Parameterization of the simulation model

Starting from three different randomized initial configurations of the cytoplasm model (all shown in Figure S1), we performed independent Brownian dynamics (BD) simulations [14] to explore diffusive behavior. A variety of energetic descriptions of intermolecular interactions were explored, ranging from a simple steric-only model – which provides an opportunity to directly test the predictions of excluded-volume ‘crowding’ theories [2],[3] – to models that include both long-range electrostatic interactions and short-range potential functions that mimic hydrophobic interactions between exposed non-polar groups. In order to determine the most realistic energy model, the long-time translational diffusion coefficients, D^L_trans, of the ‘tracer’ GFP molecules were computed from the BD simulations and compared with previously reported experimental estimates obtained by fluorescence-recovery-after-photobleaching (FRAP) analysis of GFP in the E. coli cytoplasm [15]–[18].

A comparison of the computed GFP D^L_trans values obtained with the different energy models is shown in Figure 2A. For simulations in which only steric interactions operate between macromolecules the computed GFP D^L_trans value is 3–6 times higher than the experimental estimates, and although this value decreases somewhat when electrostatic interactions between macromolecules are added, it remains 2–5 times too high relative to experiment. A more realistic model of macromolecular interactions would allow favorable short-range attractions to occur between exposed hydrophobic atoms and one simple way of approximating such interactions is to use a Lennard-Jones potential, with the well-depth of the potential, ε, being treated as an adjustable parameter (see Methods). As shown in Figure 2A, the inclusion of such a term results in computed GFP D^L_trans values that decrease monotonically as the well-depth, ε, increases in magnitude. The best agreement with experiment is obtained with ε = 0.285 kcal/mol: at this value of ε the computed value of D^L_trans – which is ∼10% of its value at infinite dilution – is within the experimental error of all in vivo estimates [15]–[18] including a very recent report for diffusion in cells growing in minimal media [18]. As noted in the Discussion, this optimal value of ε is very similar to the values obtained in our previous efforts to model the interaction thermodynamics of single-component protein solutions [19].

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Figure 2. Parameterization and sampling in the cytoplasm model.

A. Extrapolated long-time D_trans values for GFP from BD simulations performed with different energy models; ‘ε’ refers to the well-depth (in kcal/mol) of the Lennard-Jones potential used to describe hydrophobic interactions (see Methods). Solid, long-dash, short-dash and dotted lines are the experimental D_trans values from refs. 14, 15, 16 and 17 respectively. The vertical arrow indicates the energy model selected for further BD simulation. B. Average of the maximum distance moved during the 15µs of production for all molecule types plotted versus their molecular weights. Upper error bars indicate the largest value of the maximum distance moved found for any molecule of that type; lower error bars indicate the smallest value of the maximum distance moved. All distances expressed in terms of the molecular diameters (obtained by doubling the hydrodynamic radius calculated by HydroPro [88]. C. Average number of unique neighbors encountered by each molecule type as a function of simulation time; each line refers to a different molecule type. D. Average number of neighbors possessed by each molecule type at any instant, plotted versus molecular weight. E. Time constant for the slower of the two exponentials describing the rate at which neighbors are lost, plotted for each molecule type versus molecular weight. F. Average number of times that each molecule type's immediate neighbors exchange during 15µs simulation plotted versus molecular weight of each molecule type.

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Having determined that good agreement with experiment could be obtained using a so-called ‘full’ energy model that included steric, electrostatic and short-range attractive hydrophobic interactions, we extended each of three independent simulations performed with this energy model to 20µs (see Figure S2 for plots of the system's energy versus time). In order to provide a useful baseline for comparative purposes we also performed extended simulations with the purely ‘steric’ energy model (i.e. one that neglects the electrostatic and hydrophobic interactions); the latter simulations were performed for simulation times of 17.5µs. Each BD simulation using the ‘full’ energy model required more than a year (clock-time) to complete. For both energy models, snapshots taken from the last 15µs of each simulation were used for detailed analysis.

Overall characteristics of the Brownian dynamics simulations

An informative, albeit non-quantitative, impression of the simulation behavior can be obtained by viewing movies of the simulations (Supporting Information). In some respects, these movies can be considered a key result of this work: they represent, in effect, dynamic analogs of the highly influential pictorial representations pioneered by Goodsell [20]. Examination of a typical movie obtained from a simulation performed with the ‘steric’ energy model shows the simulated motion to be rapid, chaotic and obviously Brownian. For the more realistic ‘full’ model, on the other hand, motion is somewhat slower-paced, and molecules can be seen to fluctuate between engagement in short-lived associations and periods of relatively free diffusion.

We can place these observations on a more quantitative footing, and obtain an indication of the extent of sampling achieved in 15µs of simulation, from the remaining panels of Figure 2. Figure 2B shows the maximum distances moved, on average, by each molecule type during simulations performed with the ‘full’ and ‘steric’ energy models; all distances are expressed relative to the diameter of the diffusing molecule. In the case of GFP with the ‘full’ energy model, for example, each molecule travels, on average, approximately 6 molecular diameters (i.e. 320Å) from its position at the beginning of the simulation. Since the data in Figure 2B are plotted versus molecular weight it is apparent that 15µs of simulation is sufficient for the smaller macromolecules to move very significant distances, while for the largest macromolecules (the 30S and 50S ribosomal subunits), little motion away from the initial position is achieved. On this basis alone, therefore, we expect the estimates of diffusional behavior for the smaller macromolecules to be somewhat more precise than those of the larger macromolecules. A second measure of the extent of sampling achieved during each simulation period is provided by plotting the number of unique interaction partners encountered by each type of macromolecule as a function of the simulation time (Figure 2C). Encouragingly, most molecule types encounter many unique neighbors over the course of 15µs: during a typical simulation with the ‘full’ model, for example, each GFP molecule encounters ∼80 different neighbors. Just as importantly, the total numbers of unique neighbors continues to increase even toward the end of the simulation period: this indicates that the cytoplasm model remains highly dynamic and does not tend to ‘freeze’ as the simulation progresses.

