Perspective: On the importance of hydrodynamic interactions in the subcellulardynamics of macromoleculesJeffrey Skolnick Citation: The Journal of Chemical Physics 145, 100901 (2016); doi: 10.1063/1.4962258 View online: http://dx.doi.org/10.1063/1.4962258 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/145/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Multi-scaled normal mode analysis method for dynamics simulation of protein-membrane complexes: A casestudy of potassium channel gating motion correlations J. Chem. Phys. 143, 134113 (2015); 10.1063/1.4932329 Calculating hydrodynamic interactions for membrane-embedded objects J. Chem. Phys. 141, 124711 (2014); 10.1063/1.4896180 Correlating anomalous diffusion with lipid bilayer membrane structure using single molecule tracking andatomic force microscopy J. Chem. Phys. 134, 215101 (2011); 10.1063/1.3596377 Shape transformation of lipid vesicles induced by diffusing macromolecules J. Chem. Phys. 134, 024110 (2011); 10.1063/1.3530069 Dynamical model of DNA-protein interaction: Effect of protein charge distribution and mechanical properties J. Chem. Phys. 131, 105102 (2009); 10.1063/1.3216104

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THE JOURNAL OF CHEMICAL PHYSICS 145, 100901 (2016)

Perspective: On the importance of hydrodynamic interactionsin the subcellular dynamics of macromolecules

Jeffrey Skolnicka)

Center for the Study of Systems Biology, School of Biology, Georgia Institute of Technology,950 Atlantic Dr., NW, Atlanta, Georgia 30332, USA

(Received 23 June 2016; accepted 1 August 2016; published online 8 September 2016)

An outstanding challenge in computational biophysics is the simulation of a living cell at moleculardetail. Over the past several years, using Stokesian dynamics, progress has been made in simulatingcoarse grained molecular models of the cytoplasm. Since macromolecules comprise 20%-40% ofthe volume of a cell, one would expect that steric interactions dominate macromolecular diffusion.However, the reduction in cellular diffusion rates relative to infinite dilution is due, roughly equally,to steric and hydrodynamic interactions, HI, with nonspecific attractive interactions likely playingrather a minor role. HI not only serve to slow down long time diffusion rates but also causea considerable reduction in the magnitude of the short time diffusion coefficient relative to thatat infinite dilution. More importantly, the long range contribution of the Rotne-Prager-Yamakawadiffusion tensor results in temporal and spatial correlations that persist up to microseconds andfor intermolecular distances on the order of protein radii. While HI slow down the bimolecularassociation rate in the early stages of lipid bilayer formation, they accelerate the rate of large scaleassembly of lipid aggregates. This is suggestive of an important role for HI in the self-assemblykinetics of large macromolecular complexes such as tubulin. Since HI are important, questions as towhether continuum models of HI are adequate as well as improved simulation methodologies that willmake simulations of more complex cellular processes practical need to be addressed. Nevertheless,the stage is set for the molecular simulations of ever more complex subcellular processes. Publishedby AIP Publishing. [http://dx.doi.org/10.1063/1.4962258]

I. INTRODUCTION

A grand challenge for computational biophysics in the21st century is to be able to accurately simulate the biologicalprocesses occurring in living cells. If this were possible, inaddition to providing fundamental insights into how cellswork and the role of molecular specificity versus promiscuityin dictating biochemical and physiological processes, onecould model the course of diseases including infections andcancers and suggest optimal and personalized treatments bysimulating a virtual model of the cell. Just as rocket enginesare now designed in silico,1 one could design cells to becomemolecular factories.2,3 Moreover, one could model the processof cellular evolution and possibly the emergence of livingsystems from their inanimate components. Of course, now,this is but a dream, but the past decade has seen the emergenceof promising early steps towards achieving this goal.4–13

At present, the most advanced methods do not attempt tosimulate all the molecules in a cell from first principlesbut rather adopt various kinetic schemes. For example,the pioneering E-cell project14–20 has seen success inthe metabolic modeling of human erythrocytes.21 Anotherpromising series of flux balance analysis models have beendeveloped by Luthey-Schulten and co-workers.22–24 Theyadopt a microscopic kinetics view of individual moleculesthat interact in a crowded cellular environment modeled as a

a)Email: skolnick@gatech.edu

discretized lattice, where individual macromolecules can hopbetween lattice sites. These studies are a molecular realizationof the phenomenological dynamic flux estimation approachesdeveloped by Voit et al.25–30 for the analysis of metabolic timeseries data. Voit reports a number of successes including theability to reproducing hitherto unexplained observations inglycolytic time course data under anaerobic conditions.

