Diffusion within the Cytoplasm: A Mesoscale Model of Interacting Macromolecules

      Abstract

      Recent experiments carried out in the dense cytoplasm of living cells have highlighted the importance of proteome composition and nonspecific intermolecular interactions in regulating macromolecule diffusion and organization. Despite this, the dependence of diffusion-interaction on physicochemical properties such as the degree of poly-dispersity and the balance between steric repulsion and nonspecific attraction among macromolecules was not systematically addressed. In this work, we study the problem of diffusion-interaction in the bacterial cytoplasm, combining theory and experimental data to build a minimal coarse-grained representation of the cytoplasm, which also includes, for the first time to our knowledge, the nucleoid. With stochastic molecular-dynamics simulations of a virtual cytoplasm we are able to track the single biomolecule motion, sizing from 3 to 80 nm, on submillisecond-long trajectories. We demonstrate that the size dependence of diffusion coefficients, anomalous exponents, and the effective viscosity experienced by biomolecules in the cytoplasm is fine-tuned by the intermolecular interactions. Accounting only for excluded volume in these potentials gives a weaker size-dependence than that expected from experimental data. On the contrary, adding nonspecific attraction in the range of 1–10 thermal energy units produces a stronger variation of the transport properties at growing biopolymer sizes. Normal and anomalous diffusive regimes emerge straightforwardly from the combination of high macromolecular concentration, poly-dispersity, stochasticity, and weak nonspecific interactions. As a result, small biopolymers experience a viscous cytoplasm, while the motion of big ones is jammed because the entanglements produced by the network of interactions and the entropic effects caused by poly-dispersity are stronger.

      Introduction

      The cell cytoplasm is a very complex environment in which a large number of biopolymers and other molecules with variable sizes and chemical properties form a network of specific and promiscuous interactions (
      • Wirth A.J.
      • Gruebele M.
      Quinary protein structure and the consequences of crowding in living cells: leaving the test-tube behind.
      ). The influence of this network can be recognized at several levels: for instance, the high macromolecule concentrations (∼50–400 g/L) alter biopolymer folding, binding, and association rates (
      • Luby-Phelps K.
      Cytoarchitecture and physical properties of cytoplasm: volume, viscosity, diffusion, intracellular surface area.
      ) compared to dilute solution conditions, with net effects that promote either stabilization or destabilization of single structures and assemblies (
      • Wang Y.
      • Sarkar M.
      • Pielak G.J.
      • et al.
      Macromolecular crowding and protein stability.
      ). On the other hand, diffusional dynamics, necessary for protein encounter, is typically slowed down due to an increased effective viscosity. This effect is only partially explained by the volume that the surrounding macromolecules (crowders) exclude (
      • Ye Y.
      • Liu X.
      • Li C.
      • et al.
      19F NMR spectroscopy as a probe of cytoplasmic viscosity and weak protein interactions in living cells.
      ,
      • Zorrilla S.
      • Hink M.A.
      • Lillo M.P.
      • et al.
      Translational and rotational motions of proteins in a protein crowded environment.
      ). In any case, cellular function is influenced by crowders (
      • Zhou H.X.
      • Rivas G.
      • Minton A.P.
      Macromolecular crowding and confinement: biochemical, biophysical, and potential physiological consequences.
      ) in a way that is dependent on the proteome and cytoplasm compositions, biopolymer aggregation propensity, and solubility (
      • Levy E.D.
      • De S.
      • Teichmann S.A.
      Cellular crowding imposes global constraints on the chemistry and evolution of proteomes.
      ,
      • de Groot N.S.
      • Ventura S.
      Protein aggregation profile of the bacterial cytosol.
      ). A common conclusion is that biological crowding agents are not inert, like the most frequently used synthetic ones in experiments, and the inclusion of the above effects, is crucial to model a cell realistically (
      • Feig M.
      • Sugita Y.
      Reaching new levels of realism in modeling biological macromolecules in cellular environments.
      ).
      Phenomenologically, translational diffusion in cells is often observed to deviate from the normal behavior in dilute solution. In fact, diffusion can depend on the timescale, growing sublinearly or even superlinearly in the case of energy-driven processes, in which cases it is defined as “anomalous” (
      • Golding I.
      • Cox E.C.
      Physical nature of bacterial cytoplasm.
      ,
      • Höfling F.
      • Franosch T.
      Anomalous transport in the crowded world of biological cells.
      ). Normal and anomalous regimes can coexist in proportions that depend upon the cellular organization (
      • Kalwarczyk T.
      • Tabaka M.
      • Holyst R.
      Biologistics—diffusion coefficients for complete proteome of Escherichia coli.
      ,
      • Dix J.A.
      • Verkman A.S.
      Crowding effects on diffusion in solutions and cells.
      ). In addition, the measured diffusion coefficients are observed to decrease rapidly as the crowder size increases, not only as a result of the space occlusion but also because of biopolymer distribution and interactions (the latter including the solvent-mediated correlations or hydrodynamic interactions (
      • Elowitz M.B.
      • Surette M.G.
      • Leibler S.
      • et al.
      Protein mobility in the cytoplasm of Escherichia coli.
      ,
      • Wang Q.
      • Zhuravleva A.
      • Gierasch L.M.
      Exploring weak, transient protein-protein interactions in crowded in vivo environments by in-cell nuclear magnetic resonance spectroscopy.
      ,
      • Ando T.
      • Skolnick J.
      Crowding and hydrodynamic interactions likely dominate in vivo macromolecular motion.
      )).
      Although in many important cases specific interactions may dominate, nonspecific ones are indeed ubiquitous and frequent, owing to the high macromolecule density. These are generally weakly attractive, and include hydrophobicity, proportional to the area buried upon binding (
      • Eisenberg D.
      • McLachlan A.D.
      Solvation energy in protein folding and binding.
      ); dispersion forces (
      • Israelachvili J.
      Intermolecular and Surface Forces.
      ); ion-mediated charge-charge correlations (
      • Bloomfield V.A.
      DNA condensation by multivalent cations.
      ); screened electrostatics, fading at 5–7 Å from the biopolymer surfaces at physiological conditions; and transient hydrogen bonds (
      • Spitzer J.
      • Poolman B.
      The role of biomacromolecular crowding, ionic strength, and physicochemical gradients in the complexities of life’s emergence.
      ,
      • Gabdoulline R.R.
      • Wade R.C.
      On the protein-protein diffusional encounter complex.
      ). The above interactions coupled to passive diffusion (hereafter referred to as “diffusion-interaction”) drive the formation of transient encounter complexes before specific binding (
      • Bhattacharya A.
      • Kim Y.C.
      • Mittal J.
      Protein-protein interactions in a crowded environment.
      ), and thus contribute to the regulation of macromolecule organization and mobility.
      Atomistic molecular-dynamics simulations could, in principle, give a realistic representation of the cytoplasm by representing all atoms and solvent explicitly (
      • McGuffee S.R.
      • Elcock A.H.
      Diffusion, crowding and protein stability in a dynamic molecular model of the bacterial cytoplasm.
      ,
      • Mereghetti P.
      • Wade R.C.
      Atomic detail Brownian dynamics simulations of concentrated protein solutions with a mean field treatment of hydrodynamic interactions.
      ). However, the computational burden limits the accessible spatiotemporal scales, and discrepancies with experiments call into question the accuracy of atomistic force fields (
      • Andrews C.T.
      • Elcock A.H.
      Molecular dynamics simulations of highly crowded amino acid solutions: comparisons of eight different force field combinations with experiment and with each other.
      ). On the other hand, low-resolution or coarse-grained models allow simulations to reach the submicrosecond/submillisecond scales with modest resources (
      • Schöneberg J.
      • Noé F.
      READDY—a software for particle-based reaction-diffusion dynamics in crowded cellular environments.
      ), additionally preserving the accuracy, provided an ad hoc parameterization of the interactions compensates for the structural simplifications. Mesoscale coarse-grained models, treating entire macromolecules as single interacting centers (
      • Bicout D.J.
      • Field M.J.
      Stochastic dynamics simulations of macromolecular diffusion in a model of the cytoplasm of Escherichia coli.
      ,
      • Ridgway D.
      • Broderick G.
      • Ellison M.J.
      • et al.
      Coarse-grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm.
      ), can achieve these goals—and are amenable to be mixed with finer biopolymer representations to address the multiscale problems of diffusion and interaction (
      • Tozzini V.
      Multiscale modeling of proteins.
      ,
      • Trovato F.
      • Nifosí R.
      • Tozzini V.
      • et al.
      A minimalist model of protein diffusion and interactions: the green fluorescent protein within the cytoplasm.
      ,
      • Cheung M.S.
      Where soft matter meets living matter—protein structure, stability, and folding in the cell.
      ).
      In this article, a mesoscale (one-bead-per-biopolymer) model of the Escherichia coli cytoplasm is reported. Since in bacteria cytoskeletal elements (
      • Rudner D.Z.
      • Losick R.
      Protein subcellular localization in bacteria.
      ) and protein aggregates (
      • Coquel A.S.
      • Jacob J.P.
      • Berry H.
      • et al.
      Localization of protein aggregation in Escherichia coli is governed by diffusion and nucleoid macromolecular crowding effect.
      ) are scarce, and passive transport is believed to dominate, in our model macromolecules are represented as freely diffusing, weakly and nonspecifically interacting particles. As in previous works, the model cytoplasm is poly-disperse, but at variance with those, we also introduce the genetic material. The parameterization of the intermolecular interactions is the second element of novelty in this work. Prompted by growing experimental evidences on the importance of nonspecific intermolecular attraction in vivo (
      • Wang Y.
      • Sarkar M.
      • Pielak G.J.
      • et al.
      Macromolecular crowding and protein stability.
      ,
      • Sarkar M.
      • Smith A.E.
      • Pielak G.J.
      Impact of reconstituted cytosol on protein stability.
      ) we devise an integrative scheme to parameterize the intercrowders interactions accurately and include the most important biological effects. This approach combines microscopic physicochemical principles (the “bottom-up approach”) and the necessary macroscopic knowledge, i.e., diffusion coefficients and aggregation tendency (the “top-down approach”). An independent validation of the obtained optimal energies is performed comparing with experimentally determined intermolecular binding free energies. In addition, simple rules are used to make the interactions scalable as a function of the crowder size. Consequently, the model is automatically consistent with residue-level coarse-grained models and amenable to be extended to a multiscale representation, on a cell scale.
      The article is organized as follows: The model is first described for the E. coli cytoplasm, further indicating how to extend it to different systems. Then we describe the parameterization strategy and simulation results, showing that the inclusion of a carefully tuned weak attraction is necessary to reproduce the size-dependence and spreading around the average diffusion coefficients. The coexistence of normal and anomalous diffusive regimes will emerge straightforwardly from the combination of high macromolecular concentration, poly-dispersity, stochasticity, and weak nonspecific interactions.