As might be expected, the average numbers of neighbors that a macromolecule possesses at any instant scales essentially monotonically with its molecular weight: the average number of macromolecules in the immediate neighborhood of a GFP molecule, for example, is only ∼5 while for the 50S ribosomal subunit it is more than 25 (Figure 2D). The time constants for the dissociation of these neighboring interactions – which in all cases are in the microsecond range – also scale straightforwardly with the molecular weight (Figure 2E), indicating that molecules remain in the neighborhood of larger macromolecules for somewhat longer periods of time than they do with smaller macromolecules. The data shown in Figures 2C and 2D can be combined to provide an estimate of the number of times that each molecule's entire complement of neighbors is replaced during the simulations (Figure 2F). Interestingly, while the overall trend is such that smaller macromolecules encounter a more dynamic constellation of neighbors even the largest macromolecules experience a significant number of environmental changes during the 15µs simulation period. While each GFP molecule, for example, effectively ‘shed its skin’ of neighbors a total of ∼14 times, even the 50S ribosomal subunit undergoes ∼5 such transformations (Fig. 2F). This observation suggests that the limited diffusional exploration carried out by the largest macromolecules evident in Figure 2B may, in at least one important respect, give a misleadingly low indication of the extent of configurational sampling achieved in the simulations: it is in fact, possible for a completely static macromolecule to rapidly encounter widely different microenvironments simply by virtue of the dynamic exchange of its smaller, more mobile neighbors.

Translational and rotational diffusion

While it was noted above that the long-time D^L_trans value of GFP obtained with the ‘full’ energy model is in good agreement with in vivo measurements (Figure 2A), there are other aspects of diffusional behavior in the simulations that warrant examination. One question that is of interest is how the observed D_trans values of macromolecules depend on the observation interval, δt, over which their diffusion is monitored (see Methods). The answer to this question is plotted in Figures 3A and 3B for the three most abundant members of the cytoplasm model (MetE, TufA and CspC); these proteins have been chosen for closer examination because their high abundance yields the most statistically robust numbers, but very similar results are obtained for the other constituents of the model. Figure 3A plots the computed D_trans values of the three proteins versus δt for both the ‘full’ and ‘steric’ energy models. The clear variation of D_trans with δt seen for all three proteins is indicative of ‘anomalous’ diffusion [21]–; the magnitude of the anomaly is conventionally expressed by the anomality exponent, α, (Methods) which is plotted for the same proteins, again versus δt, in Figure 3B. Examination of this figure shows that with the ‘steric’ energy model, the diffusion of all three proteins progresses from being normal (α∼1), to transiently subdiffusive (α<1), to normal again as the observation interval increases from δt∼100ps to δt∼10ns to δt∼1µs. With the ‘full’ model, in contrast, macromolecules exhibit transiently anomalous subdiffusion even at the shortest observation intervals; again however, a slow, but unequivocal return toward normal diffusion occurs on a high microsecond timescale. The same qualitative features are seen for all other molecule types although, for the largest macromolecules or those with the very lowest copy numbers, it is not always clear that sampling is sufficient to be absolutely certain of a return to normal diffusion at the longest δt values. At very short values of δt however we can obtain quite precise values of α for all molecule types; when these are plotted versus molecular weight (Figure 3C) it is apparent that while there is a clear difference between the values obtained with the two energy models, and a clear size-dependence of α with the ‘steric’ model, there is no such obvious trend with the ‘full’ model.

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Figure 3. Translational and rotational diffusion in the cytoplasm model.

A. D_trans values for the three most abundant proteins plotted versus observation interval δt; error bars indicate the standard deviation of values obtained from three independent simulations; solid lines represent fits to the data obtained by integrating the analytical functions shown in the next panel. B. Computed anomality exponents, α, obtained by numerically differentiating the D_trans values shown in A; solid lines represent fits to the data using an analytical function defined in Methods. C. Anomality exponent, α, computed at the shortest accessible time interval (δt_mid = 144ps) plotted for all molecule types versus molecular weight; error bars represent standard deviations from the three independent BD simulations. D. Long-time D_trans values expressed relative to infinite-dilution values plotted versus molecular weight of each molecule type; asterisk denotes GFP. E. Short-time D_rot values expressed relative to infinite-dilution values plotted versus molecular weight of each molecule type. F. Ratio of the effective translational and effective rotational viscosities, plotted for all molecule types versus molecule weight.

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For both energy models, the plots of α versus δt fit well to an analytical function (solid lines in Figure 3B) that, when integrated, enables an asymptotic long-time translational diffusion coefficient, D^L_trans, to be estimated (see Methods). The observed D^L_trans values of all molecule types are expressed relative to their translational diffusion coefficients at infinite dilution (D⁰_trans) and plotted versus molecular weight in Figure 3D. For both energy models, the ratio D^L_trans/D⁰_trans decreases with increasing molecular weight, which is qualitatively consistent with experimental studies of tracer protein diffusion in simple single-component protein solutions [24] and of Ficoll diffusion in the cytoplasm of mouse 3T3 cells [25]. The poorer correlation obtained for the ‘full’ model (which does not appear to be solely due to incomplete sampling) suggests that translational diffusion in vivo should not be predictable with arbitrary precision solely from knowledge of molecular weight; again, this is in line with the often significant variations observed in the in vivo diffusion coefficients of similarly-sized GFP-constructs [15],[26]. It is perhaps worth noting, however, that the computed diffusive behavior of the heterologous GFP – marked by an asterisk in the ‘full’ model data points – is consistent with the general trend established by the endogenous E. coli macromolecules.

The rotational motion of macromolecules is also significantly affected by immersion in the cytoplasm model. In the case of the ‘full’ energy model, the rotational behavior can be fit equally well by either a double-exponential function or a model that describes transiently anomalous rotational diffusion [27]. Since it is the rotational behavior on a nanosecond timescale that is more relevant to experimental measurements (see Methods), we plot the short-time rotational diffusion coefficient, D^S_rot of all molecule types, relative to their rotational diffusion coefficients at infinite dilution, D⁰_rot, in Figure 3E. As would be anticipated given the translational behavior shown above, rotational diffusion is significantly slower with the ‘full’ model than it is with the ‘steric’ model.

