Thermodynamics of Macromolecular Association in Heterogeneous

Oct 31, 2016 - Hereafter, we call F(σαγ) the contact free energy. In Model 2 (see right side of Figure 1b), the product is represented as a sphere ...
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Thermodynamics of Macromolecular Association in Heterogeneous Crowding Environments: Theoretical and Simulation Studies with a Simplified Model Tadashi Ando,† Isseki Yu,‡ Michael Feig,†,§ and Yuji Sugita*,†,‡,∥ †

RIKEN Quantitative Biology Center (QBiC), Integrated Innovation Building 7F, 6-7-1 Minatojima-minamimachi, Chuo-ku, Kobe, Hyogo 650-0047, Japan ‡ RIKEN Theoretical Molecular Science Laboratory and iTHES, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan § Department of Biochemistry & Molecular Biology, Michigan State University, East Lansing, Michigan 48824, United States ∥ RIKEN Advanced Institute for Computational Science (AICS), 7-1-26 Minatojima-minamimachi, Chuo-ku, Kobe, Hyogo 650-0047, Japan S Supporting Information *

ABSTRACT: The cytoplasm of a cell is crowded with many different kinds of macromolecules. The macromolecular crowding affects the thermodynamics and kinetics of biological reactions in a living cell, such as protein folding, association, and diffusion. Theoretical and simulation studies using simplified models focus on the essential features of the crowding effects and provide a basis for analyzing experimental data. In most of the previous studies on the crowding effects, a uniform crowder size is assumed, which is in contrast to the inhomogeneous size distribution of macromolecules in a living cell. Here, we evaluate the free energy changes upon macromolecular association in a cell-like inhomogeneous crowding system via a theory of hard-sphere fluids and free energy calculations using Brownian dynamics trajectories. The inhomogeneous crowding model based on 41 different types of macromolecules represented by spheres with different radii mimics the physiological concentrations of macromolecules in the cytoplasm of Mycoplasma genitalium. The free energy changes of macromolecular association evaluated by the theory and simulations were in good agreement with each other. The crowder size distribution affects both specific and nonspecific molecular associations, suggesting that not only the volume fraction but also the size distribution of macromolecules are important factors for evaluating in vivo crowding effects. This study relates in vitro experiments on macromolecular crowding to in vivo crowding effects by using the theory of hard-sphere fluids with crowder-size heterogeneity.

I. INTRODUCTION The cellular interior is crowded with macromolecules, which is one of the most characteristic features of living cells.1 Macromolecules have evolved in the crowded environments to express their functions correctly and efficiently. It is now well recognized that macromolecular crowding influences the thermodynamics and kinetics of biological reactions, such as protein folding, association, and diffusion.2−5 Since these reactions consequently affect cellular processes, macromolecular crowding is a crucial component to understand the nature of living systems. Systematic studies of the crowding effects on the thermodynamics and kinetics of macromolecules were conducted first by Ogston6 and Laurent.7 After these studies, many experiments on macromolecular crowding have been reported.2,4 In vitro experimental studies often employ poly(ethylene glycol) (PEG), Ficoll, dextran, or soluble globule © 2016 American Chemical Society

proteins like ovalbumin, hemoglobin, or bovine serum albumin as crowding agents to mimic the crowded environments in test tubes, since they are physically and chemically well characterized and easy to handle even at high concentrations (see references cited in refs 2−4 for examples). Besides the experimental studies, theoretical models have been applied extensively to obtain a quantitative understanding of the crowding effects. Minton first evaluated the influence of macromolecular crowding on the thermodynamic activity of proteins using scaled particle theory (SPT) with a simplified geometrical representation of molecular shapes.8,9 Since then, this theory has played an important role on the theoretical studies of the crowding effects.2−4 All-atom molecular dynamics Received: June 20, 2016 Revised: October 25, 2016 Published: October 31, 2016 11856

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II. THEORY AND METHODS II.A. The Cytoplasmic Model Used in This Study. Recently, Feig et al.20 built a comprehensive atomistic model for the cytoplasm of M. genitalium by integrating genomic, proteomic, metabolomic, and structural information and applying bioinformatics and structure prediction tools.20 The cytoplasmic model used in the study contains 41 different types of macromolecules abundant in the atomistic model, including ribosomes, the molecular chaperon GroEL/ES, and many enzymes related to glycolysis and other biochemical pathways (see Figure 1a). Each macromolecule is represented by a