      Methods

      Cytoplasm model

      Crowder description

      The model cytoplasm is poly-disperse with 12 different species of crowders, distributed according to the known protein abundances (
      • Ridgway D.
      • Broderick G.
      • Ellison M.J.
      • et al.
      Coarse-grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm.
      ), here defined by the mass distribution histogram in Fig. 1 a.
      Figure thumbnail gr1
      Figure 1Integrative design of the E. coli cytoplasm model. (a) The experimental mass distribution is used to build macromolecules with a mass-dependent radius R(M). (b) Each spherical particle is decorated with elementary beads (SpoB model) interacting as shown in panel (c). (d–e) Intermolecular potentials UIJ, resulting from the integration of the elementary interactions, hydrodynamic radius Rh, and collision frequency γ0, which constitute the inputs of the simulations of the cytoplasm model in panel (f). (g) Ratio of experimental diffusion coefficients in dilute solution and within the cytoplasm (green dots) which drives the refinement of the intermolecular potential parameters ϵ0,α0 evaluated from the SpoB model (gray dots and lines) to the optimal ones ϵ,α (green), in the case of identical crowders (h). (Gray and green lines in panel h) Fit to the data calculated with the SpoB model (see Eq. S2 in the ) and after the optimization (see Eq. S4 in the ). (Green line in panel g) Fit to Eq. S11 in the . To see this figure in color, go online.
      Although the nucleoid is a long and highly dynamic polymer, it is found condensed in the interior of the cell (when not dividing) as a result of supercoiling, macromolecular crowding, and nucleoid-associated protein binding (
      • Benza V.G.
      • Bassetti B.
      • Lagomarsino M.C.
      • et al.
      Physical descriptions of the bacterial nucleoid at large scales, and their biological implications.
      ,
      • Trovato F.
      • Tozzini V.
      Supercoiling and local denaturation of plasmids with a minimalist DNA model.
      ). Accordingly, and to simplify its complex internal dynamics, it is described as a compact object. The following scaling law is used to relate any object radius to its mass,
      Ri(M)=ci(ρi)M1/3,
      (1)


      with ci(ρi) = [3/(4πρi]1/3; and ρcrwd = 1.3 g/cm3 and ρnucl = 0.075 g/cm3 account for the different densities of proteins/nucleic acids and the nucleoid supramolecular assembly, respectively. The nucleoid mass is estimated from the relationship M = ρnuclV with V/Vcell ≃ 20%, and Vcell the total volume of the simulated cell (see the Supporting Material for details). Three nucleoid particles were used compatibly with the E. coli aspect ratio of 3. Molecular details are reported in Table S1 in the Supporting Material.

      Transport properties

      The simulations are performed using periodic boundary conditions and neglecting the cytoskeletal elements. The latter, in fact, are mostly found at the cytoplasm periphery (
      • Vendeville A.
      • Larivière D.
      • Fourmentin E.
      An inventory of the bacterial macromolecular components and their spatial organization.
      ), while the confining membrane effect is known to be relevant above the millisecond timescale (
      • Coquel A.S.
      • Jacob J.P.
      • Berry H.
      • et al.
      Localization of protein aggregation in Escherichia coli is governed by diffusion and nucleoid macromolecular crowding effect.
      ). Adding the membrane or the cytoskeleton will be the subject of future articles.
      Solvent friction and collisions are implicitly introduced into the simulations using the Langevin dynamics (see Eq. S7 in the Supporting Material). This requires the knowledge of the collisional frequency γ0 for each particle, which is related to its diffusion coefficient in dilute solution D0 by the Einstein-Smoluchowski relationship D0 = kBT/0, where kB, T, M, and η0 are the Boltzmann constant, the temperature, the mass, and the viscosity in the dilute solution limit (=1 cp for water), respectively. We define a mass-dependent collisional frequency γ0(M), obtained combining the Einstein-Smoluchowski equation with Stokes-Einstein equation as
      D0(M)=kBT6πη0Rh(M).
      (2)


      where Rh is the hydrodynamic radius. The resulting expression is
      γ0(M)=6πη0Rh(M)M.
      (3)


      The hydrodynamic radius Rh is calculated as a function of the molecular size R and hydration shell δh, according to
      Rh(R)=R+δh(R).
      (4)