Notably, a comparison of Figures 3D and 3E shows that with both energy models rotational diffusion is slowed less by immersion in the cytoplasm than is translational diffusion. This can be viewed as indicating that the two kinds of motion experience different relative viscosities (η_rel^T and η_rel^R for translational and rotational diffusion respectively). Figure 3F plots the ratio of these relative viscosities, η_rel^T/η_rel^R, versus molecular weight for all molecule types. For the abundant proteins MetE, TufA, and CspC, and the less abundant GFP, we find the ratio of these relative viscosities, η_rel^T/η_rel^R, to be 3.6, 3.0, 3.2 and 2.5, respectively using the ‘full’ model; perhaps surprisingly, similar numbers are also obtained with the ‘steric’ model (Figure 3F). These computed ratios are in quite good agreement with the value of η_rel^T/η_rel^R of 2.6±0.2 obtained from in vitro data for apomyoglobin diffusion in human serum albumin [28] (see Methods) and the value of η_rel^T/η_rel^R of 2.1±0.3 reported for GFP in Chinese hamster ovary cells [29]; the lower value obtained in the latter case is consistent with the lower macromolecular concentration of the mammalian cytoplasm relative to that of E. coli.

The thermodynamics of protein stability in the cytoplasm model

In addition to the simulations providing direct views of diffusive motions in the cytoplasm, snapshots extracted from the simulations offer an important opportunity to explore the thermodynamic consequences of the cytoplasm on macromolecular stability. Using a variant of Widom's ‘particle-insertion’ method [30], the free energy change that accompanies the insertion of a molecule into the cytoplasm can be rigorously computed by subjecting the molecule to millions of randomized placements (see Methods). We used this approach to compute the cytoplasm's effects on the folding equilibria of selected proteins by performing separate insertion calculations on their native state structures and on ensembles of 1000 unfolded structures generated by a sophisticated conformational sampling method [31]. We focused initially on the only two proteins for which experimental estimates of thermodynamic stability in the E. coli cytoplasm are available: (1) a construct of the λ-repressor N-terminal domain, λ_6-85 [4], which has been found to have essentially identical stability in vivo and in vitro, and (2) the cellular retinoic acid binding protein [7],[32] (CRABP), which has been found to be thermodynamically destabilized in vivo relative to in vitro. Both of these findings – the latter in particular – are non-trivial results to capture since they are inexplicable in terms of conventional macromolecular crowding theory [2],[3],[7],[33],[34] (see below).

We performed thermodynamic calculations under a total of four different scenarios. The first scenario that we examined involved taking cytoplasm snapshots sampled during the ‘steric’ BD simulations, and computing the cytoplasm-interaction energies of the folded and unfolded conformations with the same ‘steric’ energy model: this scenario corresponds to that considered in conventional models of macromolecular crowding effects [2]. In this case, the differences between the folding free energies in vivo and in vitro are computed to be +1.3±0.0 and +2.2±0.1 kcal/mol for λ_6-85 and CRABP respectively (blue bars in Figure 4A), with the positive signs indicating that the folding free energies of both proteins are calculated to be more favorable in vivo than in vitro. When compared to the experimental values (red bars in Figure 4A), these results are in poor quantitative agreement for λ_6-85 and are qualitatively wrong for CRABP. In a second scenario, we took cytoplasm snapshots sampled during the ‘full’ model BD simulations, but computed the cytoplasm-interaction energies of folded and unfolded conformations using the simpler ‘steric’ energy model. In this case, the differences between the folding free energies in vivo and in vitro are computed to be +1.0±0.0 and +1.6±0.0 kcal/mol for λ_6-85 and CRABP respectively (cyan bars in Figure 4A). The smaller crowding effects obtained in this situation reflect the fact that during the ‘full’ BD simulations transient clustering of molecules creates bigger voids in the system; again however, these computed results are in poor quantitative agreement with experiment for λ_6-85 and are in qualitative disagreement with experiment for CRABP.

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Figure 4. Thermodynamic effects of the cytoplasm model on protein folding and association equilibria.

A. Computed stabilization of the folded state relative to the unfolded state for two experimentally-studied proteins; experimental data for Lrp (λ_6-85) and CRABP taken from refs [4] and [32] respectively. ‘steric sampling’ indicates that insertions were performed on snapshots taken from a BD simulation performed with the ‘steric’ energy function; ‘steric scoring’ etc. indicates that the ‘steric’ energy function was used to calculate the cytoplasm-interaction energies, E_int, of the inserted proteins. B. Histogram of interaction energies, E_int, obtained for all non-clashing insertions of the folded and unfolded state conformations of CRABP with snapshots sampled from the ‘full’ model BD simulations; inset shows the same for λ_6-85. C. Distribution of radius of gyration values for the 1000 unfolded conformations generated with the RCG software [31]; distributions are plotted in order of increasing molecular weight of the studied proteins. D. Same as A. but showing computed results for six other proteins, listed in order of increasing molecular weight. E. Computed stabilization of dimeric form relative to two separated monomers for eleven proteins, listed in order of increasing molecular weight. F. Computed stabilization of oligomeric form relative to separated monomers for three proteins.

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A third scenario that we examined involved taking cytoplasm snapshots sampled during the ‘steric’ BD simulations and computing the cytoplasm-interaction energies with the ‘full’ energy model. In this case, the differences between the folding free energies in vivo and in vitro are computed to be +0.1±0.5 and −1.8±1.4 kcal/mol for λ_6-85 and CRABP respectively (green bars in Figure 4A), both of which, notwithstanding the larger error bars, are in rather good agreement with the experimental results. Finally, we took cytoplasm snapshots sampled during the ‘full’ model BD simulations and computed the cytoplasm-interaction energies with the same ‘full’ energy model. In this fourth scenario – which on the basis of the diffusional properties described above would be hoped to provide the most realistic description (Figure 2A) – the computed changes in stability amount to +0.3±0.1 and −0.9±0.4 kcal/mol for λ_6-85 and CRABP respectively (yellow bars in Figure 4A); again, these results are in close quantitative agreement with the experimental results. The overall picture that emerges, therefore, is that the experimental results cannot be reproduced, even qualitatively, when the ‘steric’ energy model is used to score the interactions between the folding protein and the cytoplasm environment, but they can be reproduced – and with a perhaps surprisingly high degree of quantitative accuracy – when the ‘full’ energy model is used in the particle-insertion calculations. Furthermore, the fact that similarly good results are obtained regardless of which energy model was used in the BD simulations suggests that, for such calculations, the method of sampling the cytoplasm's configurations is perhaps less important than the nature of the energy function used to describe the protein of interest's interaction with it.