Despite the fact that all the requisite kinetic parametersare not known, kinetic modeling approaches are extremelyvaluable. They can provide many insights into cellularbehavior and are at the cutting edge in treating cellularcomplexity. However, these methods suffer from thedisadvantage that they have to assume a certain kinetic schemea priori and then examine its consequences. Thus, they requirethat the fundamental kinetic mechanisms be known in advance.In reality, this is not always the case, and unincluded emergentfeatures would not be discovered (e.g., in metabolic pathwaymodeling, suppose noncanonical pathways in aggregatecontribute a substantial component of the metabolic flux,or certain important correlated motions are ignored). Theyrely on the ability of kinetic modeler to identify all relevantprocesses and provide at least order of magnitude estimatesof the relevant kinetic parameters. Thus, there is the needfor complementary bottom up approaches that can identifythe essential fundamental processes involved in intracellularmacromolecular dynamics. It is the discussion of the state ofthe art of such bottom up, molecular simulations of subcellularand cellular processes that is the focus of this Perspective.

0021-9606/2016/145(10)/100901/10/$30.00 145, 100901-1 Published by AIP Publishing.

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100901-2 Jeffrey Skolnick J. Chem. Phys. 145, 100901 (2016)

A most striking feature of the interior of living cells isjust how crowded they are, with macromolecules occupyingfrom 20% to 40% of the total volume.31,32 Indeed, to afirst approximation, the cytoplasm closely resembles thepopulation density of Times Square on New Year’s Eve. In thecytosol, the average intermolecular distance between proteinsis of the order of their diameter. Indeed, in bacteria, yeast, andmammalian cells, recent mass spectrometric measurementssuggest that there are 2-4 × 106 proteins/µm3 (fL).33 Forsuch a situation, molecular crowding has been found toexert a significant influence on intracellular protein stability,enzymatic activity, and association kinetics.34–40 Similarly,one would expect that crowding should affect the dynamicsof macromolecular motion in the cytosol. Understanding suchelemental processes is necessary before far more complexsituations, e.g., reaction kinetics in cells, not to mentionphysiological processes could be understood from a bottomup perspective.

Diffusion is one of the most elemental and importantdynamical processes needed to describe the motion ofthe molecules themselves as well as their reaction rates.In the crowded environment of a cell, macromolecularmotion will be slowed down due to excluded volumeeffects between molecules. Over the past decade, thediffusion of macromolecules in the cytoplasm has beenexamined by a variety of experimental techniques includingsingle particle tracking,41 Fluorescence Recovery AfterPhotobleaching (FRAP),42 and Fluorescence CorrelationSpectroscopy (FCS).43–45 The diffusion coefficient of Greenfluorescent Protein (GFP) in E. coli measured by FRAP46,47

shows a reduction of about a factor of 10 relative to that atinfinite dilution in water. Thus, macromolecular motion withinthe cytosol is significantly reduced.

One possible cause of the reduction in diffusion constantis that the viscosity of the cytosol is dramatically less thanthat of water. However, this early view that the cytosol isgel-like is incorrect; rather its viscosity in eukaryotic cellsis quite close to that of pure water.48 Given the greatercomplexity of eukaryotic cells compared to prokaryotic cells,it is a reasonable expectation that this observation will holdfor prokaryotic cells as well. Thus, the difference in theviscosity of the cytosol relative to water cannot explain thereduction in macromolecular diffusion. Another possibilityconsistent with the high volume fraction of macromoleculesis steric repulsions between macromolecules. A key questionis whether excluded volume effects are sufficient or if otherphysical interactions are needed.

The early simulations of Ridgway49 and Roberts et al.,50

suggested that excluded volume effects alone could notaccount for the reduction in macromolecular diffusion. At firstglance, this is somewhat surprising given the very high volumefractions of macromolecules in the cytoplasm. To addressthese issues, in a pioneering series of studies, McGuffee andElcock4,5 performed Brownian dynamics (BD) simulations ofa model of the E. coli cytoplasm where protein molecules aredescribed at atomic detail. In Brownian dynamics, the solventis not explicitly treated, and inertial effects are ignored.51

Again, crowding/excluded volume was insufficient to describethe reduction in diffusion constant of GFP relative to infinite

dilution. Rather, they found that not only were electrostaticinteractions needed but also they had to significantly increasethe intermolecular van der Waals interaction energy to recoverthe reduced mobility. Obviously, the van der Waals parametercan be tuned to give whatever diffusion constant between zero(no motion) and the value at infinite dilution that one wants.The key question is whether such enhanced van der Waalsinteractions are physical or if there are there other interactionsat play.

Another effect that is well known both from the studyof the dynamics of polymers52 and colloidal suspensions ishydrodynamic interactions which describe the perturbationof the solvent flow around a given molecule due to thewake created by other molecules. For polymer dynamics,hydrodynamic interactions (HI) are very important. Forexample, they change the dependence of the diffusion constantof a polymer from D ∼ n in the absence of HI to D ∼ n−1/2,with n the number of beads (repeat units) in the polymer.52

Thus, intrachain dynamics can be dramatically modified byHI. Similarly, the role of HI in colloidal suspensions hasbeen studied by a variety of computational methodologiesincluding Stokesian dynamics (SD)53 (which includes bothHI and short range lubrication forces and is appropriate forlow Reynolds numbers54), lattice-Boltzmann,55 multiparticlecollision dynamics,56 and dissipative dynamics.57 Stokesiandynamics is attractive as it has been shown to reproducethe dynamical properties of high density colloids.58 However,with a few exceptions, its use has been mainly limited tomonodisperse systems.59,60 One problem associated with theuse of Stokesian dynamics is the scaling of the traditionalimplementation of the algorithm (see below),6,7 where thecomputational cost of each step is of order N3; here, N is thenumber of system particles. If one desires to simulate largesystems containing millions of proteins per fL, methods thatreduce the dependence of the scaling with N are needed.