(MD) simulations have also been applied to analyze the crowding effects on protein stability10 and hydration structure and dynamics,11 providing atomistic details that are otherwise difficult to obtain by experiments and simplified theories. A uniform size of background crowder molecules is assumed in most of the theoretical and simulation models to compare with the experiments where just one type of the crowder agents is used.4 In contrast, the size distribution of macromolecules in a living cell is diverse. For example, the average radius of proteins in a cell is ∼2 nm, but the 70S ribosome has a much larger size of ∼26 nm.12 Considering the total volume of a bacterial cell, ribosomes occupy 10% of the volume in the cell.1 Recent advances in experimental techniques allow us to measure not only diffusion of macromolecules but also protein folding and dynamics in living cells.13−15 The effect of size heterogeneity of crowder molecules is essential for fully relating in vitro experiments to in vivo crowding effects. To study macromolecular dynamics and stability in bacterial cells, several computational studies using atomistic and coarsegrained (CG) models have been performed. McGuffee and Elcock carried out Brownian dynamics (BD) simulations of a bacterial cytoplasm with atomistic models of macromolecules to analyze their diffusion and thermodynamic stability.16 In the study, a nonspecific interaction term between macromolecules was introduced and calibrated to reproduce the experimentally observed in vivo diffusion of macromolecules.16 Ando and Skolnick showed that not only excluded volume effects but also hydrodynamic interactions between macromolecules are important factors to explain the large reduction of macromolecular diffusion seen in vivo.17 Qin and Zhou developed an efficient method to evaluate the crowding effects of atomistic protein models with less computational time by using the fast Fourier transforms. They applied the method to investigate protein folding and binding in cell-like environments.18,19 In these computational studies, the inhomogeneous crowding environments were taken into account. However, how the heterogeneity of macromolecular sizes in cells alters the crowding effects has not been fully addressed. Here, we investigate the effects of macromolecular size heterogeneity on the equilibrium of macromolecular associations in a crowded environment. The system considered here composed of 41 different types of macromolecules at the physiological concentrations in the cytoplasm of Mycoplasma genitalium.20 Each type of macromolecules is represented as a sphere with a radius corresponding to its size. To compute the free energy changes upon macromolecular association, an existing theory of hard-sphere (HS) fluids, the so-called BGHLL theory,21−23 is used. The theory, which is an extension of SPT, can compute not only the free energy (chemical potential) changes but also various thermodynamic properties of a given system. The accuracy of this theory has been assessed by comparing the theoretical results with computer simulations of the systems consisting of a single component and binary mixture HS systems.24−26 For this purpose, free energy calculations based on BD trajectories were employed. This paper is organized as follows: In the next section, macromolecular models, simulation methods, and the theory of HS fluids are described. The free energy changes upon macromolecular associations in the crowded environments obtained from the theory are then compared to the BD trajectories results followed by a discussion and conclusions.

Figure 1. (a) A simulation model mimicking the cytoplasm of Mycoplasma genitalium used in this study. Macromolecules are represented as spheres with their Stokes radii. Types of macromolecules are represented by different colors. (b) Two models for macromolecular associations (Model 1 for a nonspecific binding and Model 2 for a specific binding) considered.

spherical object, whose radius a is set to the Stokes radius estimated by HYDROPRO27 based on its atomic structure.20 The total number of particles was 2305, placed in a cubic box with a size of 106.2 × 106.2 × 106.2 nm3, and the number of each macromolecule was chosen to match the concentration in the atomistic model. The radii and number of macromolecules are summarized in Table S1. The resulting system has avolume fraction of 0.54 when calculated using the Stokes radii. If radii of gyration are used, the volume fraction is calculated to be 0.24. This system corresponds to 302 g/L, which is a typical concentration for macromolecules in a bacterial cytoplasm at physiological conditions.3 It is worth noting that three very large molecules, the chaperonin GroEL/GroES complex as well as the 50S and 70S ribosome complexes, contribute 50% of the volume fraction of the system, even though the number fraction of these large molecules is only 3%. Hereafter, we call this system the “cytoplasmic model”. II.B. Size of the Most Abundant Protein in the Cytoplasmic Model. The distribution of protein chain lengths from 22 species whose genomes have been fully sequenced is known to vary according to the gamma distribution:28 P(L) =

L χ − 1e−L / θ Γ(χ )θ χ

(1)

where L is the protein chain length, Γ(χ) is the gamma function, and χ and θ are parameters to fit the observed distribution of protein chain lengths to the gamma distribution. Given the distribution function, the average chain length ⟨L⟩ and the most probable chain length L* can be calculated as ⟨L⟩ = χθ 11857

and

L* = (χ − 1)θ

(2)

DOI: 10.1021/acs.jpcb.6b06243 J. Phys. Chem. B 2016, 120, 11856−11865

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The Journal of Physical Chemistry B For the cytoplasmic model for M. genitalium, we find ⟨L⟩ ≈ 365 and χ = 2.02, so L* = 184.28 Protein sizes (measured via the radius of gyration) are found to follow the power law based on the analysis of over 37 000 protein structures from the PDB:29 R g(L) = 2.24L0.392

where ϕ is the volume fraction of particles in the system and Mn is the nth moment of the diameter distribution of particles, which is defined by N