Histograms of the computed interaction energies of the folded and unfolded state with the cytoplasm explain why the predictions of the ‘full’ model successfully reproduce experiment, and deviate so significantly from the predictions of the purely steric model: for both proteins, but especially so in the case of CRABP, the unfolded state conformations are computed to have somewhat more favorable energetic interactions with the cytoplasm than the folded state conformations (Figure 4B). The consequence is that while the excluded-volume (crowding) effect experienced by both proteins undoubtedly significantly stabilizes their folded states relative to their unfolded states (e.g. see the blue and cyan bars in Figure 4A), the effect is counterbalanced by the more favorable energetic interactions engaged in by the unfolded state conformations.

To explore the potential generality of this latter result, we performed identical calculations for a number of other monomeric proteins using snapshots taken from the ‘full’ model BD simulations; histograms illustrating the size distributions of the unfolded states of the tested proteins are shown in Figure 4C. The computed changes in their folding free energies are plotted in order of increasing molecular weight in Figure 4D. As before, when the ‘steric’ energy model is used to compute the cytoplasm-interaction energies the proteins' stabilities are computed to increase (white bars in Figure 4D); the computed stability changes scale broadly with the molecular weight of the protein, reflecting the greater relative difference between folded and unfolded state dimensions of larger proteins. In contrast, when the ‘full’ energy model is used to compute the cytoplasm-interaction energies, the molecular weight dependence is lost (dark grey bars in Figure 4D): some proteins are computed to be stabilized and others destabilized in vivo relative to in vitro (in no case however is the extent of destabilization sufficient to predict that the proteins will be predominantly unfolded in vivo). These results suggest that differences between the in vitro and in vivo thermodynamic stabilities will vary significantly with the identity of the protein.

The thermodynamics of protein-protein interactions in the cytoplasm model

We performed similar calculations to explore the potential thermodynamic effects of immersion in the cytoplasm on a variety of protein-protein associations. For the formation of homo-dimeric complexes (Figure 4E), we again find that the excluded-volume crowding effect, which alone stabilizes dimers relative to separated monomers by on average 1.1±0.3 kcal/mol, is largely cancelled by the more favorable energetic interactions that the monomers form with the cytoplasm constituents: when the ‘full’ energy model is used the stabilization of the dimeric forms by the cytoplasm is computed to be, on average, only 0.1±0.3 kcal/mol. For the assembly of the trimeric nucleus [35] of the bacterial cytoskeletal protein ParM from three separated monomers, we find that the stabilization predicted with the ‘full’ energetic model is also significantly lower than that predicted from the crowding effect alone (Figure 4F); again, the smaller value appears more consistent with the close similarities between the polymerization behavior of ParM observed in vitro and in vivo [36]. Finally, we performed calculations on the assembly of two published (but putative) structural models of amyloid-like aggregates [37],[38], each formed by association of 8 monomer units (Figure 4F). For one of these two cases, the aggregation of an SH3 domain [37], we find that the use of the ‘full’ model predicts a slightly greater stabilization than that predicted solely on the basis of the crowding effect; the additional stabilization observed in this case results from the protein's interactions with the cytoplasm being dominated by repulsive electrostatic interactions, which, on average, are diminished in the aggregated state (see Figure S3).

Selection of the constituents for the cytoplasm model

When this work was initiated, the only large-scale quantitative study of the E. coli proteome was that reported by Link et al. [12] who experimentally measured levels of >200 of the most abundant proteins present in E. coli. A number of other quantitative proteomic studies of E. coli have since been reported [40],[70],[71], and, since this work was completed, quantitative estimates of metabolite concentrations have also become available [72]. Restrictions on computer memory (4GB of RAM for all servers used) meant that the total number of different types of macromolecules that could be realistically modeled was limited to 51: these would be 50 types of E. coli macromolecule plus the Green Fluorescent Protein (GFP). Although including only 50 different types of macromolecules means that the model underestimates the structural diversity of the cytoplasm, it is important to note that the macromolecules selected for inclusion account for 85% (by number of protein chains) and 86% (by mass) of all the cytoplasmic proteins quantified and identified in Table 4 of Link et al. [12].

Of the 50 types of E. coli macromolecules to be included in the model, 45 would be proteins. These were selected by working down the list identified by Link et al. in order of decreasing abundance, selecting only those proteins (a) for which high-resolution structures were then available in the Protein Data Bank [13] (PDB) or for which reasonable homology models could be constructed (see below), and (b) for which the cytoplasm was unambiguously identified as the major cellular location in the EcoCyc [73] and/or CCDB [74] databases. A full list of all potentially cytoplasmic proteins identified and quantified in Table 4 of Link et al. (under minimal media conditions), arranged in decreasing order of chain-abundance, is shown in Table S1; asterisks in the columns headed ‘Mod.’ denote those proteins included in our cytoplasm model. It is an indication of the tremendous coverage of the structural proteome that has been achieved by the structural biology community that we were able to obtain, or build, reasonable structural models for all of the 30 most abundant cytoplasmic proteins identified by Link et al. [12]. In addition to the 45 different types of proteins, 5 types of macromolecule were RNAs or RNA-protein complexes: these were the two ribosomal subunits (50S and 30S), and three typical tRNAs for which structures were available: (tRNA-Gln, tRNA-Phe and tRNA-Cys). It is to be noted that we did not model complete (translating) 70S ribosomes owing (a) to the inherent difficulties in modeling the flexible mRNA, and (b) to the absence – at the time this work was begun – of a three-dimensional structure showing the arrangement of multiple 70S ribosomes in a polyribosome [75].

