive crowder/diffuser interactions is intensified, partic-ularly as the accessible volume fraction is reduced. Similarly, theenhancement of diffusion due to attractive interactions is moreapparent at lower κH values. These trends are consistent withthe ionic strength dependence of diffusion rates reported inexperiment.71,72 In sharp contrast, for the randomizedensembles of negatively and positively charged crowdersintroduced in the previous section, Deff is almost entirelyindependent of κH, except at very low κH where minorenhancement is observed. Together, these data suggest that forrandomly distributed cytoplasmic crowders, the excludedvolume effect overwhelmingly determines the effective diffusionrate of small diffusing molecules; however, for regions with adisproportionate share of electropositive or electronegativecrowders, as might be expected for compacted DNA in thenucleus,74 effective diffusion rates may significantly deviatefrom estimates based on excluded volume alone.For the latter scenario, we propose a Deff relationship based

on an effective crowder radius determined by the Debye length,λ. For this approximation, we use the functional form of eq 7,but assume ϕ is dependent on λ as given by

ϕπ γλ

= −±⎛

⎝⎜⎞⎠⎟

R NL

1( )D

2

2(9)

where +γ and −γ are corrections used for repulsive andattractive interactions, respectively. In contrast to eq 8, which ismost accurate at low ionic strength (e.g., long Debye lengths),this approximation performs well at moderate to high ionic

strengths as shown in Figure 9a,b. The γ parameter scalesapproximately linearly with surface potential independent ofaccessible volume fraction as illustrated in Figure S9. Therevised analytical bound (line), eq 9, agrees well with thediffusion coefficient calculated using HSE for moderate to highκH when γ is 0.45 and 0.66 for repulsive and attractiveinteractions, respectively, and it is independent of the accessiblevolume fraction. The small deviation at ϕ = 0.717 when κH <2.5 can be attributed to strong DL overlap, for which eq 8 is amore appropriate bound.

Limitations. Our HSE simulations of small moleculediffusion through matrices of weakly charged cytosolic proteinsoffer insight into the influence of protein shape, size, charge,and density on transport within the cell. The fundamentalassumption for homogenization theory is that there exists anunderlying periodic structure. In biological media, thisassumption can only be satisfied approximately, as thedistribution of proteins is not strictly periodic. Nevertheless,it is apparent from this study and others45,75 that homoge-nization-based predictions of effective diffusion tensors are inexcellent agreement with experimental measurements. Weattribute this success to the fact that the position of eachinclusion does not strongly deviate from the “typical” unit cell,or similarly, the statistical distribution within a given unit cell isconsistent across all cells. In fact, this is the basis for ahomogenization scheme valid for nonperiodic materials,76

wherein it is assumed that periodic spherical inclusions areremapped to nonperiodic positions that remain strictlyconfined within the unit cell. Additionally, for examination ofsystems with periodicity-breaking defects, such as the presenceof intracellular organelles in a cytoplasmic protein lattice,periodic correctors for localized perturbations may beappropriate.77,78

There are a number of additional model improvements thatmay be considered, particularly for describing highly chargedsystems. For the crowder surface potentials considered in thisstudy (|ψ| ≤ 25 mV), the linearized PB was sufficient todescribe the diffuse layer about charged crowders as well astheir overlapping DLs. For more highly charged diffusers orcrowders (|ψ|≫ 25. mV), a numerical solution of the nonlinearPB will be necessary to adequately model the electricpotential.79 More importantly, stronger attractive electrostaticinteractions will introduce several complications that did notarise in our study. As illustrated by the Boltzmann equation, ρ =ρbulk exp(−βqψ), attractive interaction energies as low as 3kcal/mol are sufficient to increase the local diffuser

Figure 9. Normalized effective diffusion coefficient (via HSE) for (a) repulsive and (b) attractive electrostatic interactions as a function of κH andvarying crowder surface distances (H). Results for H = 2.57, 5.2, 9, or 14 nm are denoted with crosses, circles, diamond, triangles, and squares,respectively. Evaluation of MG bound for each volume fraction is represented by a pink star. Solid lines represent empirical fits based on “effective”volume fractions from eq 9 (see Figure S9).