      Two approximations are considered for δh to explore different surface hydrations, in relation to the biopolymer physicochemical properties, as follows:
      In the first, the minimum hydrodynamic radius Rh(min) and collisional frequency γ0(min) are obtained considering a one-molecule-thick layer of water, i.e., δh = rw, with rw = 1.4 Å as the radius of a water molecule (
      • Trovato F.
      • Nifosí R.
      • Tozzini V.
      • et al.
      A minimalist model of protein diffusion and interactions: the green fluorescent protein within the cytoplasm.
      ,
      • Tozzini V.
      • Trylska J.
      • McCammon J.A.
      • et al.
      Flap opening dynamics in HIV-1 protease explored with a coarse-grained model.
      ).
      In the second approximation, hereafter the “shape-corrected hydration pattern”, the information on nonspherical biopolymer shapes and increased viscosity due to surface irregularities is reintroduced. The value γ0(sc) is evaluated as a function of γ0(min) by fitting experimentally measured E. coli sedimentation coefficients S according to the relationship γ0(sc) = (Smax/S)γ0(min), where Smax is the sedimentation coefficient corresponding to the minimal frictional coefficient (
      • Erickson H.P.
      Size and shape of protein molecules at the nanometer level determined by sedimentation, gel filtration, and electron microscopy.
      ). The expression of Smax and details of the calculation are reported in Table S3. From the fit, γ0(sc)/γ0(min) = Rh(sc)/Rh(min) ≃ 1.3 is obtained. The fit indicates that E. coli biopolymers deviate from ideal spheres, having a 30% larger surface friction that is almost independent from the size of most protein crowders. Systematically higher values are found for DNA-containing systems (see Table S3), probably due to their complex internal structure and dynamics. Because only a few data with large uncertainties on DNA-containing systems are available, we considered the uniform value given above, implying δh(R) ≃ 0.3R + 1.3rw.

      Intermolecular potentials

      The total potential energy of the system is
      U=INIJNJUIJ(DIJ),
      (5)


      where NI, NJ, and UIJ are the numbers of an intermolecular potential (UIJ) between molecules I and J at distance DIJ, accounting implicitly for solute-solvent interactions (see below). In this section, the bottom-up and top-down parameterization approaches and their integration are described.

      Bottom-up approach

      The intermolecular potential UIJ in Eq. 5 is calculated according to the sphere-of-beads (SpoB) representation (
      • Trovato F.
      • Nifosí R.
      • Tozzini V.
      • et al.
      A minimalist model of protein diffusion and interactions: the green fluorescent protein within the cytoplasm.
      ), as depicted in Fig. 1 b. Briefly, each object I, J is decorated with smaller beads of mass 0.12 kDa, randomly distributed within the spherical volume at the mass density ρcrwd. Two different bead flavors are considered: h (hydrophobic) and p (polar), as shown in Fig. 1 c. The h-beads are weakly attractive and mediate soft and nonspecific interactions. The p-beads mediate purely repulsive interactions, thus penalizing any steric overlap at short distances and accounting for the roughly equal propensity of forming hydrogen bonds with surrounding water and amino acids. Although this model does not explicitly depend on the crowder sequence, its chemical composition is implicitly accounted for by the relative amount of h- and p-beads (
      • Fukuchi S.
      • Nishikawa K.
      Protein surface amino acid compositions distinctively differ between thermophilic and mesophilic bacteria.
      ) (see the Supporting Material).
      The h- and p-beads interact according to single-well (hereafter “elementary”) potentials, the functional form of which is a Morse function
      uμν(dμν)=ϵμν{[eαμν(dμνdμν0)1]21},


      with parameters {d0μν, ϵμν, αμν} depending on the bead type μ,ν = p, h. The parameters were optimized based on an experimental dataset of crystallographic structures via a Boltzmann-inversion-related procedure. The details of the procedure used to evaluate the elementary interactions were described in our previous article (
      • Trovato F.
      • Nifosí R.
      • Tozzini V.
      • et al.
      A minimalist model of protein diffusion and interactions: the green fluorescent protein within the cytoplasm.
      ). For any given couple of crowders with radii RI, RJ, the effective intermolecular potential UIJ(DIJ) is calculated by averaging the sum of type-dependent uIJ over different relative crowder-crowder orientations, at the intermolecular distance DIJ (see Eq. S1 in the Supporting Material). The UIJ(DIJ) is then refitted with a Morse function (see Fig. 1 d) with parameters {D0IJ, ϵIJ, αIJ} for J = I (“homo-crowder” interaction), whereas in the case IJ the mixing rules in Eq. S3 in the Supporting Material are applied, the latter accurate within a few percent of the direct calculation. The size-dependent parameters, ϵ0(R) and α0(R), for the homo-crowder potentials, are obtained by fitting the corresponding quantities previously evaluated with the SpoB model. The resulting ϵ0(R) and α0(R) functional forms are reported in Eq. S2 in the Supporting Material and plotted in Fig. 1 h (gray lines). The values ϵ0(R) and α0(R) constitute a first estimate, later subject to a refinement phase driven by experimental data (see Top-Down Approach, below).
      The nucleoid-nucleoid interaction energy ϵnucl,nucl was evaluated by rescaling the value of ϵcrwd,crwd to account for its different physicochemical properties. By assuming the same elementary interaction potentials as in the crowder case, the nucleoid average mass density and an elementary nucleoid mass, 〈mnucl〉 = 0.12–0.72 kDa, the homo-crowder energy is estimated using Eq. S5 in the Supporting Material, which returns the value ϵnucl,nucl ≃ (0.2–0.4) ϵcrwd,crwd (see the Supporting Material for details). All other parameters, including the ones related to the nucleoid-crowder interactions, are evaluated according to the SpoB model or by using the mixing rules defined in Eq. S3 in the Supporting Material.

      Top-down approach

      The interaction potentials obtained from the bottom-up approach are refined to include experimental data on diffusion in crowded environments in two distinct phases of growing complexity. In the first, the diffusion-interaction dynamics of a tracer protein embedded in a mono-disperse cytoplasm of identical crowders (Cr) is analyzed to evaluate the optimal interaction parameters needed to reproduce the tracer diffusion in absence of entropic effects due to poly-dispersity. The tracer protein is chosen to mimic the green fluorescent protein (GFP) because the GFP interacts with the cytoplasm components nonspecifically (
      • Elowitz M.B.
      • Surette M.G.
      • Leibler S.
      • et al.
      Protein mobility in the cytoplasm of Escherichia coli.
      ) and because its diffusion properties are experimentally known. Molecular details are reported in the Supporting Material.
      With extensive simulations of the GFP-Cr system the diffusion coefficient DGFP is evaluated and mapped onto a phase diagram as a function of the interaction parameters ϵGFP,Cr and αGFP,Cr, indicating additionally the aggregation propensity (see Fig. S1 in the Supporting Material). From the phase diagram the optimal parameter values were selected to reproduce the experimental diffusion coefficient DGFP = 5–10 μm2/s and no aggregation or normal diffusion over the entire simulation. The size-dependent homo-crowder parameters ϵ(R) and α(R) (input of the simulated system in Fig. 1 f) were evaluated consistently with the refined parameters obtained from the phase diagram. The details of the procedure are described in the Supporting Material.
      To account for the effects that the cytoplasmic macromolecular size distribution have on diffusion (
      • Asakura S.
      • Oosawa F.
      On interaction between two bodies immersed in a solution of macromolecules.
      ), and similarly to the mono-disperse system discussed before, the optimization of the intermolecular parameters was driven by minimizing aggregation in the poly-disperse cytoplasm, compatibly with known experimental data on diffusion. The calculated ln(D0/D) for a small protein (the GFP) and big macromolecules (the ribosome and the nucleoid) were matched by using a curve best-fitted on experimental data describing the same quantity as a function of the crowder hydrodynamic radius Rh, namely the scale-dependent viscosity reference curve (sdVRC (Rh) (
      • Kalwarczyk T.
      • Tabaka M.
      • Holyst R.
      Biologistics—diffusion coefficients for complete proteome of Escherichia coli.
      )) (see Eq. S11 in the Supporting Material). Note that the original sdVRC was reparameterized to include recent data and to be consistent with our model of diffusion-nonspecific interactions. With the above requirements, the parameters ϵ(R) and α(R) were refined leading to the optimal size-dependent ϵ(R) and α(R) plotted in Fig. 1 h (green lines). The expressions of ϵ(R) and α(R) are reported in Eq. S4 in the Supporting Material.