The total number of molecules in the simulations was set to 1008 (eight copies of GFP and 1000 E. coli macromolecules). This number was chosen so that the eventual assembled cytoplasm model would be large enough to provide a reasonable representation of the environment while still allowing simulations of up to 20µs to be performed (albeit over the course of more than a year clock-time). The linear dimensions of the final modeled system (808.4Å in each of the x, y and z directions) correspond to approximately one-twelfth of the diameter of a typical E. coli cell [76]. A summary of the macromolecules selected, their subunit compositions, the PDB codes of their originating structures, and the degree of sequence coverage achieved by the structural models, is presented in Table S2. Using composition estimates provided by Neidhardt et al. [76] as a guide, we set the total concentration of macromolecules in the model (excluding the ‘tracer’ GFP) to 275 g/l; this is slightly on the low side of the rough values of 300–340 g/l estimated independently by Zimmerman and Trach [77]. Of this, 55g/l (i.e. 20% of the total) is contributed by RNA, with 15% of the RNA dry weight contribution being made by tRNA and the remainder being made by ribosomal RNA [76]. mRNA, which accounts for only ∼4% of the total dry weight of RNA in the cell, is omitted from the present model. The remaining 219g/l (i.e. 80%) of the model is contributed by proteins; this percentage is deliberately set somewhat higher than the 55% contribution to the dry weight of the entire cell estimated by Neidhardt et al. [76] in order to compensate for the missing volume of components that are not explicitly represented in the model (DNA, mRNA, lipid, lipopolysaccharide, murein, and glycogen). If one takes the specific volumes of proteins and RNA to be 0.73ml/g and 0.58ml/g respectively [77], the total volume fraction occupied by macromolecules in the model is 0.19; if instead, an ‘effective’ specific volume of macromolecules suggested by Zimmerman and Trach is used [77] (1.0ml/g), the total volume fraction occupied by the macromolecules in the model amounts to 0.27.

Preparation of the macromolecular structures for simulation

Structures for all selected proteins were identified by performing a BLAST search [78] of the protein's FASTA sequence (as reported in the EcoCyc database) against the entire PDB and selecting the structure with the closest identity to the query sequence using the program BioEdit [79]. The quaternary structure of each selected structure was determined using the PQS web server [80] and was verified, where possible, with the EcoCyc database; it should be noted that correct identification of a protein's quaternary structure is a non-trivial undertaking, and the PQS predictions are unlikely to be 100% reliable [80],[81]. Homology modeling was used for all proteins for which either no E. coli structure was directly available in the PDB, or for which a significantly greater coverage of the sequence could be obtained through the use of a non-E. coli structure. All homology modeling was performed using the SWISS-MODEL web server [82] via the so-called “First Approach mode”; for oligomeric proteins each individual chain was homology-modeled independently.

Any sidechains missing from a structure were built in using the molecular modeling program WHATIF [83]. Hydrogens were then added, and partial charges and radii were assigned to atoms using the program PDB2PQR [84]. For proteins, partial charges and atomic radii were taken directly from the PARSE parameter set [85]. For nucleic acids, which are not represented in the PARSE parameter set, partial charges were instead assigned from the CHARMM23 parameter set [86]; partial charges for the modified bases of tRNAs, such as pseudouridine, were assigned based on similarity to functional groups already represented in the parameter sets. The protonation states of all protein ionizable residues were assigned using the fast empirical algorithm PropKa [87]; for these calculations, the pH was set to 7.6, the estimated pH of the E. coli cytoplasm [76]. With each structure complete, infinite-dilution translational and rotational diffusion coefficients – which are necessary input parameters for BD simulations [14] – were calculated with the program HYDROPRO [88] using default parameters. For the latter calculations we assumed a solvent viscosity, η, of 0.89cP, which corresponds to the viscosity of pure water at 25°C; given that the most recent estimate of the total metabolite concentration in the E. coli cytoplasm is ∼300mM [72] we do not anticipate, based on what we currently know, that the viscosity of the solvent environment will be hugely altered from the pure water value.

The final stage of preparation for each molecule involved the calculation of electrostatic potential grids; these were computed in all cases by using the APBS software [89] to solve the non-linear Poisson-Boltzmann (PB) equation [90]. As in our previous BD study of single-component protein solutions [19], two cubic electrostatic potential grids were computed for each type of macromolecule: (a) a comparatively fine grid, of spacing 2Å, with dimensions sufficient to encompass a 20Å shell around the macromolecular surface, and (b) a coarse, long-range grid, of spacing 4Å, that extends at least 50% further in each direction than the smaller grid. The use of a 2Å grid spacing for the higher resolution grids, rather than the 1Å grid spacing used in our previous simulations [19], was necessary in order to fit all potential grids into the available 4GB of RAM. This spacing is, however, sufficiently detailed that at least two grid points always intervene between interacting atoms even when they are at the closest possible separation distance (4.5Å); significant numerical instabilities in the calculated electrostatic forces do not, therefore, arise. In all PB calculations the solvent dielectric was set to 78.0 and the internal dielectric of the macromolecule was set to 12.0, with the boundary between the two being determined by the cubic-spline surface [91] implemented in APBS [89]. Use of an internal dielectric of 12.0 is intended to provide a simple, averaged description of the different dielectric responses of macromolecular interiors and exteriors [19],[92],[93]. The ionic strength in all PB calculations was set to 150mM. With the electrostatic potentials in hand, ‘effective charges’ were computed for each molecule type using the procedure established by Gabdoulline & Wade [94],[95]. Finally, as in our previous work [19], simulations were accelerated by retaining, in addition to the effective charges, only those non-hydrogen atoms that were solvent-exposed: these atoms were identified using the ACC tool within APBS [89], with a 4Å solvent probe.

Brownian dynamics simulation protocol

The BD software used for the simulations is an extension of the methodology developed and tested in our previous work on pure protein solutions [19]. Modifications were made to the software to improve memory usage so that 102 electrostatic potential grids could be simultaneously held in memory; in addition, toward the end of this study, loop-level parallelization of a number of key loops was implemented with OpenMP (http://www.openmp.org) to accelerate computations by a factor of ∼4.

All simulations were performed under periodic boundary conditions [96] in a cubic cell with edges of 808.4Å. The initial configuration of each system had eight GFP molecules evenly positioned at the center of the eight octants of the cell; all other macromolecules were initially positioned by performing random translations and rotations within the cell subject to the requirement that there was at least a 10Å separation between the surfaces of all neighbors. Three independent configurations were generated in this way by use of different random seeds; views of each system before and after 15µs of simulation are shown in Fig. S1. As in our previous work, BD simulations were conducted using the Ermak-McCammon algorithm [97] with a time step of 2.5ps, with additional algorithmic measures being taken to ensure that no atom-atom distances at the completion of each timestep were less than 4.5Å. For subsequent analysis of the simulations, the 3D translational vector and the 3×3 rotational matrix necessary to specify the position of each macromolecule were recorded every 100ps.