Mn ≡

Here, xα is the number fraction of species α and N is the number of particle species in the system. The parameter q in eq 8 varies for different theories; q = 0 for the Percus−Yevick theory,32 q = 3/4 for the scaled particle theory,33−38 and q = 1/ 2 for the more accurate expression, introduced by Boubliḱ 21 and, independently, Grundke and Henderson22 and Lee and Levesque23 (BGHLL). From the BGHLL theory, the excess chemical potential μα of HS particle species α in the system is calculated with β = 1/kBT as26

(4)

Combining eq 3 with eq 4, the Stokes radius can be expressed as a function of the chain length: a(L) = 3.25L0.392

(5)

βμα = − ln(1 − ϕ) +

For L = 184, the most probable chain length in the cytoplasmic model, the corresponding Stokes radius becomes 25.1 Å. This value is used to compare the crowding effects in the homogeneous system with those in the cytoplasmic model (the inhomogeneous system). II.C. Models for Different Macromolecular Association Modes. In this study, two different macromolecular association modes are considered (Figure 1b): In Model 1 (see left of Figure 1b), the product is a tangent dimer, which may correspond to a nonspecific binding or a transient binding with very low affinity. In this model, the crowding effects on the nonspecific binding mode between reactant species α and γ are evaluated by a free energy defined as F(σαγ ) = −kBT ln[gαγ (σαγ )]

2⎤ ⎫ ⎡ ⎪ 4qϕ2 ⎛ M 2 ⎞ ⎥ ϕ M1 1 ⎢ + + + ⎜ ⎟ (2aα)3 ⎬ 2 ⎪ ⎢⎣ 3M 2 1 − ϕ M3 3(1 − ϕ) ⎝ M3 ⎠ ⎥⎦ ⎭

(10)

A physical interpretation for each term in the right-hand side of eq 10 may be given like in ref 39: The first term simply accounts for the volume fraction of particles. The second term accounts for the curvature of the particles. The third and fourth terms describe the dependence on the particle surface area and its volume, respectively. All terms are always positive, meaning that the introduction of a macromolecule into the crowded solution is unfavorable. In this study, we use the BGHLL theory, i.e., q = 1/2 in eqs 8 and 10, to calculate Fαγ(σαγ) for Model 1 and μα for Model 2. II.E. Energy Functions for BD Simulations. In BD simulations, we considered only repulsive interactions between particles to take into account the excluded volume effects, which are described as a half-harmonic potential

(6)

⎧1 2 ⎪ k(rij − ai − aj − Δ) if rij < ai + aj + Δ 2 Vij = ⎨ ⎪0 if rij ≥ ai + aj + Δ ⎩

+q

(7)

4aαaγ M 2 ϕ 1 3 + 2 1−ϕ 2 (1 − ϕ) σαγM3

2 ϕ2 ⎛ 4aαaγ M 2 ⎞ ⎜ ⎟ ⎜ ⎟ (1 − ϕ)3 ⎝ σαγM3 ⎠

(11)

where k is the force constant, rij is the distance between particles i and j, and Δ is an arbitrary parameter representing a buffer distance between particles.17 In this study, values of Δ = 1 Å and k = 10 kBT/Δ2 were used, which translates into Vij = 5 kBT at the contact distance.17 II.F. BD Simulations and the Contact Free Energy from the BD Trajectories. In the simulations of the cytoplasmic model system, we employed a second-order BD algorithm introduced by Iniesta and de la Torre,40 which is based on the original first-order algorithm developed by Ermak and McCammon.41 Since we focus on the thermodynamic properties of the system, hydrodynamic interactions between particles are not considered in the BD simulations, which affect only kinetics and not thermodynamics.41 All BD simulations were performed under periodic boundary conditions at 298 K. A time step of 8 ps was used, which roughly corresponds to 0.0005 × a2/D for the particles with the smallest radius in the

Hereafter, we call Δμαγ the binding free energy. II.D. Theoretical Estimate of the Contact and Binding Free Energies. There are several analytical approaches to compute structural and thermodynamic properties of not only a single-component system but also multicomponent mixtures of HS liquids.25 Given a HS mixture, the values of the RDF at the contact distances σαγ for a particle pair α and γ are calculated as gαγ (σαγ ) =

3ϕ M 2 1 − ϕ M3

⎧ ⎡M ⎪ M2 ⎤ 3ϕ ⎥(2aα)2 × ⎨2aα + ⎢ 1 + ⎪ − ϕ 2(1 ) M M ⎣ 2 3⎦ ⎩

where kB is the Boltzmann constant, T is the temperature, and gαγ(σαγ) is the value of the pair radial distribution function (RDF) for particle species α and γ at the contact distance σαγ = aα + aγ with reactant radii aα and aγ. Hereafter, we call F(σαγ) the contact free energy. In Model 2 (see right side of Figure 1b), the product is represented as a sphere having the same volume as the sum of two reactants’ volumes. The product radius aαγ is expressed as aαγ3 = aα3 + aγ3. This association mode may correspond to specific binding as discussed later. In this model, the excluded volume effects on the specific binding between two reactant species α and γ are evaluated by the difference between the excess chemical potentials of the product, μαγ, and the sum of two reactants, μα + μγ: Δμαγ = μαγ − (μα + μγ )