      Simulation protocol and analysis

      Simulation protocols and data analysis (calculation of MSD, D, α, Rg and sdVRC) are described in the Supporting Material.

      Results and Discussion

      Structural and diffusive properties

      The nucleoid sphere radius Rnucl ≃ 40 nm, evaluated using Eq. 1, is in good agreement with experimental (
      • Kim J.
      • Yoshimura S.H.
      • Takeyasu K.
      • et al.
      Fundamental structural units of the Escherichia coli nucleoid revealed by atomic force microscopy.
      ,
      • Romantsov T.
      • Fishov I.
      • Krichevsky O.
      Internal structure and dynamics of isolated Escherichia coli nucleoids assessed by fluorescence correlation spectroscopy.
      ) and theoretical (
      • Kalwarczyk T.
      • Tabaka M.
      • Holyst R.
      Biologistics—diffusion coefficients for complete proteome of Escherichia coli.
      ) estimates in the range of 35–42 nm. Dilute solution diffusion coefficients for the nucleoid and the GFP, calculated with Eq. 2 and the shape-dependent correction to the hydrodynamic radius in Eq. 4, are D0,nucl ≃ 4.2 μm2/s and D0,GFP ≃ 80 μm2/s, and they also agree well with the corresponding experimental values 1.3–6.5 μm2/s (
      • Romantsov T.
      • Fishov I.
      • Krichevsky O.
      Internal structure and dynamics of isolated Escherichia coli nucleoids assessed by fluorescence correlation spectroscopy.
      ,
      • Cunha S.
      • Woldringh C.L.
      • Odijk T.
      Polymer-mediated compaction and internal dynamics of isolated Escherichia coli nucleoids.
      ) and 87 μm2/s (
      • Terry B.R.
      • Matthews E.K.
      • Haseloff J.
      Molecular characterization of recombinant green fluorescent protein by fluorescence correlation microscopy.
      ) (for additional data, see Table S1). Conversely, by using the minimum hydrodynamic radius, D0 is systematically overestimated.
      To have a reference curve for the comparison with the ln(D0/D) values calculated from the simulations, where D is the diffusion coefficient in the cytoplasm, we reparameterized the sdVRC compared to the original work (
      • Kalwarczyk T.
      • Tabaka M.
      • Holyst R.
      Biologistics—diffusion coefficients for complete proteome of Escherichia coli.
      ) to account for the different scaling law R(M), by adding very recent data and excluding all measured diffusion coefficients dominated by specific interactions (see details in the Supporting Material). For both the minimum and shape-corrected approximations of the hydrodynamic radius, the corresponding sdVRC, plotted in Fig. 2, was obtained by fitting Eq. S11 to the data in Table S4 in the Supporting Material. The values sdVRC (Rmin) and sdVRC (Rh,sc) differ for a uniform rescaling along the Rh axis because Rh,sc/Rh,min ≃ 1.3, and also differ from the original sdVRC (RKalw) in the plateau region, i.e., when Rh > 100 Å. Newly added data (orange circles in Fig. 2), not present in the original dataset, seem to confirm the lower plateau we find. The difference between the optimized and original sdVRC values can be ascribed to the specific interactions of the corresponding macromolecules, absent in our model which includes only weak nonspecific interactions.
      Figure thumbnail gr2
      Figure 2Scale-dependent viscosity reference curves sdVRC(Rh)=(ξ2/L2+ξ2/Rh2)a/2 as a function of the minimal (continuous black line) and shape-corrected (green line) hydrodynamic radius, obtained by fitting the corresponding experimental values ln(D0/D) (dots). Recent data not considered in the original function sdVRC of Kalwarczyc et al. (
      • Kalwarczyk T.
      • Tabaka M.
      • Holyst R.
      Biologistics—diffusion coefficients for complete proteome of Escherichia coli.
      ) (dashed black line) are highlighted (orange contour) and included in the fit (green dots, orange line). Experimental data are reported in . To see this figure in color, go online.

      Energetic properties and comparison with experiments

      The size-dependent energetic parameters ϵ0(R) and ϵ(R) in Fig. 1 h are obtained from the SpoB model and after the two refinement phases described, respectively, in the Bottom-Up Approach and the Top-Down Approach sections. Both increase with the object radius R (see Eqs. S2 and S4 in the Supporting Material), but whereas the slope of ϵ0 depends exclusively on the chemical composition and physical details of the biopolymer representation, the slope of ϵ also includes experimental information about diffusion coefficients in E. coli. Thus, at least to a first approximation, ϵ0(R) and ϵ(R) represent the binding free energies in dilute solution and within the cytoplasm, respectively. The monotonically increasing bilinear shape of ϵ(R), with an inflection point at rs ≃ 40 (average E. coli crowder size), is a direct consequence of the different mass densities ρcrwd and ρnucl.
      To better understand the biophysical content of ϵ0 and ϵ, we compared them to binding free energies ΔG of a set of biopolymers dominated by nonspecific interactions. Among the most promiscuous proteins in the set there are the chaperones, which recognize mostly unstructured or misfolded proteins and proteins that interact with DNA fragments embedded in the nucleoid. Binding in the latter case often induces DNA bending. To account for the higher number of stabilizing contacts found in complexes involving unstructured proteins and DNAs, compared to our model of compact proteins, the experimental binding free energies ΔG were corrected with a scaling constant, yielding ΔG = kΔG. From the dissociation constants of the chaperone GroEL (
      • Zahn R.
      • Perrett S.
      • Fersht A.R.
      Conformational states bound by the molecular chaperones GroEL and secB: a hidden unfolding (annealing) activity.
      ) in complex with the folded and unfolded states of the same protein (barnase) and from proteins binding the DNA, we estimated k ≃ 1/3. Details on the correction procedure are reported in the Supporting Material.
      In Fig. 3, the correlation between ΔG and ϵ0,ϵ are presented for both in vitro and in vivo data (the dataset including also molecular details and a list of references is reported in Table S2). For ΔG and ϵ0, the calculated correlation coefficient R ≃ 0.75 and the ratio 〈ΔG/ϵ0〉 = 1.07, where 〈…〉 indicates the average over the molecule dataset, demonstrate that the interactions obtained with the SpoB model (the top-down approach) agree with experimental free energies in vitro. Deviations from this average (see also Fig. S4) indicate that the experimental intermolecular interactions are stronger (ΔG/ϵ0 > 1) or slightly weaker (ΔG/ϵ0 < 1) compared to the ones calculated for complexes involving unstructured proteins or DNAs, respectively. Although we cannot rule out that better corrections for ΔG exist, we interpret the average excess energy 〈ΔG/ϵ0〉 = ∼0.2–0.5 kcal/mol as a specific signature of the chaperone-protein and DNA-protein energetics, beyond the one captured by our model of compact biopolymers (see also the Supporting Material for an estimate of the range of chaperone-protein binding energies).
      Figure thumbnail gr3
      Figure 3Correlations between experimental and calculated binding free energies in the following cases: in vitro-ϵ0,IJ (black); in vitro-ϵIJ (green); and in vivo-ϵIJ (red). ΔG are the experimental binding free energies, corrected to account for more intermolecular contacts compared to our model (see text). Calculated energies ϵ0,IJ, ϵIJ are calculated via Eq. S2, Eq. S4, and the mixing rules in Eq. S3 in the . (Dashed line) Regression line for in vitro ϵ0,IJ. To see this figure in color, go online.
      The same comparison was carried out between the ΔG in vivo and the optimal parameter ϵ (in the sense of the transport properties in Fig. 1 g), although the paucity of experimental data prevented us from building a reliable statistical analysis. The few cases found are reported in Fig. 3. Although the correlation is not meaningful, the ratio 〈ΔG/ϵ〉 = 1.1 suggests that there is a good correspondence between experimental and optimal energies. This is attributed to two factors: 1), the larger stoichiometry of the complexes involved (DnaJ+DnaK chaperones and repressor proteins+nucleoid) and 2) ϵ is representative of in vivo interactions. Details on these systems are reported in Table S2. Fig. 3 reports also the comparison between the corrected free energies ΔG in vitro and the optimal parameter ϵ, useful for understanding the effect of macromolecular crowding on the nonspecific interactions. Although the calculated average ratio 〈ΔG/ϵ〉 = 1.9 indicates that the in vitro conditions differ from the crowded ones, the high correlation coefficient R ≃ 0.79 suggests a nonnegligible dependence between the optimal ϵ and the free energy measured between any two macromolecules in isolation.
      Among the potential microscopic mechanisms at the origin of the decrease ϵ0ϵϵ (see Fig. 1 h and Fig. S2 for a plot of ϵ), there is the distribution of solvent molecules around biopolymers, responsible for the weakening of hydrophobic interactions, through water-mediated contacts. Entropic effects due to poly-dispersity also play an important role, altering the crowder mobility via two different mechanisms: 1), a more efficient crowder packing; and 2), an attractive contribution (known as depletion force (
      • Marenduzzo D.
      • Finan K.
      • Cook P.R.
      The depletion attraction: an underappreciated force driving cellular organization.
      )) to the total force experienced by big molecules. The decrease of the intermolecular energies after the optimization phase compensates for these effects, as explained in the Supporting Material, where an estimate is provided.
      From this analysis, we conclude that, for diffusion to be observed in crowded environments without the formation of extensive aggregates (see Top-Down Approach), the intermolecular interactions must decrease compared to those derived in isolation. Indeed, cellular crowding and hydration are known to exert strong constraints on the evolution of proteomes, by reducing nonspecific interactions to avoid aggregation while preserving the potential to form transient encounter complexes (
      • Levy E.D.
      • De S.
      • Teichmann S.A.
      Cellular crowding imposes global constraints on the chemistry and evolution of proteomes.
      ,
      • Gabdoulline R.R.
      • Wade R.C.
      On the protein-protein diffusional encounter complex.
      ).