In what follows, we discuss a series of simulations ofconcentrated macromolecular models of the cytosol designedto elucidate the key factors governing intracellular dynamics.Key questions we address in this Perspective are the following:(1) What is the state of the art of Brownian dynamics basedmethods to simulate crowded macromolecular solutions? (2)What level of detail is required to simulate translational androtational motion in crowded macromolecular environments?Specifically, under what conditions can proteins be treated asspheres and when does one need to treat them at atomicdetail? (3) What is the relative importance of excludedvolume interactions, long and short range (lubricationforces) hydrodynamic interactions, and nonspecific attractiveinteractions in cellular diffusion? (4) If HI are important, howdo they influence the short and long time dynamics of crowdedmacromolecular solutions? (5) Is there a distance scale, justlike in semidilute polymer solutions, where hydrodynamicinteractions are screened?61 (6) What is the distance scalewhen the granularity of the solvent is important and isa continuum model of the cytosol a good approximation?(7) How does the finite size of a cell affect intracellularmacromolecular motion? (8) What role do HI play in theassembly of lipid bilayers that comprise the cell membrane?Finally, we summarize the outlook for the future.

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100901-3 Jeffrey Skolnick J. Chem. Phys. 145, 100901 (2016)

II. SIMULATION METHODOLOGIES

There are a number of algorithms that are routinely usedto simulate a collection of Brownian particles. These aresummarized in Table I.

In what follows, we present a summary of the salientfeatures of these algorithms, where for simplicity only particletranslations and forces are considered, and rotations andtorques are ignored.64

A. Stokesian dynamics

The translational motion of N Brownian particles isrepresented by the Langevin equation53,65

mdudt= fH + fP + fB. (1)

Here, m is the 3N × 3N mass tensor, u is the particletranslational velocity vector of length 3N , and the forcesacting on the particles consist of the hydrodynamic forces,fH, the deterministic non-hydrodynamic forces, fP, thatinclude electrostatic and van der Waals contributions, andthe stochastic Brownian forces fB. All are vectors of length3N . For low Reynolds number and without bulk flow, thehydrodynamic forces are given by

fH = −Ru = −D−1u/kBT. (2a)

R (D) is the 3N × 3N configuration dependent hydrodynamicresistance (diffusion) tensor with

kBT R−1 = D. (2b)

The Brownian forces fB arising from thermal fluctuations arecharacterized by the fluctuation-dissipation theorem66

fB� = 0;

fB (0) fB (t)� = 2kBT Rδ (t) . (3)

The angle brackets denote the ensemble average, kB isBoltzmann’s constant, T is the absolute temperature, t istime, and δ(t) is the Dirac delta function.

The time evolution of the particle coordinates given byErmak and McCammon51 is obtained by integrating Eq. (1)over a time step ∆t larger than the inertial relaxation time(τB = m/6πηa, for a particle of radius a and mass m, and

solvent viscosity η) but smaller than the time over which theforces relax,

r (t + ∆t) = r (t) + kBT (∇ · D)∆t + DfP∆t + x(∆t). (4)

r is the 3N length position vector, x(∆t) is a randomdisplacement due to Brownian noise. x is calculated from amultivariate Gaussian distribution whose mean and covarianceare given by

⟨x(∆t)⟩ = 0; ⟨x (∆t) x (∆t)⟩=2D∆t. (5)

If z is a standard normal vector, then

x =√

2∆tBz, (6a)

where

D = BBT . (6b)

Any B that satisfies Eq. (6b) can be used in Eq. (6a). Thus,all techniques that solve Brownian dynamics require theprincipal square root or symmetric factorization of D. Thesimplest way of doing this is to use a Cholesky factorization,an O(N3) operation.6 This is the origin of the O(N3) scalingfor each time step mentioned above. Another alternativeintroduced by Fixman is to employ a Chebyshev polynomialapproximation.67 Recently, we suggested using Krylovsubspace methods that are related to the Chebyshev methodbut do not require eigenvalue estimates and scale roughly asO(N2).6 In that regard, a recent comparison of Chebyshevand Krylov subspace methods for polymer dynamics is ofnote.68

B. Hydrodynamic interactions

The matrix D contains the hydrodynamic interac-tions that are often implemented using the Rotne-Prager-Yamakawa, RPY,63 mobility tensor between the ith andjth spheres at a distance Rij defined by the followingequation:

TABLE I. Description of simulation algorithms for crowded macromolecular solutions.

Algorithm Description Methodology

SD: Stokesiandynamics

Includes both many-body far-field HI andthe near-field lubrication forces at lowReynolds number.54

Original SD algorithm requires an O(N 3)matrix inversion and Cholesky factorization ofdense matrices.