(9)

α=1

(3)

where Rg(L) is the radius of gyration of a protein with its chain length L. The Stokes radius of a protein with a given chain length, a(L), can be approximated to30,31 a(L) = 1.45R g(L)

∑ xα(2aα)n

(8) 11858

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system, where D is the diffusion constant (D = kBT/6πηa with the viscosity of water η). Ten independent simulations were performed over 1 ms with different random seeds resulting in different randomly generated initial configurations. Coordinates of particles and energies of the system were saved every 10 ns. RDFs gαγ(rαγ) for a pair of particle species α and γ were computed from the trajectories after excluding the first 50 μs for Model 1 (see Figure S1), where the value is average of gij(rij) with i ∈ α and j ∈ γ. We note that the positions of the first peaks in the RDFs obtained from the BD simulations were not necessarily the contact distances σαγ = aα + aγ used in the BGHLL theory due to the use of a half-harmonic potential with the buffer distance for repulsive interactions between particles described according to eq 11. However, deviations between the RDF peaks and contact distances were less than 1 Å for all pairs. Therefore, we simply used the first RDF peak positions from the BD simulation as the contact distance σαγ in the calculation of the contact free energy F(σαγ) via eq 6. All BD simulations were performed with the GENESIS molecular dynamics and modeling software.42 II.G. Binding Free Energy from the BD Trajectories. The Bennett acceptance ratio (BAR) method can estimate the free energy difference (ΔF = F1 − F0) between two canonical ensembles generated by dynamics simulations with potential energy functions V0(x) and V1(x) for a given configuration of the system x.43 The excess chemical potential μα to be computed in the study corresponds to the transfer free energy from an infinite dilution to the crowded environment for one particle of species α. The accuracy in the calculation of the free energy differences by BAR (as well as other free energy estimation methods, such as free energy perturbation and thermodynamic integration) strongly depends on the extent of overlap between the two ensembles of interest. Therefore, a number of intermediate states connecting the two states are typically introduced, which are parametrized using a coupling parameter λ (0 ≤ λ ≤ 1). As λ is changed from 0 to 1, the potential energy varies from V0 to V1. In the BAR method, the chemical potential difference between λm and λm+1 values, δμm, for the case of an equal number of sampled configurations in the two states is estimated as

III. RESULTS III.A. Crowding Effects on the Nonspecific Binding. The correlation between the contact free energies of Model 1 for all possible particle pairs in the cytoplasmic model estimated by the BGHLL theory, F(σαγ)theory, and by the simulation, F(σαγ)simul, is shown in Figure 2. A good agreement is found for

Figure 2. Correlation between F(σαγ)theory and F(σαγ)simul values for Model 1. In these calculations, all possible pairs in the cytoplasmic model were examined. The diagonal is shown as a dashed green line. The correlation coefficient (R2) between F(σαγ)theory and F(σαγ)simul is 0.98.

= ⟨[1 + exp(β {Vλm + 1(x m + 1) − Vλm(x m + 1) + δμm })]−1 ⟩λm + 1 (12)

where ⟨...⟩λ indicates an ensemble average at a particular value of λ, m is the index of the states, and xm represents configuration of the system generated by the simulation with λm. Then, the total free energy differences between states with λ = 0 and 1 is obtained as the sum of δμm

all possible pairs, indicating the applicability of the theory described in eqs 8 and 9 for multicomponent systems at high density. The theory gives F(σαγ) values consistent with the BD simulations but at a much reduced cost and within the spherical approximation of macromolecules. In Figure 3a, the contact free energies F(σαγ)theory calculated from the theory for all possible macromolecule pairs in the crowding system are shown as a function of the Stokes radii of the two interacting molecules. F(σαγ)theory values decrease with particle radii, indicating that the association of larger macromolecule pairs is more stable than for smaller macromolecules according to Model 1. For a pair with aα of 25 Å, which is the most abundant protein size in M. genitalium (see section II.B), a value of F(σαγ)theory = −1.5 kBT is obtained. For the largest

n−1

∑ δμm m=0

(14)

A total of 14 states (n = 14) were used with values of λm = 0, 0.0001, 0.0005, 0.016, 0.004, 0.008, 0.015, 0.026, 0.063, 0.13, 0.24, 0.41, 0.66, and 1. For each λm value, ten independent BD simulations over 10 μs (1 250 000 steps) were performed after energy minimization using different random seeds and different initial configurations for each λ value. Coordinates were saved every 10 ns (1250 steps). For the BAR analysis, the last 500 coordinate frames were used. The same procedure was performed for estimating μγ of the reactant species γ. For estimating μαγ of the product species αγ, the same computational procedure explained above was applied to the systems where two particles, one from the reactant species α and another from species γ, were removed from the original system and one new particle αγ was introduced. Hereafter, for clarity, quantities obtained from the BD simulations and the BGHLL theory will be described with superscripts “simul” and “theory”, respectively.