      Excluded volume fails to describe biological media

      In this section, we analyze those effects of macromolecular crowding, which affect the transport properties and viscosity of the E. coli cytoplasm by occluding the space available for diffusion. We consider the crowders interacting with purely repulsive potentials, allowing us to describe their steric hindrance. The effect of adding nonspecific attraction will be discussed in the next section.
      Fig. 4, ac (purely repulsive case), reports the mean-square displacement MSD(t) as a function of the simulation time t, calculated from the crowder trajectories using Eq. S13 in the Supporting Material. For heterogeneous, dense, and interacting systems, such as the cytoplasm, the MSD can show deviations from linearity and can be phenomenologically expressed as MSD(t) ∝ tα. Thus, two quantities can be extracted from the MSD, namely the apparent diffusion coefficient via the relationship D = MSD/6t, and the anomalous exponent α, from the slope of the MSD in the log-log plot. The value D is then combined with the corresponding diffusion coefficient in dilute solution, D0 (see Eq. 2), to obtain the quantity, ln(D0/D) = ln(η/η0), measuring the increase of apparent solution viscosity η relative to the one in dilute conditions, η0. The scale-dependent viscosity reference curve (sdVRC) defined in Eq. S11 in the Supporting Material and reported in Fig. 5 c (black and green lines for the two different hydration patterns) is used as a metric to quantify the differences between the expected and calculated ln(D0/D) in E. coli.
      Figure thumbnail gr4
      Figure 4Time-averaged single particle (black) MSDs in the purely repulsive (upper row) and weakly attractive (lower row) intermolecular potential regimes for the smallest crowder (a and d), the ribosome (b and e),and the nucleoid (c and f). (Green lines) Ensemble- and time-averaged MSDs on which the listed anomalous exponents α-values are fitted. Normal diffusion is reported for comparison (dashed blue line). (f) (Dotted green line) Simulation with repulsive crowders—nucleoid, attractive otherwise. To see this figure in color, go online.
      Figure thumbnail gr5
      Figure 5Transport properties within the E. coli cytoplasm. (a) Time evolution of ln(D0/D) = ln(η/η0) calculated from simulations with purely repulsive and (b) weakly attractive intermolecular interactions. The values D and η are the long-time diffusion coefficient and the viscosity, respectively. (Magenta, blue, and orange lines) Smallest, ribosome, and nucleoid species, respectively; (gray) all the others. (c) Comparison between the two sdVRCs obtained after the parameterization procedure (black and green lines as in ) and the ln(D0/D) values (symbols) evaluated in the last chunk of the simulations for different combinations of intermolecular potentials and hydration patterns. Pure repulsion with nucleoid at 0.0 % vol and Rhmin (Δ); pure repulsion with nucleoid at 20 % vol and Rhsc (); attraction with nucleoid at 0.0% vol and Rhmin (▲); attraction between crowders, repulsion crowder/nucleoid with nucleoid at 20 % vol and Rhsc (); attraction with nucleoid at 20% vol and Rhmin (●); attraction with nucleoid at 20 % vol and Rhsc (). (d) Diffusion coefficient in cytoplasm as a function of the mass. Lines in panel d are the reference curves evaluated as D0(ρ)exp(−sdVRC), where ρ is the mass density and sdVRC matches the curves in panel c. Because Dcyto depends on ρ, two double bands are visible, corresponding to ρ = ρcrwd (upper) and ρ = ρnucl (lower). (Gray) Experimental DGFP range. To see this figure in color, go online.
      As a first test of the accuracy of our coarse-grained simulations, the calculated ln(D0/D) values, averaged over the crowder types, were compared to the results of atomistic simulations in implicit solvent with purely repulsive potentials (
      • McGuffee S.R.
      • Elcock A.H.
      Diffusion, crowding and protein stability in a dynamic molecular model of the bacterial cytoplasm.
      ). A very good agreement is found between our simulations (open triangles in Fig. 5 c) and the atomistic ones, whose ln(D0/D) values are reported in Fig. 3 of Kalwarczyk et al. (
      • Kalwarczyk T.
      • Tabaka M.
      • Holyst R.
      Biologistics—diffusion coefficients for complete proteome of Escherichia coli.
      ).
      A more in-depth analysis of the crowder trajectories shows that the ln(D0/D) value averaged over the different crowders is ∼1.1–1.2, weakly varying with the object size (see Fig. 5 a). Ribosomes and nucleoid show the larger splitting from the others and additionally the nucleoid’s curve increases with time, a signature of anomalous diffusion. Compared to the expected ln(D0/D) value of ∼0.7 for mono-disperse spheres without hydrodynamic interactions (the formula D/D0 = 1-2ϕcrwd is used with ϕcrwd = 0.25), (
      • Bernadó P.
      • García de la Torre J.
      • Pons M.
      Macromolecular crowding in biological systems: hydrodynamics and NMR methods.
      ), the higher 〈ln(D0/D)〉 value is a footprint of poly-dispersity.
      Microscopically, two effects are responsible for the observed behavior:
      • 1.
        The depletion force, which is an effective attractive force between big particles, originating from the entropy gain of small particles when the bigger ones are close together (
        • Marenduzzo D.
        • Finan K.
        • Cook P.R.
        The depletion attraction: an underappreciated force driving cellular organization.
        ). This force is also present in systems with a lower degree of poly-dispersity, such as mono-disperse colloidal solutions in which few bigger particles are immersed.
      • 2.
        An effective smaller volume available to the crowders due to the nucleoid large mass and steric hindrance (Veff = VVnucl ≃ 0.8V): In fact, diffusion of smaller crowders is faster when the nucleoid is absent, as evidenced by comparing the simulations with and without the nucleoid (open green circles and black triangles in Fig. 5 d).
      The anomalous exponent α, obtained by fitting tα onto the ensemble- and time-averaged MSDs shown in Fig. 4, ac, measures the deviations from normal diffusion. The nucleoid and all the crowders display α ≃ 0.9–1.0, practically indicating that diffusion is slightly anomalous or normal in the repulsive regime. On the contrary, because anomalous diffusion has been observed experimentally for a number of macromolecules (
      • Golding I.
      • Cox E.C.
      Physical nature of bacterial cytoplasm.
      ,
      • Javer A.
      • Long Z.
      • Cosentino Lagomarsino M.
      • et al.
      Short-time movement of E. coli chromosomal loci depends on coordinate and subcellular localization.
      ,
      • Weber S.C.
      • Spakowitz A.J.
      • Theriot J.A.
      Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm.
      ,
      • Parry B.R.
      • Surovtsev I.V.
      • Jacobs-Wagner C.
      • et al.
      The bacterial cytoplasm has glass-like properties and is fluidized by metabolic activity.
      ), we conclude that excluded volume alone cannot explain fully the reduction of diffusion in the bacterial cytoplasm.
      The effect of cytoplasm heterogeneity on the transport properties is evident from the distribution of the single-particle MSDs in Fig. 4, ac. In the first-half of the trajectories, the spreading of the MSDs is small, and practically does not depend on the mass of the diffusing objects. In the second-half, the spreading grows because of the smaller statistics (see Eq. S13 in the Supporting Material). Hence, in the repulsive regime, poly-dispersity affects the single-particle diffusion coefficients only slightly compared to the average, contrary to the large variations found experimentally. The spreadings around the average diffusion coefficients are emphasized when attraction is added to the intermolecular potentials, as we will show below.