BD: Browniandynamics

Includes HI at the level of Oseen62 or RPYtensors.63

Algorithm developed by Ermak andMcCammon.51

FD: Freedraining

BD without HI; the diffusion matrix isdiagonal and constant.

Limiting case of BD.51

FLD: Far fieldversion of SD

Far-field hydrodynamic matrix isapproximated by a diagonal matrix, butlubrication forces are explicitly considered.

Developed in Ref. 7. The algorithm scalesO(N ).

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100901-4 Jeffrey Skolnick J. Chem. Phys. 145, 100901 (2016)

D (i, j) =

kBT8πη

*,

1Ri j

I +Ri jRi j

R3i j

+-+

2a2

3R3i j

*,I −

Ri jRi j

R3i j

+-

if Ri j > 2a

kBT6πηa

(I − 9

32Ri jI/a

)+

332a

Ri jRi j

Ri j

if Ri j ≤ 2a

. (7)

Here, a is the radius of each spherical particle, and I isthe 3 × 3 identity matrix. The RPY tensor for D is oftenused because it is positive definite even for overlappingparticles51,69,70 and ∇ · D = 0. It contains two-body and long-range contributions to the particle mobility up to order 1/r3. Fordilute systems, two-body interactions may suffice. However,for concentrated systems, far-field many-body HI as well asnear-field lubrication forces are needed. Thus, SD not onlyincludes the far-field HI but also the many-body and near-fieldHI contributions.53,54,71 Here, the resistance tensor R is thesum of the far-field HI and near-field lubrication interactionsgiven by

R =(D∞)kBT

−1

+ Rlub (8)

with

Rlub = R2B − R∞2B. (9)

D∞ is the contribution of many-body, far-field interactionsgiven by the RPY expression of Eq. (7). Rlub contains the near-field lubrication forces of the two-body HI. R2B represents theexact two-body HI, which includes both near-field and far-field interactions and is calculated using the exact two-bodysolution of Jeffrey and Onishi.72 R∞2B is the resistance tensorthat represents two-body, far-field interactions and is obtainedby inverting the two-body diffusion tensor matrix containingterms to order 1/r. The far-field part has already been includedin (D∞)

kBT

−1. To avoid over counting, we must subtract these

two-body interactions. This corrects Rlub.For periodic boundary conditions, D∞ can be estimated

by the Ewald summation of the RPY tensor.73 Due to the long-range nature of HI, use of the Ewald summation technique isoften necessary not only for accuracy but also for obtainingpositive definite matrices. D∞ is a dense matrix, and itsexplicit construction and inversion require O(N2) and O(N3)computations, respectively. Due to the fact that lubricationforces occur at short distances, Rlub is sparse, and using acut-off method, its construction requires O(N) operations.

III. SIMULATION OF MACROMOLECULAR MOTIONIN THE CYTOPLASM

A. Level of detail required to simulatethe E . coli cytoplasm

For a concentrated solution of polydisperse proteinmolecules designed to model the cytoplasm, if one is interestedin translational diffusion, what level of detail is required toreproduce the translational self-diffusion constant, D? For the

ith molecule, D is given by(ri(t) − ri(0))2= 6D(t)t. (10)

The long time diffusion constant, DL, is the limit of Eq. (10)at long times. Similarly, we define the short time diffusioncoefficient, DS, as the initial value of D. To address this issue,Ando and Skolnick8 initially performed Brownian dynamicssimulations in the absence of hydrodynamic interactions in amodel of the E. coli cytoplasm based on the data of Ridgwayet al.49 Thus, the observed size distribution and relativeconcentration of macromolecules in the E. coli cytoplasmare reproduced in the simulation. Both molecular-shapedand equivalent sphere systems were considered where onlysteric interactions were allowed. For each macromolecule, thediffusion constants at infinite dilution, D0, were calculated atatomic detail from rigid particle theory.74–76

Examination of the ratio of DL/D0 shows that theequivalent sphere and the atomic representations of themacromolecules are well correlated over the entire rangeof particle radii (20 Å-115 Å) for overall macromolecularconcentrations of 300 mg/ml and 350 mg/ml. Onlywhen exceptionally high concentrations of 400 mg/ml ofmacromolecule are present does the diffusivity of thespherical system become slightly lower than the atomicrepresentation. Thus, with respect to translational motion,treating the macromolecules by their equivalent sphere is agood approximation. This is important in that it enables verylarge number of particles in the cytoplasm to be simulated.We note that this conclusion likely does not hold for veryasymmetric molecules nor for rotational diffusion.10,77

B. Are steric interactions betweenmacromolecules dominant?

For probe GFP molecules, experiments find that DL/D0is in the range of 0.06-0.09.78,79 In the model system withjust steric interactions, simulated values of DL/D0 for GFPfrom molecule shaped and equivalent sphere systems in 300mg/ml and 350 mg/ml are 0.41 and 0.38, which is 4-5 timeslarger than experiment. Even for the molecular-shaped systemat 400 mg/ml, for GFP the simulations find that DL/D0 is still2 times larger than experiment. Thus, excluded volume effectsonly account for at most half of the reduction in the diffusionconstant in the cytosol. This result is consistent with the workof McGuffee and Elcock.5