⟨[1 + exp(β {Vλm + 1(x m) − Vλm(x m) − δμm })]−1 ⟩λm

μ=

(m = 0, 1, ..., 13)

(13)

Here, n is the number of states generated between λ = 0 and 1. For estimating μα in eq 7, we first selected a particle i which belongs to species α in the simulation system. In the free energy calculations, the state with λ = 0 corresponds to the dilute solution where only particle i exists. The state with λ = 1 corresponds to the crowded solution where the particle i is surrounded by other particles. The force constant k in eq 11 for pairs between the particle i and the other particles as a function of λm are then given by 11859

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Figure 3. (a) Crowding-induced contact free energy F(σαγ)theory for Model 1 calculated by BGHLL theory for all possible pairs in the cytoplasmic model. (b) F(σαα)theory values for homodimers in three different systems. The values for the cytoplasmic model correspond to the energies along the black diagonal line in (a).

particle pair, two ribosome 70S particle with a radius of 126.7 Å, the crowding-induced contact energy becomes −3.0 kBT. In Figure 3b, the contact free energies induced by crowding for homodimer pairs calculated in the system with a unified crowder radius ac are compared with the values in the cytoplasmic model. If ⟨a⟩ = 33.44 Å is used for ac and ϕ = 0.54, the same volume fraction as in the cytoplasmic model, the contact free energies are slightly overestimated for the entire range of radii. If ac and ϕ are treated as fitting parameters, values of ac = 38 Å and ϕ = 0.5 gave the best agreement in terms of a minimum root-mean-square deviation compared to F(σαγ)theory calculated in the cytoplasmic model. The fit between F(σαα)theory and the free energies with optimized uniform crowder parameters are very good for aα ≤ 80 Å. However, for aα > 80 Å, the contact free energy becomes more favorable compared to the fitted uniform crowder model. These results suggest that it is not straightforward to relate experiments (or theory) considering a single uniform crowder size to heterogeneous cytoplasmic environments. III.B. Crowding Effects on the Specific Binding in Model 2. The change in binding free energy Δμαγ upon associating a pair of particles obtained according to the BGHLL theory is compared with BD simulation in Figure 4. Here, a total of 15 pairs of particles with radii a = 16.26, 26.2, 34.9, 45.25, and 84.89 Å were examined. The computed Δμαγsimul values for the examined particle pairs were in good agreement with the theoretical values Δμαγtheory, although numerical errors increase for the larger particles in the chemical potential calculations from the simulation. This result clearly demonstrates that the theory described in eqs 9 and 10 is able to estimate the chemical potentials in heterogeneous crowding systems. In Figure 5a, the crowding effects on the binding free energies Δμαγtheory according to the BGHLL theory are shown for all possible pairs in the cytoplasmic model. Δμαγtheory values decrease with increasing radii of pair particles, the same trend as seen above in F(σαγ). We note that the absolute values of Δμαγtheory are larger than those of F(σαγ)theory because a single particle with a combined volume is much easier to insert into a crowded system than two adjoined spherical particles with the same total volume. This has been noted already for the case of monodisperse crowder suspension.44 The effects of crowder size heterogeneity on Model 2 are shown in Figure 5b. Δμααtheory values calculated with uniform

Figure 4. Comparison of changes in the binding free energy for pairwise association according to Model 2 calculated from theory (Δμαγtheory) and BD simulation (Δμαγsimul). In these calculations, all combination of pairs involving particles with radii of 16.26, 26.2, 34.9, 45.25, and 84.89 Å were examined. A diagonal line is shown with green broken line. The correlation coefficient (R2) between Δμαγtheory and Δμαγsimul is 0.99.

crowder radii ac again overestimate binding free energies if ac = ⟨a⟩ = 33.4 Å and the same volume fraction as the original system ϕ = 0.54 are chosen. Optimization of ac and ϕ with ac = 23 Å and ϕ = 0.34 gives an excellent fit with the cytoplasmic model for most radii (aα ≤ 90), but the optimized values to fit Model 2 were different from those in Model 1. These results indicate again that the crowding effects on specific binding are sensitive to the size distribution of crowder particles in the system at a given volume fraction. Judging from the results of the two binding models, there are no optimal uniform radius and volume fraction that reproduce the values for the cytoplasmic particle distributions. III.C. Comparison with Experiments. Finally, to address the applicability of our simple model to a real system, Δμαγtheory calculated by the BGHLL theory in Model 2 were compared to available experimental values for protein binding in crowded environments. Batra et al. systematically studied the crowding effects on binding between the ε and θ subunits of E. coli DNA polymerase III core in the presence of dextran crowders with varying molecular mass at 6, 40, 70, 100, and 150 kDa, each at a concentration of 100 g/L (a volume occupancy of 0.15 was assumed) using a spectroscopic method.45 In their study, the 11860