The form of the energy model used to describe intermolecular interactions was identical to that used in our previous work [19]: the effective charge method [94] was used to calculate electrostatic interactions, and a Lennard-Jones potential (comprising 1/r¹² and 1/r⁶ terms) was used to provide a simple combined description of steric, van der Waals and hydrophobic interactions. To accelerate the simulations, the combined non-electrostatic interactions were computed only between atom pairs separated by less than 12Å; a list of all such pairs was continually updated every 40 timesteps (i.e. every 100ps). As in our previous work, we treated the strength of these non-electrostatic interactions, which are determined by the well-depth, ε_LJ, of the Lennard-Jones potential, as the only adjustable parameter of the model. In order to determine the best setting, three independent BD simulations of at least 6µs duration were performed with each of the following ε_LJ values: 0.190, 0.285, 0.3325 and 0.380 kcal/mol. Finally, for comparison purposes, two additional sets of three BD simulations were also performed: these were (a) simulations in which the only the repulsive (1/r¹²-dependent) steric interactions operated (these are the ‘steric’ simulations discussed in the main text) and (b) simulations in which only steric plus electrostatic interactions acted.

Analysis – translational diffusion coefficients

The effective translational diffusion coefficients, D_trans, of molecules were calculated from the simulations using the Einstein equation:(1)where < δr² > is the mean-squared distance traveled by the molecular center of mass in the observation interval, δt; all D_trans values reported in Results are mean values for each molecule type averaged over the number of copies of each type. In cases of ‘normal’ diffusion, the computed D_trans values are independent of δt; in certain cases of diffusion in vivo and in vitro however, anomalous sub-diffusion is observed [8], [21]–[23],[66]; in such cases, the apparent D_trans value is dependent on δt, decreasing with increasing δt. A common way of describing anomalous diffusion involves writing it in the form:(2)where the apparent translational diffusion coefficient D_trans is now written to indicate that it depends on the observation interval and α is the so-called anomalous diffusion (anomality) exponent; α = 1 corresponds to normal diffusion since it leads to D_trans being independent of δt, and α<1 indicates anomalous (sub)diffusion. Taking logarithms and differentiating with respect to log (δt) allows us to write:(3)This enables us to obtain α by numerically differentiating D_trans values computed over a range of δt values; in practice we computed D_trans at δt values of 100, 200, 300, 600, 1000, … ps, and obtained α at the logarithmic mid-point, δt_mid, of these time-intervals, δt_mid = 141, 245, 424, … ps.

Plots of α versus log (δt_mid) for macromolecules simulated with both the ‘steric’ and ‘full’ energy models all indicated that α itself was dependent on δt_mid, thus signifying that diffusion was transiently anomalous. To our knowledge, there is no explicitly derived functional form that describes the expected dependence of α on δt for transient anomalous diffusion. We found however that the data fit well to the following empirical functional form (see Fig. 3B):(4)where α₀ is a constant, a and b are parameters that describe the amplitude of the δt-dependent changes to α, and τ_short and τ_long are, respectively, the timescales over which α first decreases, and then returns to one, with increasing δt. Plots of α versus δt for all molecule types were fit to the above functional form with SigmaPlot [98]: fits were performed using all datapoints from the shortest δt_mid value up to the first datapoint that had a percent error exceeding ∼25% (obtained by comparing the α values computed from the three independent BD simulations), or that deviated qualitatively from the trend. To ensure that the latter criterion did not drastically affect the results, the fits were repeated retaining even those datapoints that qualitatively deviated; essentially the same behavior was obtained but with slightly greater values of τ_long. Regressed values of τ_short and τ_long are plotted versus molecular weight for all molecule types in Figs. S4 and S5 respectively.

Having fit a function to the observed dependence of α on δt, it was numerically integrated to obtain an extrapolated, asymptotic long-time D_trans value using the D_trans value at δt = 100ps as the starting point for the integration. The quality of fits of the integrated D_trans values (for the most abundant proteins) is indicated by the solid lines in Fig. 3A.

Analysis – rotational diffusion coefficients

Effective rotational diffusion coefficients were computed from the time-dependent behavior of the 3×3 rotational matrix recorded every 100ps for every molecule during the simulations. For each of the three rotational axes, an autocorrelation function, θ (δt), was calculated as:(5)where e (0) and e (δt) are unit vectors pointing along one of the rotational axes at time t = 0 and t = δt respectively, and the brackets indicate an average over all possible initial timepoints; the three computed autocorrelation functions were averaged to give a single decay function consistent with the isotropic rotation that we assumed for all molecule types at infinite dilution. Since the resulting averaged autocorrelation function for the ‘full’ energy model did not fit well to a single-exponential decay, and given that translational diffusion was clearly transiently anomalous, we decided to use the following functional form proposed recently for transiently anomalous rotational diffusion [27]:(6)where θ₀ is the value of the autocorrelation function at δt = 0 (always 1), a is a parameter, τ_rot is a long-time rotational correlation time (which dominates as δt→∞), and τ_rel is the timescale over which a faster, short-time rotational relaxation gives way to the slower rotation characterized by τ_rot. The above functional form was fit to computed values of θ for each molecule type over a range of δt values up to 1µs; the r² values for these fits were all in excess of 0.999. An example of such fits for the most abundant proteins is shown in Fig. S6. The long-time rotational diffusion coefficient, D^L_rot, is then obtained using the relationship:(7)and the short-time rotational diffusion coefficient, D^S_rot, is obtained from [27]:(8)The computed ratios D^L_rot/D⁰_rot and D^S_rot/D⁰_rot obtained with the ‘full’ energy model are plotted for all molecule types versus their molecular weights in Fig. S7; a plot of the parameter a versus molecular weight shows no obvious relationship (not shown).

Analysis – literature estimates of relative translational and rotational viscosities

Comparison of the simulated translational and rotational diffusion coefficients with the infinite-dilution values that are input parameters for the simulations provides an indication of the relative viscosities experienced during the two types of motion. From studies of GFP diffusion in Chinese hamster ovary cells, the Verkman group reports [29] a relative viscosity experienced by translational motion, η_rel^T = 3.2±0.2, and a relative viscosity experienced by rotational motion, η_rel^R = 1.5±0.1. Combining these numbers gives a ratio, η_rel^T/η_rel^R of 2.1±0.3, indicating that the effective relative viscosity experienced by translational motion is roughly twice that experienced by rotational motion in mammalian cells.