      Diffusion-interaction with weak attraction

      Adding nonspecific attraction to the purely repulsive intermolecular potentials (treated in the previous section) has the effect of decreasing the particle mobility (compare Fig. 4, df, and Fig. 4, ac) and of emphasizing the splitting among the calculated ln(D0/D) values (Fig. 5 b) compared to the purely repulsive case (Fig. 5 a). As a consequence, for both approximations of the biopolymer hydrodynamic radii (minimum and shape-corrected hydration patterns), the reference curves in Fig. 5 c are reproduced more accurately than with purely repulsive potentials. Between the two hydration models, the one reintroducing the information on biopolymer shapes performs better because the corresponding reference Dcyto is closer to the experimental diffusion coefficients. For example, the calculated GFP and nucleoid’s Dcyto values agree with experimental data in the ranges 5–10 and 10−3–10−2 μm2/s, respectively (
      • Elowitz M.B.
      • Surette M.G.
      • Leibler S.
      • et al.
      Protein mobility in the cytoplasm of Escherichia coli.
      ,
      • Javer A.
      • Long Z.
      • Cosentino Lagomarsino M.
      • et al.
      Short-time movement of E. coli chromosomal loci depends on coordinate and subcellular localization.
      ,
      • Weber S.C.
      • Spakowitz A.J.
      • Theriot J.A.
      Nonthermal ATP-dependent fluctuations contribute to the in vivo motion of chromosomal loci.
      ) (see also Table S4). On the contrary, without the shape-dependent correction, the calculated Dcyto is overestimated because D0 is larger. Removing also the nucleoid additionally increases the calculated diffusion coefficients of the remaining crowders with a visible disagreement with the reference curves (Fig. 5 d, compare solid circles and triangles). Thus, the intermolecular attraction, the nucleoid, and the biopolymer shape-dependent correction are necessary to reproduce the transport properties in the cytoplasm.
      Hydrodynamic correlations have been demonstrated to affect biomolecule mobility (
      • Ando T.
      • Skolnick J.
      Crowding and hydrodynamic interactions likely dominate in vivo macromolecular motion.
      ,
      • Mereghetti P.
      • Wade R.C.
      Atomic detail Brownian dynamics simulations of concentrated protein solutions with a mean field treatment of hydrodynamic interactions.
      ,
      • Trovato F.
      • Nifosí R.
      • Tozzini V.
      • et al.
      A minimalist model of protein diffusion and interactions: the green fluorescent protein within the cytoplasm.
      ) and interactions (
      • Elcock A.H.
      A molecule-centered method for accelerating the calculation of hydrodynamic interactions in Brownian dynamics simulations containing many flexible biomolecules.
      ). They were not included in our treatment because structural rearrangements caused by the intermolecular interactions dominate on the long-time diffusion, i.e., when the simulation time is larger than the structural relaxation time τIR2/D0 (
      • Bernadó P.
      • García de la Torre J.
      • Pons M.
      Macromolecular crowding in biological systems: hydrodynamics and NMR methods.
      ). Neither did we include small molecules or specific interactions into the model. The absence of hydrodynamics, low-molecular-weight compounds, and specific interactions in our simulations can explain the discrepancies between the simulated and the reference sdVRCs. For example, for the fastest/smallest particles we expect that hydrodynamics would reduce their diffusion, leading to a better agreement with the reference curve. However, assessing the role of these effects relative to the nonspecific interactions here studied would require to model them explicitly, and to have reference curves able to distinguish between specific and nonspecific interactions. Whereas the first is computationally demanding, the second is even a harder task, experimentally. As more data will be produced, especially in the less populated regions in Fig. 2, such a comparison will become feasible.
      The effect of cytoplasm heterogeneity on the transport properties can be studied with the distribution of single particle trajectories or, equivalently, with the distribution of diffusion coefficients. The error bars in Fig. 5 c show the spreadings around the value 〈ln(D0/D)〉 averaged over a given species (fixed Rh). The agreement between the calculated and experimental spreadings is remarkable, as evidenced by comparing the error bars in Fig. 5 c with the amplitude of point scatter in Fig. 1 g. The variability of diffusion coefficients for crowders of a given Rh can be ascribed to the different interactions a given macromolecule of that type experiences as it probes different environments throughout its trajectory. Both poly-dispersity and the nonspecific intercrowder attractions are responsible for the calculated spreadings of the diffusion coefficients. In fact, we do not observe such effects in mono-disperse solutions of crowders with attractive interactions (see Fig. S3) or in poly-disperse mixtures of purely repulsive crowders (see Fig. 4, ac).
      Diffusion of small particles is almost normal at all timescales with an anomalous exponent of α ≃ 0.95, whereas for the nucleoid and the big macromolecules such as the ribosomes the anomalous exponent is αnucl ≃ 0.85 and αribo ≃ 0.75, respectively (Fig. 4, df). In the nucleoid’s case, subdiffusion is caused mainly by those entropic effects that act also in the purely repulsive case. In fact, both simulations with repulsive or weakly attractive potentials show that the nucleoid’s ln(D0/D) value increases slowly with time (i.e., D decreases) in a similar fashion (Fig. 5, a and b), demonstrating that a similar trapping mechanism is expected to hinder nucleoid’s motion, i.e., the depletion force.
      The coupled contributions of attraction and repulsion play an important role, especially for the biggest crowders. Indeed, when weakly attractive intermolecular potentials are considered, a network of energetic traps is formed around the nucleoid and the ribosomes, which synergistically cooperate with steric exclusion to constrain diffusion on all timescales probed by our simulations. The importance of attraction is evident when it is turned off only among crowders-nucleoid (the purely repulsive cytoplasm model has been discussed in the previous section), i.e., the nucleoid is treated as inert. In this case we find that Dnucl(t) decreases very rapidly (see Fig. S5), showing an anomalous exponent αnucl = 0.45, the lowest found in our simulations (see Fig. 4 f). Such low value originates from an augmented nucleoid-nucleoid attraction by depletion forces, the latter stronger as a consequence of pure repulsion, consistently with theoretical calculations (
      • Egorov S.A.
      Effect of repulsive and attractive interactions on depletion forces in colloidal suspensions: a density functional theory treatment.
      ).
      Although in the hybrid regime described above αnucl = 0.45 is in agreement with experimental evidences (
      • Javer A.
      • Long Z.
      • Cosentino Lagomarsino M.
      • et al.
      Short-time movement of E. coli chromosomal loci depends on coordinate and subcellular localization.
      ,
      • Weber S.C.
      • Spakowitz A.J.
      • Theriot J.A.
      Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm.
      ), we exclude that the nucleoid is inert for two reasons: 1), we observe cluster formation in this case; and 2), it is known that such a low anomalous exponent is partially explained by the nucleoid’s internal dynamics (
      • Javer A.
      • Long Z.
      • Cosentino Lagomarsino M.
      • et al.
      Short-time movement of E. coli chromosomal loci depends on coordinate and subcellular localization.
      ,
      • Weber S.C.
      • Spakowitz A.J.
      • Theriot J.A.
      Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm.
      ), which was ignored in our simulations. On the contrary, the model that includes attraction among all species shows a higher anomalous exponent αnucl = 0.85, which is likely not to be an inaccuracy of the model, but instead the correct value accounting for the viscoelastic nature of the cytoplasm only.
      In the case of subdiffusive species such as the ribosomes and the nucleoid, the apparent diffusion coefficient is time-dependent. Experimentally, values of the diffusion coefficients similar to those evaluated at the end of our simulations (∼0.3–0.4 ms) occur from fractions of seconds to tens of seconds. The generally faster timescales observed in our simulations have different origins: the first is that simulations probe different conditions compared to the experimental ones, with the former ignoring the flexible nature of some macromolecules. The second is related to the coarse-graining procedure, which, by softening the intermolecular interactions compared to the same atomistic system, accelerate the escape of each crowder from the cage formed by its surroundings by orders of magnitudes (
      • Depa P.
      • Chen C.
      • Maranas J.K.
      Why are coarse-grained force fields too fast? A look at dynamics of four coarse-grained polymers.
      ). Although it is clear that these two factors contribute in accelerating the system dynamics, their quantitative evaluation is beyond the scope of this work.
      The simulation results so far described allow us to identify the microscopic mechanisms behind macromolecular diffusion-interaction within the bacterial cytoplasm. Frequent hoppings drive the macromolecules in and out the locally distributed caging traps, with jump frequencies that depend on solvent collisions, clashes with the repulsive barrier of the intermolecular potential and constrained motion due to direct or depletion-induced attraction (see also Movie S1 in the Supporting Material).