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Figure 5. (a) Crowding-induced binding free energy Δμαγtheory for Model 2 calculated by BGHLL theory for all possible pairs in the cytoplasmic model. (b) Δμααtheory values for homodimers in three different systems. The values for the cytoplasmic model correspond to the energies along the black diagonal line in (a).

reasonable agreement with the experiments, although the theory slightly underestimated the crowding effects for Mw = 6 kDa. If the ε subunit and HOT protein are introduced into our cytoplasmic model, the crowding effect on the binding reaction was estimated to be −2.5 kBT by the BGHLL theory, which is 4 times stronger than the experimental results for dextran with Mw = 40−150 kDa.

crowder size in each experiment is homogeneous. To the best of our knowledge, there is no experimental data analyzing the influence of size heterogeneity of crowders on crowding effects. Therefore, we chose their systematic experiments as a reference to show the applicability of the theory for explaining the experimental results, although heterogeneity of the crowder size is not examined in this test. Using the crystal structure of the ε-HOT complex46 (HOT is a homologues protein of the θ subunit) (PDB ID 2IDO), we obtained Stokes radii of the ε subunit (chain A in 2IDO), the HOT protein (chain B in 2IDO), and the complex from HYDROPRO as 22.5, 18.7, and 25.7 Å, respectively. These calculated Stokes radii of the protein chains satisfy the assumption used in Model 2, aαγ3 = aα3 + aγ3: 25.73 = 1.7 × 104, 22.53 + 18.73 = 1.8 × 104. For calculating Δμαγtheory values in the presence of dextran at varying molecular weights, ϕ = 0.15 as well as aα = 22.5 and 18.7 Å for the ε subunit and the HOT protein, respectively, were used in eq 10. The radius of dextran was calculated as adextran = 8.26Mw1/3 Å with the molecular weight Mw given in kDa.45 Calculated values and experimental values are compared in Figure 6. The Δμαγtheory values for Mw = 40−150 kDa were in

IV. DISCUSSION The primary objective of the present study is to investigate the thermodynamic effects of molecular crowding on macromolecular associations in the inhomogeneous environment of living cells using the simulations and theory of spherical objects. It is clear that the spherical approximation of macromolecules used here and considering only volume exclusion greatly simplifies the full crowding effects in vivo. However, the simple model provides useful information on the thermodynamics of macromolecular associations in vivo as a first-order approximation. We have demonstrated that the BGHLL theory and the BD simulations combined with BAR method give consistent values of crowding effects for both two macromolecular association models in the cytoplasmic model, in which the effects are measured by F(σαγ) for Model 1 and Δμαγ for Model 2. The calculated Δμαγtheory values for the ε-HOT binding in the presence of dextran of various molecular weights were also in reasonable agreement with experiments. Thus, we believe that the BGHLL theory using the HS approximation is useful to provide a baseline for further studies of crowding effects as a function of macromolecular shapes, attractive/repulsive interactions between molecules, and flexibility. As shown in Figures 3 and 5, the heterogeneity of macromolecular sizes in a cell decreases or weakens the effect of crowding when compared to the use of uniform crowders with the same mean size at a fixed volume fraction of the system. It is worth noting that according to the theory heterogeneity always decreases the effect of crowding at the same volume fraction with respect to a system with uniform crowders of the same mean size, since the ratios of the moments Mn in eqs 8 and 10 for a heterogeneous system are always smaller than those in the corresponding homogeneous crowder system. In Figure 6, we show the crowding effect from BGHLL theory on binding between the ε and θ subunits of E. coli DNA

Figure 6. Comparison of the crowding effect on the binding of ε and θ subunits of E. coli DNA polymerase III core evaluated by experiments (−ΔΔGαγ) in ref 45 and the BGHLL theory (−Δμαγtheory) in the presence of dextran with different molecular weights. Parameters used in the theoretical calculation are described in the main text. 11861