A second estimate of the η_rel^T/η_rel^R ratio can be obtained from the work of Zorrilla et al. [28],[99]: these authors have reported measurements of the translational diffusion coefficients of apomyoglobin (17kDa) using fluorescence correlation spectroscopy (FCS) measurements [28] and have compared them with rotational diffusion coefficients that they had previously measured [99] for the same system using time-resolved fluorescence depolarization experiments. They report measurements for two different background proteins, RNaseA and human serum albumin (HSA); we focus on the data reported for the latter since its molecular weight (67kDa) is much closer to the number-averaged molecular weight of the macromolecules in our cytoplasm model (87kDa), than is the molecular weight of RNaseA (14kDa).

The data reported by Zorrilla et al. are expressed relative to the macroscopic viscosity, η_m, of the protein solution (measured with an Ostwald viscometer). They report that η_m fits to the following functional form, η_m = η₀ exp (Ac/(1−Bc)), where η₀ is the viscosity of pure water, c is the background protein's concentration in mg/ml, and A and B are background-dependent constants: A = 2.7×10⁻³ ml/mg and B = 1.3×10⁻³ ml/mg for HSA [99]. Using these values we obtain a macroscopic viscosity for a 275 mg/ml HSA solution of 3.155 η₀. Using the data given in Table 2 of ref. 49, the effective viscosity experienced by the translational motion of apomyoglobin in HSA is expressed as η_rel^T = (η_m/η₀)^1.28, which from above means that we can write η_rel^T = 3.155^1.28 = 4.35; following similar calculations the effective viscosity experienced by the rotational motion is η_rel^R = (η_m/η₀)^0.44 = 3.155^0.44 = 1.66. Together, these numbers translate into a value of η_rel^T/η_rel^R of 2.6±0.2.

As noted in the main text, we find that both the translational and rotational diffusion coefficients of molecules vary with the time interval, δt, over which diffusion is observed. While the observation of this transient anomalous diffusion is significant in its own right it takes on added significance when comparing the relative viscosities experienced by translational and rotational motion. This is because the timescales over which the two types of experiments are conducted are quite different: translational diffusion coefficients are obtained from FCS experiments by fitting to an autocorrelation function over a timescale extending from microseconds to seconds [21],[22],[66] while rotational diffusion coefficients are obtained from fits to data obtained over a nanosecond timescale [28],[29]. We therefore compare the experimentally derived relative viscosities quoted above with diffusion coefficients computed from the BD simulations on the same timescales, i.e. we compare with the ratio of the long-time translational diffusion coefficient D^L_trans and the short-time rotational diffusion coefficient, D^S_rot (see Fig. 3F).

Analysis – monitoring of intermolecular contacts

The intermolecular contacts engaged in by each molecule were recorded every 100ps during the BD simulations and subsequently analyzed to determine: (a) the average number of neighbors of each molecule type at any given time, (b) the number of unique neighbors encountered by each molecule type during the course of the entire simulations, and (c) the rate of dissociation of intermolecular interactions. The definition of ‘neighbor’ was kept somewhat loose in order to detect all molecules in the immediate environment of the molecule being probed: molecules were assigned as neighbors if any of their atoms were within ∼12Å of each other. The rates at which the neighbors of a particular molecule dissociated were obtained from plots of the fraction of its neighbors, initially present at t = 0, that remained after some time t = δt, averaged over all possible initial timepoints. In order to obtain the characteristic neighbor-decay rate for each particular type of molecule, such plots were averaged over all molecules of that type. The resulting plots are found to follow biexponential kinetics: (a) a very fast decay process (τ_fast) that typically has an amplitude of ∼0.7 and is due to loss of neighbors that interact only peripherally with the molecule of interest, and (b) a slower decay process (τ_slow) that has an average amplitude of ∼0.3 and is due to loss of those neighbors that form bona fide intermolecular contacts. Typical fits for these data are shown in Fig. S8.

Method for calculating thermodynamics in the cytoplasm

The effects of immersion in the cytoplasm on the thermodynamics of protein folding and protein-protein association were computed using the particle insertion technique first outlined by Widom [30]. For small perturbations, the free energy change, ΔG, for transferring a molecule from an environment free of any interacting macromolecules to the cytoplasm environment can be rigorously expressed as:(9)where E_int is the interaction energy of the molecule with the constituents of the cytoplasm, R is the Gas constant, T is the temperature, and the brackets indicate an average over randomly selected insertion positions and configurations of the cytoplasm environment. In order to assess the likely effects of the cytoplasm on a thermodynamic process (such as protein folding) therefore, separate particle-insertion calculations are required for both the initial state (e.g. unfolded protein) and the final state (e.g. folded protein). Such calculations give the free energy changes for the vertical processes in the thermodynamic cycle shown below:(10)Since free energy is a state function, the difference between the free energy changes of the horizontal processes is equal to the difference between the free energy changes of the vertical processes. We can therefore write the difference between the free energy change for the process in vivo and in vitro, ΔΔG, as:(11)The effect of the cytoplasm on the free energy change for a process can therefore be calculated without needing to know the actual value of the free energy change for the process in vitro. A conceptually similar but different approach to computing thermodynamics in crowded solutions has recently been outlined by Zhou and co-workers [100]. Code for performing particle-insertion calculations was generated by modifying the existing BD simulation program; prior to performing large-scale explorations of protein folding and association thermodynamics, the code's correctness was first checked by comparing its predictions for the free energy cost of placing a sphere into a solution of spheres with the corresponding predictions of scaled particle theory [101],[102].

Cytoplasm effects on protein folding equilibria

Calculations of the cytoplasm's thermodynamic effects initially focused on protein folding equilibria. In addition to calculating the folding thermodynamics of six proteins already present in the cytoplasm model (Adk, Bcp, CspC, Efp, GFP and PpiB), we examined two other proteins that have been subject to direct experimental study in vivo: these were the 80-residue λ_6-85 construct studied experimentally by Ghaemmaghami and Oas [4] and the 136-residue cellular retinoic acid binding protein (CRABP) investigated by Ignatova, Gierasch and co-workers [7],[32]. The structure of the folded state of λ_6-85 was taken from its crystal structure in complex with operator DNA (pdbcode: 1LMB [103]); the G46A & G48A mutations present in the experimental construct were made using the rotamer-sampling method SCWRL3 [104]. The structure of the folded state of CRABP (pdbcode: 1CBI [105]) was altered to include the R131Q mutation used in the experimental construct [7], but in the absence of direct structural information no attempt was made to model the experimentally-incorporated fluorophore.