      Consequences for bacterial cytoplasm mobility and organization

      Which is the relative contribution of intermolecular steric repulsion and nonspecific attraction within the simulated bacterial cytoplasm? How are mobility and organization affected by these forces? We addressed these questions by calculating a number of properties as a function of the biopolymer size, as shown in Fig. 6.
      Figure thumbnail gr6
      Figure 6Consequences of nonspecific interactions on cytoplasm mobility and organization. Biopolymer size corresponds to 2R where R is the radius evaluated through Eq. 1. Anomalous exponent (α) and ergodicity breaking (EB) parameter in panel a. Gyration radius (Rg) of the trajectory in panel b with slow (Rg < 40 nm) and fast (Rg > 40 nm) particles delimited (dashed line). Each point in panel b represents the traveled distance of one macromolecule. Color-coding for panels a and b: (black dots) Values obtained from simulations with repulsive potentials, nucleoid at 20 % vol and Rhsc; (green dots) simulations with attractive potentials, nucleoid at 20 % vol and Rhsc. (Lines connecting the points are a guide for the eye.) Distribution of Rg in panel c (see Eq. S15 in the ) follows the same color-coding as in panels a and b. The distribution in panel d is rescaled by a factor of 1000. To see this figure in color, go online.
      With purely repulsive intermolecular potentials, diffusion is almost normal within the calculated spreading, although a propensity to anomalous diffusion is clearly visible (α ≃ 0.85–1.0 in Fig. 6 a), especially for the nucleoid. For smaller/faster particles, this is the result of a reduced free volume, whereas the depletion force is the main determinant in the case of bigger/slower particles. Additionally, caging effects are limited because no ergodicity breaking (EB, defined in Eq. S14 in the Supporting Material) is detected, as expected for a collection of dynamic hard spheres at cytoplasmic volume fractions (
      • Cipelletti L.
      • Ramos L.
      Slow dynamics in glasses, gels and foams.
      ) (Fig. 6 a). When attraction is switched on, the anomalous exponent α decreases as a function of the biopolymer size, reaching the minimum value at ≃19 nm, and finally increases in correspondence of the nucleoid to the same value found in the repulsive regime. The opposite holds for the EB parameter. The spreadings around the average α and EB values are also larger compared to the purely repulsive case, in analogy with what we found for the diffusion coefficients (Fig. 5). This demonstrates that caging effects play a stronger role compared to repulsion alone and originate from the combination of crowder size distribution and direct and depletion-induced intermolecular attractions.
      For each diffusing particle a measure of the traveled distance is the trajectory radius of gyration Rg, defined in Eq. S15 in the Supporting Material and plotted in Fig. 6 b. In the purely repulsive regime, the distribution of Rg (see Fig. 6 c) is characterized by a broad single peak, located in the region of fast diffusion, whereas it is bimodal when attraction is included. In the latter case the fraction of slow particles is ∼55% of the total and additionally fast particles probe the cytoplasm less thoroughly compared to the repulsive case.
      Together with the splitting of the sdVRC levels in Fig. 5 we conclude that the attractive intermolecular potentials are responsible for a biphasic behavior of the cytoplasm, more fluid at short length scales and reminiscent of a jammed or glassy state at longer length scales. A similar conclusion was recently reported in Parry et al. (
      • Parry B.R.
      • Surovtsev I.V.
      • Jacobs-Wagner C.
      • et al.
      The bacterial cytoplasm has glass-like properties and is fluidized by metabolic activity.
      ), in which the authors tracked fluorescently labeled protein aggregates called “inclusion bodies” (IBs). The most striking feature of these experiments is the anomalous exponent α, increasing from 0.1 (IB size 50 nm) to ∼0.8 for the biggest IB (150 nm). This trend is somewhat counterintuitive based on other experiments and our simulations, both indicating that big biopolymers show a stronger subdiffusion compared to small ones caused by macromolecular crowding and interactions. It can be nevertheless worthwhile to compare our results with these experiments because the anomalous exponent reflects physicochemical features, while being robust upon minor variations (
      • Golding I.
      • Cox E.C.
      Physical nature of bacterial cytoplasm.
      ,
      • Weber S.C.
      • Spakowitz A.J.
      • Theriot J.A.
      Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm.
      ). If the experimentally measured IB sizes and apparent diffusion coefficients are reported on the plot Dcyto(M) in Fig. 5 (inclusion bodies are densely packed (
      • Margreiter G.
      • Messner P.
      • Bayer K.
      • et al.
      Size characterization of inclusion bodies by sedimentation field-flow fractionation.
      ) such as the protein interior at mass density ∼1.3 g/cm3), the smallest IB falls on the reference curve M,Dexpt ≃ 2.3 104 kDa, 2 10−2 μm2/s, whereas the biggest IB is well above it at M,Dexpt ≃ 6.3 105 kDa, 1.5 10−3 μm2/s. This analysis suggests that the smallest IBs favorably interact with the surroundings whereas nonspecific interactions among the biggest IBs and the endogenous cytoplasmic components are reduced. This explanation, based on our model of interacting biopolymers, agrees with the general idea that IBs evolve from a highly reactive nucleus until forming a relatively inert structure (
      • Baneyx F.
      • Mujacic M.
      Recombinant protein folding and misfolding in Escherichia coli.
      ).
      Macromolecular organization and mobility are affected by interactions and poly-dispersity in such a way that small crowders (Region I) experience large entropic effects originating from an effective volume reduced by bigger particles and the low volume fractions, whereas direct attraction is relatively small. A similar trend is seen for particles bigger than the ribosomes, such as, for example, the nucleoid (Region III). But, at variance with the mechanisms affecting Region I, the molecules in Region III experience a strong depletion, induced by smaller particles. At intermediate sizes (Region II), including ribosomes and m-t- RNAs, diffusion is strongly hampered by enthalpic effects (see also Fig. S6), while the reduced entropic contributions help to avoid unspecific aggregation (
      • Coquel A.S.
      • Jacob J.P.
      • Berry H.
      • et al.
      Localization of protein aggregation in Escherichia coli is governed by diffusion and nucleoid macromolecular crowding effect.
      ). We hypothesize that this could favor both the formation of frequent mRNA-ribosomes encounter complexes (before specific recognition) and the high fidelity of polypeptide translation. The biological significance of subdiffusion and attraction has also been recently pointed out for the enzyme EcoRV (
      • Sereshki L.E.
      • Lomholt M.A.
      • Metzler R.
      A solution to the subdiffusion-efficiency paradox: inactive states enhance reaction efficiency at subdiffusion conditions in living cells.
      ).
      Consistently with a number of experimental investigations on biopolymer diffusion-interaction the phenomenology emerging from our simulations points to a partially jammed bacterial cytoplasm, where macromolecular crowding and soft interactions entangle particle motion (
      • Golding I.
      • Cox E.C.
      Physical nature of bacterial cytoplasm.
      ,
      • Elowitz M.B.
      • Surette M.G.
      • Leibler S.
      • et al.
      Protein mobility in the cytoplasm of Escherichia coli.
      ,
      • Sarkar M.
      • Smith A.E.
      • Pielak G.J.
      Impact of reconstituted cytosol on protein stability.
      ,
      • Javer A.
      • Long Z.
      • Cosentino Lagomarsino M.
      • et al.
      Short-time movement of E. coli chromosomal loci depends on coordinate and subcellular localization.
      ,
      • Weber S.C.
      • Spakowitz A.J.
      • Theriot J.A.
      Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm.
      ,
      • Parry B.R.
      • Surovtsev I.V.
      • Jacobs-Wagner C.
      • et al.
      The bacterial cytoplasm has glass-like properties and is fluidized by metabolic activity.
      ,
      • Nenninger A.
      • Mastroianni G.
      • Mullineaux C.W.
      Size dependence of protein diffusion in the cytoplasm of Escherichia coli.
      ). Although a unified picture of the bacterial cytoplasm diffusion-interaction has emerged from our simulations, a number of factors is not explicitly included in this model cytoplasm. Among the most important factors believed to regulate intracellular mobility and organization there are the energy sources such as ATP that fluidize the cytoplasm independently from motor proteins (
      • Parry B.R.
      • Surovtsev I.V.
      • Jacobs-Wagner C.
      • et al.
      The bacterial cytoplasm has glass-like properties and is fluidized by metabolic activity.
      ), hydrodynamic correlations that are able to reduce diffusion (
      • Ando T.
      • Skolnick J.
      Crowding and hydrodynamic interactions likely dominate in vivo macromolecular motion.
      ), and processes that control protein solubility and unwanted interactions (
      • Levy E.D.
      • De S.
      • Teichmann S.A.
      Cellular crowding imposes global constraints on the chemistry and evolution of proteomes.
      ), hence homeostasis (
      • Ciryam P.
      • Tartaglia G.G.
      • Vendruscolo M.
      • et al.
      Widespread aggregation and neurodegenerative diseases are associated with supersaturated proteins.
      ). Refinements of the input intermolecular interaction parameters or frictional coefficients can improve the description of the diffusion-interaction. Alternatively, the parameterization of more complex potential functional forms or the explicit inclusion of hydrodynamics can be employed at the cost of computational speed (
      • Trovato F.
      • Nifosí R.
      • Tozzini V.
      • et al.
      A minimalist model of protein diffusion and interactions: the green fluorescent protein within the cytoplasm.
      ,
      • Trovato F.
      • Tozzini V.
      Supercoiling and local denaturation of plasmids with a minimalist DNA model.
      ,
      • Elcock A.H.
      A molecule-centered method for accelerating the calculation of hydrodynamic interactions in Brownian dynamics simulations containing many flexible biomolecules.
      ).