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−1.5 kBT, for pairs with a = 25 Å. For the largest macromolecule pair, 70S ribosomes, F(σαγ) = −3.0 kBT. Recent experiments using cryoelectron tomography and a templatematching approach shows that polysomes of E. coli have a unique structure where neighboring ribosomes are densely packed and exhibit preferred orientations.48 The crowding effect examined in Model 1 would explain a preferential stabilization of bacterial polysomes in vivo. In Model 2, reactants are assumed to combine their volume to form a spherical product corresponding to specific binding like the ε-HOT complex of the E. coli DNA polymerase III core. For small proteins with a = 25 Å in the modeled cytoplasm, the degree of stabilization by the crowding effects according to the Δμαγ estimate is moderate, −3.1 kBT (Figure 5). Binding affinity of protein−protein interactions calculated from nonredundant set of 144 protein−protein complexes is −19 kBT on average with standard deviation of 5 kBT, though the affinity diverse in a range between −12 kBT and −32 kBT.49 Therefore, for many of protein−protein complexes in the cell, crowding effects of −3.1 kBT are likely to be overwhelmed by specific interactions between them. For larger protein pairs with a > 60 Å, much stronger stabilization of more than −10 kBT is expected (Figure 5). Binding between 30S and 50S ribosomes generates 70S ribosome, which is the largest oligomeric complex in a bacterial cell. This complex formation is well described using Model 2. Stabilization by crowding of the association in the modeled cytoplasm estimated by our calculation with the BGHLL theory was about 20 kBT if Stokes radii of 81, 115, and 127 Å for 30S, 50S, and 70S ribosome, respectively, are used, which is 2 times smaller than estimates by Qin and Zhou as a preliminary calculation with their theory.50 In most of the experimental and theoretical studies, just one type of crowder agent is typically used for analyzing crowding effects. Additionally, recent experiments show that common crowding agents, like PEG and dextran, have only a small effect on protein−protein associations.51 A similar conclusion has been obtained from experimental and theoretical analysis by another group.45,52 In order to obtain insights into in vivo crowding effects from the experiments, examining differences between chemically generated crowded environments and in vivo space may be necessary. In this study, we observed significant deviations in the crowding effects on macromolecular associations estimated by Δμαγ from those in the modeled heterogeneous intracellular environment if a single particle type with the radius of the average over examined macromolecular radii was used as crowder particles (Figure 5b). Optimization of the unified crowder radius ac and volume fraction is possible to obtain better agreement with the results for the heterogeneous environment but very different optimal values of were found for reproducing F(σαγ)theory in Model 1 and Δμαγtheory in Model 2, respectively. Furthermore, even with the optimized values there are deviations for large particles with radii of larger than ∼80 Å (Figures 3b and 5b). Therefore, it is difficult to relate experiments and theory based on uniform crowder particles to describe in vivo crowding effects where there is a highly heterogeneous crowder size distribution. For the crowding or excluded volume effect, it is commonly believed that the larger molecule is the stronger crowder. However, the SPT theory predicts actually that the larger molecule is a poorer crowder. As pointed out by Sharp, this misconception may be due to the fact that the relevant theory has not been applied consistently.39 There is also a weak

polymerase III core in the presence of various sizes of crowders (mimicking dextran used in the experiments) at the same volume occupancy of 0.15. The theory well reproduced two important features: (1) the smaller crowders show stronger crowding effects, and (2) its dependence on crowder size is weak if the molecular weight of the crowder is larger than 40 kDa. A physical interpretation of the first feature is that the entropic cost required for avoiding overlaps with a few large molecules is less than that for the many small molecules.39 To obtain deeper insights to the observed features of the crowding effect, we discuss scaling properties of the chemical potential on size of crowder. For different crowder sizes at the same volume fraction, varying quantities in eq 10 are coefficients of three terms: M2/M3 in the linear term, M1/M3 and (M2/M3)2 in the surface area term, and 1/M3, (M1M2)/M32, and (M2/M3)3 in the volume term, where Mn defined in eq 9 is function of crowder radius adextran, Mn ∝ (adextran)n. Since adextran is proportional to one-third of its molecular weight Mw, Mn is also described as a function of Mw, according to Mn ∝ Mwn/3. Therefore, the coefficients in eq 10 for the linear, area, and volume terms scale as Mw−1/3, Mw−2/3, and Mw−1, respectively. In the calculation of Δμαγtheory calculation using thermodynamic cycle, contribution of the volume term to the crowding effect is zero because aαγ3 = aα3 + aγ3 is assumed. So, the crowding effect Δμαγtheory arises from the linear and surface area terms. This is the reason why the crowding effect decreases with molecular weight of crowder and why it is insensitive to size of crowder if Mw > 40 kDa. In the model presented here, Stokes radii of macromolecules were used for their particle radii. Based on Stokes radii, the volume fraction of the modeled bacterial cytoplasm reached 0.54. If radius of gyration is used for radius of modeled particle, the volume fraction is calculated to be 0.24, which is about 2 times smaller than that calculated with Stokes radii. If radii of gyration are used, the crowding effects on binding free energies Δμαγ and contact free energies F(σαγ) are much smaller than those calculated with Stokes radii (Figures S2 and S3). Previous results of Stokesian dynamics, BD simulations with not only farfield but also near-field hydrodynamic interactions, of a model of E. coli cytoplasm conducted by Ando and Skolnick have shown that the use of a particle model based on Stokes radii reproduces the diffusion coefficients of green fluorescent protein seen in experiment. 17 In BD simulations of concentrated solutions of myoglobin and hemoglobin using atomistic details performed by Mereghetti and Wade, the first peaks of RDF for the molecules have been seen at the distances corresponding to their hydrodynamic diameters.47 Consequently, the use of Stokes radii for describing the particle size when comparing simulations and theories may be a reasonable choice when studying the crowding effects within the spherical approximation. However, a comparison of atomistic BD results with HS theory is more complicated since the shape of the RDF contact peaks in atomistic BD simulations significantly broadens as shown by Mereghetti and Wade. The use of a smaller force constant k in eq 11 for repulsive interactions in the BD simulation are also expected to broaden the peaks of the RDF, which would decrease the binding and contact free energies due to the crowding. Therefore, the values estimated in this study may correspond to the upper bounds of the excluded volume effects in vivo. In Model 1, where the reactants form a tangent dimer corresponding to nonspecific binding, the contact free energy F(σαγ) induced by the crowding is close to the thermal noise, 11862