C. Are hydrodynamic interactions important?

As shown in Figure 1, using full Stokesian dynamicswith steric and hydrodynamic interactions, for GFP, theBD simulation in 350 mg/ml (a value is consistent with

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100901-5 Jeffrey Skolnick J. Chem. Phys. 145, 100901 (2016)

FIG. 1. Reduction in the long-time diffusion constant, the relative diffusionconstant, DL/D0, as a function of radius in the nonspecific interaction modelwith attractions and steric repulsions (diamonds) compared to the modelwith steric repulsions and HI (solid circles). The solid triangles show theexperimentally observed DL/D0 values adapted from Figure 4B of Sourjiket al.81 who measured the diffusion of membrane and DNA binding proteinsin E. coli. The plus signs are the experimentally observed values of DL/D0for a series of engineered GFP multimers adapted from Figure 2 of Ref. 80.The concentration of macromolecules in the simulations is 350 mg/ml, andthe dashed line denotes the value for DL/D0 of GFP.

experiment31) with HI provides a predicted DL/D0 of 0.08.This result is in agreement with experiment: DL/D0 rangesfrom 0.09 in DH5α,46 0.07 in BL21(DE3),47 to 0.06 in theK-12 strain47 grown in rich medium. These results stronglysuggest that excluded volume effects and HI are the twomajor factors which decrease the intracellular diffusivity ofmacromolecules. As also shown in Figure 1, the predictedDL/D0 is in agreement with the experimentally observedvalues of DL/D0 for a series of engineered GFP multimersadapted from Figure 2 of Ref. 80. However, they do notagree with the experimental values obtained from Sourjiket al.81 who measured the diffusion constants of membraneand DNA binding proteins in E. coli. Interestingly, they findthat the mobility of DNA binding proteins depends on theirbinding specificity, indicative of the possible role of attractiveinteractions. In all cases, the composition of the simulatedcytosol remains fixed.

Qualitatively different results are seen for the shorttime diffusion constant depending on whether or not HIare included. For a system without HI, DS/D0 = 1. A givenmacromolecule does not know if it is in a concentratedmedium or not until it experiences collisions with othermacromolecules. In contrast, with HI, DS/D0 ranges from 0.2for particles with a radius of 20 Å to about 0.1 for particleswith a radius of 100 Å. This is an important signature ofthe presence of hydrodynamic interactions that has beenquantitatively confirmed in an elegant study of the self-diffusion of Bovine Serum Albumin (BSA) in a crowdedsolution at biologically relevant concentrations.82

D. Role of nonspecific attractive interactionsin subcellular diffusion

Another possible source of the reduction in diffusivity ofmacromolecules in the cytosol might be nonspecific attractiveinteractions as employed in Refs. 5 and 77. To further examinethis issue, a number of types of short range nonspecificattractive terms were introduced, and their effects on DL/D0explored. As shown in Figure 1, as might be expected,

larger particles having stronger attractive interactions (opendiamonds) experience a larger reduction in their long timediffusivity than is found when just HI with steric interactions(solid circles) are included. This is another qualitativedifference between systems where HI are important and onewhere attractive interactions dominate. Further experimentsare required to establish the relative importance of nonspecificattractive interactions by measuring the diffusion constants fordifferent size macromolecules. However, we do note that theexperiments of Sourjik et al. show an even greater reductionin mobility than that provided by the simulations. This mayreflect the fact that some proteins are in the membrane,whereas others might interact with DNA.

In that regard, real proteins mainly form complexeswith interactions across a relatively planar interface, withthe number of such distinct protein-protein interfaces on theorder of about 1000.83 While it was traditionally believed thatsuch interfaces comprised a minor fraction (roughly 1/3) ofa protein’s surface, recent work finds that the majority of aprotein’s surface (>75%) has the requisite geometry to engagein such nonspecific protein-protein interactions.84 However,in addition to having nonspecific attractive interactions, thereare residue specific interactions which may be repulsive,neutral, or attractive. The fact that most proteins do notstrongly interact (at least as assessed by yeast two hybridexperiments85) suggests that nonspecific attractive interactionsmight be expected to play a fairly minor role in dictatingmacromolecular dynamics.

E. Hydrodynamic interactions introduce long rangespatial and temporal correlations

When HI are included, the short time diffusion coefficientratio DS/D0 in concentrated solutions is macromolecular sizedependent. This suggests that the perturbation in solventmotion about a given macromolecule affects individualparticle dynamics and raises the question if it induces spatialand temporal correlations. To analyze this correlation, if any,we calculated

Cij (d0, t) = ��

ri (t) · r j (t)� δ(d0 − dij)� �(ri (t)) δ(d0 − dij)�2 ��r j (t)� δ(d0 − dij)�(11)

between particles i and j at a distance d0 at a time t.As shown in Figure 2, for pairs of GFP molecules andRNA polymerases, respectively, in a system with just stericrepulsions the correlations are weak and anti-correlated, withGFP experiencing stronger anti-correlations. In contrast, withHI both types of molecule pairs exhibit long lived temporalcorrelations up to at least 100 ns and spatial correlations upat least to 10 Å. We suspect that such correlations may playan important functional role in cellular processes. Finally, forthe nonspecific binding model, the smaller pair has very weakspatial and temporal correlations, while the larger pair ofmolecules has much stronger, long lived temporal correlationsthat persist out to 8 Å (that is, are spatially close to thesurface of the molecule), rather than at least 10 Å when HIare allowed.