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H

concentration by 2 orders of magnitude relative to bulk. For the1 mM AMP concentration assumed in our time-dependentmodeling, the local AMP concentration near crowders couldapproach that of the bulk KCl and would likely significantlyenhance electrostatic shielding. This contribution wouldmandate including both the diffuser concentration and bulkelectrolyte (KCl) in the determination of the electrostaticpotential; Poisson−Nernst−Planck (PNP) approaches,80

whereby the diffusion and PB equations are solved iterativelyand concurrently, are routinely used to describe suchphenomena. Here, protocols to homogenize the PNP wouldbe a natural extension of our current work.48 It may also beadvantageous to consider the finite sizes of electrolyte anddiffuser particles, which are well-known to attenuate apparentconcentrations of counterions along charged surfaces comparedto theories that assume point charge electrolytes.81−83 Alongthese lines, significant surface interactions would likely give riseto adsorption phenomena that may require the consideration ofadsorption isotherms when defining the model boundaryconditions.84 Consideration of additional surface interactions,such as hydrophobic interactions and colloidal aggregationforces, may be advantageous for larger diffusing species, and inmany cases could be accommodated by augmenting the qψterm of eq 2 with Lennard-Jones or DLVO potentials,79

respectively.We assumed that the diffuser was substantially smaller than

the crowders, which justified our use of immobile proteinlattices. Inherent in this assumption is that small moleculesdiffuse much faster than the crowders and thereby rapidlyachieve a local steady state before significant displacement ofthe crowders. A recent Brownian dynamics study from Putzel etal.7 examining particle diffusion among larger crowders was alsobased on this assumption; however, in their Supplement, theyalso reported effective diffusion rates for systems composedeither entirely or partially of mobile crowders that were 20% to40% higher than the fixed system. Hence, we anticipate fairlymodest increases in diffusivity under the assumption of mobilecrowders. However, for larger diffusers than the smallbiomolecules considered here, hydrodynamic effects andsubdiffusive behavior would likely demand a mobile crowdertreatment.24,25,85

■ CONCLUSIONSWe investigated the effective diffusion rates for small, chargedbiomolecules within crowded, submicrometer-scale cytoplasmdomains as functions of crowder shape, size, distribution,electrostatic interaction energies and ionic strength. Our HSEapproach recapitulated observations that neutral diffusersexhibit hindered diffusion as the accessible volume fraction isdecreased due to crowding. However, our studies furtherindicate that in the absence of electrostatic interactions,crowder shape, size, and distribution variation have minoreffects on the effective diffusion coefficient at physiological freevolume fractions (ϕ ≈ 0.8). Our method also demonstrates thatelectrostatic interactions between crowders and diffuserssubstantially alter Deffs in manners highly dependent onwhether interactions are attractive versus repulsive, the surfacepotential amplitude, and solvent ionic strength. Generally,media presenting exclusively repulsive interactions tend tohinder diffusion, whereas enhanced diffusion is observed forweakly attractive interactions. In the more likely crowdedconfiguration consisting of randomly distributed and randomlycharged crowders, our homogenization results indicate that

typical intracellular accessible volume fractions and crowderinteraction strengths are sufficient to reduce effective diffusionrates within an order of magnitude, which is in line withconventional estimates.3 Surprisingly, partitioning of like-charged crowders into subcellular domains can strongly perturbdiffusion in localized regions of the cell. This codependence ofDeff on ϕ and electrostatic interactions can be empiricallymodeled by slight modifications of the MG bound for neutralcrowders (see eq 8 and eq 9). In our opinion, this raises thepossibility that nature may exploit such perturbations to controlthe efficiency and kinetics of reactions in small, subcellularcompartments, as is postulated for intracellular nucleotide-,Na+- and Ca2+-dependent “nanodomain” signaling.62,86,87

Moreover, given the strong agreement between Deff estimatesfrom the time-dependent and homogenized electro-diffusionequations, the adaptability of the homogenized Smoluchowskiequation to arbitrary potentials of mean force (see eq 2) couldfacilitate examination of how additional prominent physicalphenomena, including adsorption and buffering,41 shapeintracellular communication in crowded, biological systems.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jpcb.6b03887.

Details on the methods including a table of simulationparameters and descriptions of Monte Carlo protocol,volumetric mesh generation, Poisson−Boltzmann model,and an effective diffusion rate estimation, resultspertaining to the effective diffusion rate, as well as ninefigures (PDF).

■ AUTHOR INFORMATIONCorresponding Authors*E-mail: pkekeneshuskey@uky.edu, (859) 257-4741.*E-mail: selcukatalay@uky.edu, (859) 218-5406.NotesThe authors declare no competing financial interest.All code written in support of this publication is publiclyavailable at https://bitbucket.org/pkhlab/pkh-lab-analyses.Simulation input files and generated data are available uponrequest.

■ ACKNOWLEDGMENTSPKH is indebted to Andy McCammon for his mentorship in allthings that diffuse and more importantly, for his foresight toapply these insights to wide-ranging problems in biophysics andphysiology. His rigor and creativity remain an inspiration asPKH launches his career. This work used the Extreme Scienceand Engineering Discovery Environment (XSEDE), which issupported by National Science Foundation Grant No. ACI-1053575.88 We are grateful to Adrian Elcock and his lab forsharing their cytoplasm simulation trajectories from ref 24 aswell as helpful suggestions.

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The Journal of Physical Chemistry B Article

DOI: 10.1021/acs.jpcb.6b03887J. Phys. Chem. B XXXX, XXX, XXX−XXX

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