      Conclusions and Perspectives

      A simplified poly-disperse model of the E. coli cytoplasm, capable of describing the diffusion and organization of macromolecules sizing from 3 to 80 nm, was presented. A weak nonspecific attraction among all species was included, and shown to be capable of accurately reproducing the different transport regimes, normal for small biopolymers and anomalous for ribosomes and the nucleoid.
      Within submillisecond-long simulations of thousands of macromolecules, at different intermolecular potentials regimes, we were able to dissect the microscopic determinants that affect diffusion within the cell. Frequent hoppings drive the macromolecules in and out of the locally distributed caging traps, with jump frequencies that depend on solvent collisions, clashes with the repulsive barrier of the intermolecular potentials, and constrained motions due to direct or depletion-induced attraction. From the interplay among the above factors, macromolecules sizing from ∼9 to 40 nm experience the strongest attraction and anomalous diffusion. We speculate that this is a necessary requirement to meet the high fidelity essential to processes such as translation and transcription, by discouraging entropy-driven compaction.
      The observed dynamical heterogeneity is the hallmark of a jammed state of the bacterial cytoplasm, observed also in nonbiological colloidal systems. At length scales <7 nm the cytoplasm behaves as an almost viscous fluid whereas at larger length scales it has glassylike features. Although these conclusions are not immediately transferable to eukaryotic cells, where specific interactions are not negligible and the cytoskeleton is believed to entangle and direct the macromolecule motion, the innovative parameterization here presented is transferable and flexible enough to be applied to different contexts—whether they be of biological interest or relevant to nanotechnological applications, such as the design of self-assembling materials from inorganic colloids (
      • Xia Y.
      • Nguyen T.D.
      • Kotov N.A.
      • et al.
      Self-assembly of self-limiting monodisperse supraparticles from polydisperse nanoparticles.
      ).
      Although we condensed the main nonspecific interactions (electrostatics, van der Waals, and the hydrophobic effect) in an economical single-well-potential functional form, more refined expressions can be employed, increasing the accuracy of the physical picture and number of parameters. Multi-well potentials can be employed to improve the microscopic description of solvent displacement upon binding and the dependence of electrostatics on the ionic strength and pH (
      • Trovato F.
      • Nifosí R.
      • Tozzini V.
      • et al.
      A minimalist model of protein diffusion and interactions: the green fluorescent protein within the cytoplasm.
      ,
      • Trovato F.
      • Tozzini V.
      Supercoiling and local denaturation of plasmids with a minimalist DNA model.
      ). Thus, considering also that a backmapping rule to switch from the one-bead-per-biopolymer to the one-bead-per-amino-acid-granularity exists, the cytoplasm model is automatically consistent with finer resolution biopolymer models. This is particularly relevant to the area of multiscale simulations, because biopolymers’ internal fluctuations and structural transitions, binding reactions, and translational and rotational diffusion can be addressed simultaneously in an in vivo-mimicking environment.
      The design strategy and results discussed, going beyond the recognized limitations of the hard-sphere approximation of biomacromolecular crowding, pave the way for new and exciting developments that complement difficult and costly in vivo experiments. For example, using single-bead-per-amino-acid or nucleotide coarse-grained models, it would be feasible to study protein aggregate formation and its consequences on proteostasis or the nucleoid internal dynamics, using models able to grasp the interplay between DNA supercoiling and association with cellular factors. Fundamental for the above aims is to properly tune the relative contribution of nonspecific and specific interactions. In fact, whereas nonspecific interactions are typically weak and frequent, owing to the high cytoplasmic density, specific interactions are stronger and relatively infrequent. Their combined action is nevertheless necessary to address the full problem of passive diffusion-interaction and subcellular localization in the near future.
      We thank Paolo Mereghetti, Joanna Trylska, and Filip Leonarski for stimulating discussions.

      Supporting Material

      Supporting Citations

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