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100901-6 Jeffrey Skolnick J. Chem. Phys. 145, 100901 (2016)

FIG. 2. Normalized pair correlation function from Eq. (11) averaged over pairs of GFP and RNA polymerase molecules for the three different simulation modelsat 300 mg/ml. The surface distance is the distance between the surfaces of the pair of proteins.

F. Comparison of hydrodynamic interactionsand nonspecific binding models of diffusion

We next summarize the differences when HI as opposedto nonspecific binding dominate the dynamics.

If HI dominate: If nonspecific binding dominates:• Decay like 1/r; lubrication forces

give repulsion on approach andattraction as pairs of moleculesmove apart.• Short time diffusion constant, DS,

is significantly reduced from D0.• DS/D0 depends on the molecule’s

radius.• Long time diffusion constant DL

has a much weaker dependenceon particle radius.• Spatial and temporal correlations

for all size macromolecules; weakdependence on particle size.

• Attractive interactions decay like1/r2 for spherical macromoleculesfilled with small van der Waalsspheres.• Short time diffusion constant is the

same as at infinite dilution, D0.• DS/D0 is independent of molecular

radius.• DL is strongly size dependent, with

long lived clusters formed withlarger macromolecules.• Significant, strongly radius depen-

dent spatial and temporalcorrelations.

G. Are hydrodynamic interactions in concentratedsolutions effectively screened?

Since the long range contribution to HI scales as 1/rbetween particles, brute force calculations scale as O(N3),with N the number of particles. This can be reducedto roughly O(N2) if Krylov subspace methods are used.6

Can the physics of concentrated macromolecular systems

help further reduce the scaling? For semidilute solutions ofrandom polymers, de Gennes has shown that HI are screenedbeyond a polymer concentration dependent length, much likeelectrostatic interactions in ionic solutions are screened athigher salt concentrations; in such a case, the free draining(FD) limit holds, and polymer dynamics becomes Rouse-like.61 However, other work suggested that hydrodynamicscreening does not occur for freely diffusing particles.86 Ifhydrodynamic interactions were strongly screened, then thefar field HI matrix becomes diagonal, and the RPY expressiongiven in Eq. (7) reduces to

D∞(i, i) = kBTI6πηFa

, (12)

where ηF is the “effective” far field viscosity, and a is theparticle radius. Since SD is being employed, and Rlub containsall higher terms, the lubrication term includes the reductionin effective viscosity. Thus, the uncorrected viscosity of thesolvent should be used for ηF in Eq. (12). This approximationhas been previously employed in fast SD algorithms forboth Brownian87–90 and non-Brownian particles91 and wastermed Fast Lubrication Dynamics, FLD by Bybee.89 Whilemonodisperse systems were originally used for algorithmdevelopment, Ando et al. considered the validity of FLD fora polydisperse concentrated solution that is a model of theE. coli cytoplasm. In their implementation, the computationalcost per time step is of O(N).7

The total neglect of all HI as provided by the freedraining (both long and short range HI are turned off)

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100901-7 Jeffrey Skolnick J. Chem. Phys. 145, 100901 (2016)

FIG. 3. Normalized pair correlation function, Ci j from Eq (11) for the 24 Å - 24 Å, GFP, and 66 Å - 66 Å, RNA polymerase pairs of molecules obtained fromSD, FLD, and FD simulations. The times and distances are normalized by a2/D0 and a, respectively.

approximation grossly overestimates the short and long timediffusion constants by a factor of 4-10. In contrast, the diagonalapproximation to D∞ overestimates the short time diffusionconstant DS by at most 12% for the smallest particles, witha similar overestimate of at most 15% for DL. Thus, weconclude that with respect to global diffusion at both shortand long times, the long distance component of hydrodynamicinteractions (see Eq. (7)) is essentially screened. However, asseen in Figure 3, FLD causes a dramatic loss in spatial andtemporal correlations, where time and surface distance arenormalized by a2/D0 and a, respectively. For SD simulations,correlated motions are obvious for both pairs. For small pairs(GFP) of particles, the correlated motions (Ci j > 0.2) rangeup to 2 in time and 0.75 in surface distance (correspondingto 110 ns and 18 Å), respectively. For large particle pairs, thecorrelation spans 12 in time and 0.4 in surface distance (timesof 14 µs and 26 Å, respectively). Ci j from FLD simulations aremuch smaller for SD and are intermediate between the originalSD and FD simulations. This underestimation is obvious forthe small particle pairs, as they can readily diffuse away fromeach other as compared to large particles, and their time tostay near to each other is relatively short. They are sensitiveto the absence of cross coupling terms of the far-field HI.This is a clear limitation of using a diagonal approximationfor D∞.