The unfolded states of all eight proteins were modeled as ensembles of 1000 unfolded conformations generated using the conformational sampling method developed by the Sosnick group [31]; the code was kindly made available by Dr. Abhishek Jha. This method has been shown to produce models with dimensions in good agreement with experimental estimates [31]. Prior to calculations, the structures of all conformations were completed by adding sidechains with SCWRL3 [104] and by adding hydrogens with the PDBTOPQR utility [84] of APBS [89]. In order to ensure consistency between the BD simulations and the Widom particle-insertion calculations, effective charges and electrostatic potential grids were calculated for all conformations (both folded and unfolded) using the exact same protocol employed with the rigid protein models of the cytoplasm model (see above).

For each protein, a large number of random trial positions were attempted with both the single, folded state structure and the 1000 unfolded state conformations; each trial consisted of a different randomly selected translation and rotation. For the folded state structure, a total of 25 million trials were attempted; for the unfolded state, 250,000 trials were attempted for each of the 1000 conformations (to give a total of 250 million trials for each cytoplasm ‘snapshot’ studied). For each trial position, the interaction energy of the protein with the surrounding cytoplasm was calculated with (a) the ‘full’ energetic model, which includes electrostatic, steric and hydrophobic contributions, and (b) the ‘steric’ energetic model. To simplify the latter calculations, only two possible energies were allowed: the interaction energy, E_int, was set to +∞ if any of the protein's atoms came within 4.5Å of any of the cytoplasm atoms, and was set to zero if not; this binary scoring method is effectively identical to that used in most examinations of excluded-volume (crowding) effects. Due to the very significant computational expense associated with the particle-insertion calculations, they were applied only to the final ‘snapshot’ of the three independent BD simulations performed with the ‘full’ and ‘steric’ models. Error bars for all reported free energy changes were therefore calculated as the standard deviation of the computed values obtained from the three different system ‘snapshots’. The total number of unfolded and folded-state trial positions that were accepted and rejected for each protein, for each of the three ‘full’ model cytoplasm ‘snapshots’ are listed in Table S3.

Cytoplasm effects on protein association equilibria

A very similar protocol was used to calculate the effects of the cytoplasm on a variety of protein association reactions. Calculations on each assembled protein complex were performed exactly as described above. Calculations on each disassembled complex – e.g. two separated protein monomers in the case of a dimerization reaction – were carried out by performing insertions of all components simultaneously; importantly, each randomized placement was first screened to ensure that there were no steric clashes between any of the inserted components before their interactions with the cytoplasm were evaluated. As might be expected, the requirement of simultaneously placing multiple molecules into the cytoplasm meant that in some cases very large numbers of trial positions were required in order to obtain reasonably converged results. Owing to the significant computational expense, therefore, calculations were only performed on snapshots taken from BD simulations performed with the ‘full’ energy model. In addition, since the Boltzmann-weighting of the sampled interaction energies can contribute significant noise in cases where the number of accepted placements are comparatively low, the cytoplasm-interaction energy distributions were first smoothed by fitting to sums of three Gaussians using SigmaPlot [98] (see Fig. S9 for a typical fit). The total numbers of accepted and attempted insertions for the various association reactions studied are listed in Table S4.

Dimerization equilibria were investigated by performing separate particle-insertion calculations on the dimeric forms and the monomeric forms; for such calculations it was assumed that no structural change (e.g. unfolding) occurs when the two monomers are separated. The trimerization equilibrium of ParM was investigated in analogous fashion, by performing calculations on a trimer extracted from the ParM filament model (pdbcode: 2QU4 [106]). The aggregation of a poly-Q-inserted RNaseA to form an amyloid fiber was studied using the theoretical model developed by Eisenberg and co-workers (pdbcode: 2APU; [38]). The model deposited in the PDB contains 56 aggregated monomeric units; the largest aggregate for which we could obtain reasonably precise free energy estimates however contained eight monomeric units (Fig. 4F). Since formation of the amyloid structure involves a significant change in conformation, the use of monomeric structures extracted without modification from the aggregate model would be inappropriate. Instead, the structure of the monomeric poly-Q-inserted RNaseA was taken from the crystal structure reported by the Eisenberg group (pdbcode: 2APQ [38]). In order to ensure sequence-consistency with the amyloid model, a A131H mutation was made with SCWRL3 [104]. In addition, since the monomeric structure has no resolved coordinates for the inserted GQQQQQQQQQQGNP stretch this region was model-built using the loop-building program Loopy [107]. The second aggregate structure studied was a theoretical model of SH3 domain aggregation proposed by the Shakhnovich group [39] and kindly made available to the authors by Dr. Feng Ding (UNC; personal communication). This structure contains only C_α atoms so complete backbone coordinates were first constructed using the SABBAC webserver [108] (http://bioserv.rpbs.jussieu.fr/cgi-bin/SABBAC) before sidechain positions were constructed using SCWRL3. Owing to the structure's origins being a C_α-only model we were unable to add sidechains in such a way that the assembled aggregate model was free of internal steric clashes; this, however, does not significantly affect our ability to estimate the model's interaction with the cytoplasm environment. As with the RNaseA amyloid model, it would be inappropriate to assume that the conformations of unaggregated monomeric units are identical to those found in the amyloid model; instead therefore the conformation of the monomeric SH3 domain was taken from the crystal structure (pdbcode: 1NLO [109]).

Two movies, each showing 1.8µs of simulation, are provided as separate Quicktime .mov files. Video S1 shows a BD simulation performed with the ‘full’ energy model; Video S2 shows a BD simulation performed with the ‘steric’ energy model. File size restrictions at the PLoS website have limited the size and resolution of the uploaded movies to be used for review. Higher resolution movies are available to readers at the authors' website: http://dadiddly.biochem.uiowa.edu/Elcock_Lab/Movies.html.