H. Effects of confinement on models of intracellularmacromolecular dynamics

Macromolecules in the cytosol are not in an infinitenor periodic system but rather they are confined within themembrane of the cell. How does such confinement affectintracellular macromolecular dynamics? Can we entirelyignore their effects including the modification of dynamicsin the neighborhood of the membrane wall? To address thesequestions, Chow and Skolnick92 simulated a suspension ofmonodisperse particles in a viscous fluid confined within aspherical shell composed of closely spaced particles that areconstrained by a harmonic potential to lie near the rest distanceof the spherical shell. For BD, confinement is achievedprimarily by steric interactions between the cytoplasmic andthe wall particles. For SD, confinement is primarily due tolubrication forces that prevent particles from overlapping. Inthis idealized model, the radii of all particles are the same(this does not qualitatively change the results, see below).Mimicking the cell interior, the volume fraction of cytoplasmicparticles is φ = 0.3.

Simulations were performed with 1000 cytoplasmparticles in a spherical shell with a radius of approximately14.4, where the radius of an individual particle is 1. Selectedsimulations were also done with 20 000 cytoplasm particles in

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a spherical shell with a radius of approximately 40.0 as wellas a system of 500 000 particles with a cell to particle ratio of119. They also considered a bidisperse system comprised of arelatively small number of particles of radius 4 surrounded bya sea of particles of radius 1, constructed such that the overallvolume fraction remains at 0.3. All results were qualitativelythe same for these different systems. BD and SD simulationswere also performed for periodic conditions to approximatean infinite, unconfined system. Some BD simulations werealso performed without HI, to examine the role played by HI.

Particles in the cell interior have qualitatively similarmotions to particles in a periodic box with no cell wall. Farenough from the wall, there are no long range correlations dueto the wall itself. However, there is an overall slowdown indiffusion due to confinement, which is likely dependent on cellradius. Nevertheless, the qualitative effects of motion in thecell interior can be effectively modeled as an infinite periodicsystem. Overall, the major effects of confinement are steric.However, the qualitative behavior of systems with and withoutHI is different. HI generate correlated behavior both far fromas well as near the cell wall. SD simulations further showsome qualitatively different features than BD simulations withHI, such as a slowdown in diffusion, a reduction in correlatedmotions of nearby particles, and structure in the velocityprofiles that persist farther into the interior of the cell. SDsimulations more accurately model the crowded interiors ofcells than BD simulations, because they account for lubricationforces.

The results of these simulations might have functionalimplications. For example, the fact that the model proteinsnear the model cell wall tend to diffuse along the wallis consistent with the conjecture that there is an increasein macromolecular concentration near the wall.93 Thesesimulations show that the cause is primarily steric and doesnot require any specific interactions for it to happen. Thetendency of protein molecules to diffuse along and remainlocalized near the membrane might give additional time forsignal transduction across a membrane to occur.

IV. IS THE DESCRIPTION OF HYDRODYNAMICINTERACTIONS IN STOKESIANDYNAMICS ACCURATE?

In an important study, Morrone et al.94 examinedthe spatial dependence of the friction coefficient in theBrownian limit for two nonpolar bodies. They find that thefriction coefficient deviates from continuum hydrodynamicpredictions at small separations and depends on the nature ofsolute-solvent interactions. For purely repulsive spheres, thefriction coefficient has a peak at the critical dewetting distanceand decreases as the intersolute region dries. For attractivesolutes, water is expelled by steric repulsions, and the effectsof solvent layering cause a non-monotonic dependence of thefriction coefficient on separation. These observations couldprovide for a better description of the lubrication forces atshort distances.

A very important complementary study95 compared theRotne Prager (RP) approach to HI with that obtained intwo and three body systems with many centers of friction

that accurately account for the many body nature of HI.Surprisingly, when a sufficient number of frictional elementsare included, at short distances, then the far field RPapproximation provides a description that is in quantitativeagreement with the essentially exact numerical scheme. Whilethis is a continuum solvent approach, it does suggest that acombination of modified lubrication forces (which in classicalSD non-physically diverge at short distances) and multiplefriction centers might provide an accurate description of HI.The disadvantage is the significantly increased computationalcost due to multiple centers of friction that might restrictthe size of tractable cellular simulations. Clearly, these issuesneed to be further addressed in the near future.

V. ROLE OF HYDRODYNAMIC INTERACTIONSIN THE KINETICS OF CELL MEMBRANE ASSEMBLY

As shown above, hydrodynamic interactions (HI) giverise to collective motions between molecules in concentratedsolutions. The next logical question is to explore their rolein the biological self-assembly of large scale molecularaggregates. For example, Ando and Skolnick12 examinedthe importance of HI on the kinetics of lipid membrane self-assembly. Using Brownian dynamics (BD) simulations, 1000coarse-grained lipid molecules in periodic simulation boxeswere allowed to assemble into stable bilayers in the presenceand absence of intermolecular HI. HI reduce the monomer-monomer association rate by 50%. In contrast, the rate ofassociation of lipid clusters is much faster in the presence ofintermolecular HI, with the membrane self-assembly rate 3-10times faster than that without intermolecular HI. These resultssuggest that HI greatly influence self-assembly kinetics, andthat simulations without HI will significantly underestimateassembly rates. Similar conclusions were reached by Li et al.regarding the importance of HI on the kinetics of assemblyof two plates of varying hydrophobicity.96 Thus, we suspectthat HI may play important roles in many other biologicallyimportant self-assembly processes.

VI. CONCLUDING REMARKS AND OUTLOOKFOR THE FUTURE

As mentioned in the Introduction, one of the outstandingcomputational challenges in the 21st century is the simulationof living cells at molecular detail. However, this is a non-trivial problem. Consider the distance and time scales thatare involved. Even for a single cell, one has to describeprocesses that range over 6-7 orders of magnitude in spaceand over 12 orders of magnitude in time and involve manymillions of macromolecules. Clearly, depending on what isbeing studied, different levels of molecular description areneeded to make the simulations practical and yet retainthe appropriate resolution. One of the most elementaldynamical processes is subcellular diffusion. If one cannotadequately represent diffusive motion, it is highly unlikelythat more complex biological phenomena will be accuratelydescribed. Thus, understanding the nature of macromoleculardiffusion in cells is clearly a necessary first step towards the

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developing the ability to simulate biological cells at moleculardetail. In this Perspective, in the context of Brownian andStokesian dynamics, we explored the relative importanceof steric interactions, nonspecific attractive interactions,and hydrodynamic interactions on macromolecular diffusiveprocesses within model cells.

Since 20%-40% of the total volume31,32 in a cell isoccupied by macromolecules, due to steric repulsions withother macromolecules, a slowdown in diffusion relative to thatat infinite dilution is anticipated. At first glance, one mightguess that this would be the dominant effect. Surprisingly, itonly accounts for roughly half of the observed reduction indiffusivity. Next, one might imagine that nonspecific attractiveinteractions between proteins might dominate. However, anumber of factors likely mitigate the importance of this effect.Proteins generally interact across quasi-planar interfacesthat have to be spatially and chemically complementary.In practice, for a pair of randomly selected proteins, weestimate that only a tiny fraction of randomly chosenprotein pairs (about 10−5) likely have strongly favorableinteractions consistent with such stereochemical interfacecomplementarity.97 Thus, we expect that such interactionsplay a minor role in subcellular dynamics. However, this is notto say that specific interactions between macromolecules arenot important, as of course they are. These specific interactingpartners might well be enhanced by subcellular localizationeffects. On the other hand, hydrodynamic interactions arefound to be as important as steric interactions in slowingdown both the long and short time diffusive processes relativeto infinite dilution. They induce cooperative motions betweenmolecules that persist on the order of microseconds and fordistances on the order of considerable fraction of a protein’sdiameter. HI are also predicted to play an accelerating role inthe assembly process of the lipid bilayer and probably in theassembly of large scale molecular aggregates such as tubulinand actin filaments. They might also play important rolesin subcellular signaling and in the dynamics of molecularmotors, but these effects need to be demonstrated.

Key to the continued success is the development ofimproved SD sampling methodologies. Since the diffusionmatrix is dense, naive implementation of SD scales asO(N3). Some promising, essentially exact methodologies thatdecrease the computational cost include Krylov subspacemethods that scale roughly as O(N2). To further reduce thescaling to O(N), a naive implementation of hydrodynamicscreening in SD was developed that simply truncates thelong range part of the RPY tensor. This approximationintroduces a rather minor error (<15%) in the calculationof the long and short time self-diffusion constants. However,it results in the loss of the dynamic correlations that arelikely to be functionally important. One possible way toaddress this loss of correlation is to employ various cutoffideas applied to treat electrostatic interactions in moleculardynamics simulations.98 Such issues need to be explored inthe near future.

What are the likely next steps for molecular simulationsof subcellular dynamics? One interesting process that cancertainly be addressed is the simulation of macromolecularmotion through the E. coli nucleoid, the bacterial region

containing chromosomal DNA. Similarly, while simulationssuggest that HI play a minor role in protein diffusion alongDNA in dilute solution (HI act to retard protein motion alongDNA by about 30% relative to the case where they areignored).13 In the nucleoloid, they may significantly modifythe motion of proteins (e.g., transcription factors) along DNA.Another area where collective motions could be important isin the mechanism of G-protein signaling across membranes.A third very promising area is to examine the role playedby HI in kinesin walking along tubulin in particular andin the mechanism of force generation by molecular motorsin general. The key is to identify processes whose generalcharacteristics are sufficiently robust that inaccuracies inthe molecular force field and implemented molecular coarsegraining have a minor effect on the qualitative behavior. Withthe advent of GPUs and the needed development of improvedsampling approaches, the outlook for simulating ever morecomplex subcellular process is extremely promising.

ACKNOWLEDGMENTS

The support for the writing of this perspective wasprovided by NIH Grant No. GM-118039. The author alsothanks his collaborators, Dr. Tadashi Ando, Dr. EdmundChow, and Dr. Mu Gao for many insightful discussions.

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