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experimental result. Additionally, absolute values of the stabilization effect due to volume exclusion are 2 orders of magnitude higher than the experimental values. As seen above, there is the large discrepancy between crowding effects on the binding evaluated by experiment and those calculated by the SPT in the presence of water particles. To fill this gap, as introduced in the paper of Graziano’s work,54 enthalpic contributions to crowding effects may need to be considered. Recalculating solvent density may be another way to fix the matter of SPT in the presence of water particles. Berg proposed that water density should be calculated by imposing that the hard sphere pressure of crowder solution is equal to the hard sphere pressure of pure water based on the Gibbs−Duhem relation at constant pressure and temperature.53 For a case of sugar-induced protein structure stabilization, the SPT with Berg’s treatment was sufficient to describe quantitatively the stabilization effect of sugars on protein structure measured by experiments. However, Sharp pointed out that this change in solvent density requires unrealistically large volume change in solvent volume.39 In contrast to the SPT with explicit water particles, the standard BGHLL theory well predicts the weak dependence on crowder size, and the obtained values are much closer to the experimental values than the values from the SPT with explicit water particles without considering additional factors. On the basis of these test calculations, we believe that the standard SPT and BGHLL theory is sufficient for explaining the weak dependence on crowder size and the decrease of the crowding effect with crowder size.

dependence of crowding effect on crowder size. To reconcile the theory with the observations, Sharp recently treated water and ions on an equal footing as the macromolecular crowders when calculating crowding effects using HS fluid theory.39 This idea was originally proposed by more than 20 years ago.53 Using the idea of explicit water treatment, the effects of sucrose addition on the thermal stability of a protein were carefully analyzed by Graziano.54 Although they used SPT in the presence of explicit water particles for estimating crowding effects, there are a few differences between each other in calculations of solvent density and volume fraction of crowders. We finally discuss about their idea by comparing with the experimental data used for validation of the BGHLL theory. In the idea introduced by Sharp,39 a water molecule is treated as an explicit particle with a diameter of 2.8 Å within the theory of a HS fluid. A fixed packing fraction for water of ξ = 0.383 was assumed, based on the density of water of 1 mg/mL and the given water diameter. The packing fraction ξ corresponds to volume fraction ϕ in eqs 8 and 10. Then, this packing fraction consists of crowder and water molecules, like 33% for crowder and 67% for water by volume. Under these conditions at the constant packing fraction ξ of 0.383, the SPT is applied to estimate thermodynamic properties of the HS mixture. According to the SPT with the treatment of explicit water particles of Berg, we again computed the crowding effect on the binding of ε-HOT complex in the presence of various sizes of dextran molecules (shown in Figure S4). The calculations were done for a constant solvent composition of 15% crowder dextran and 85% water by volume. The radii of the proteins and crowders were taken as already described in section III. Since the water particles are explicitly treated in the SPT, crowding effects on binding due to addition of crowder molecules to water solution are evaluated by ΔΔμαγ = Δμcrowder − Δμwater αγ αγ , crowder is the difference of chemical potential upon where Δμαγ biding in crowder solution and Δμwater is the difference of the αγ excess chemical potential upon biding in pure water. The model of Sharp predicts a weak dependence on crowder size and that the small dextran is a good crowder. However, the ΔΔμαγ value computed with the model of Sharp is positive and large, indicating the crowding of crowder molecules disfavors binding of proteins. This is inconsistent with the experimental results. Next, we examine the model of Graziano used for analyzing sucrose-induced stabilization of a protein structure.54 A difference in treatment from Sharp’s model is that packing fraction varies in Graziano’s model. Graziano calculated the packing fraction using an effective hard sphere diameter of water and sucrose, 2.8 and 8.1 Å, respectively, and density of sucrose solution determined by experiments. For example, packing fraction ξ = 0.469 for 1 M sucrose is calculated with density of 1127 g/L. The addition of sucrose to water causes an increase the stabilization effects of a compact native conformation of protein mainly due to the increase of the packing fraction. Using Graziano’s model, we again computed the crowding effects on the binding of ε-HOT complex in the presence of various sizes of dextran molecules (shown in Figure S5). In the calculation, a density of 1034 g/L for 10% (100 g/ L) dextran solution was used, which are determined by experiments.55 The radii of the proteins and crowders were taken as already described in section III. The SPT with model of Graziano can predict that the addition of dextran into solution stabilizes the protein binding and that the small dextran is a good crowder. However, the idea gave a strong dependence on crowder size, which is inconsistent with the

V. CONCLUSIONS We have presented a quantitative analysis of the effects of macromolecular crowding on the macromolecular association in a modeled intracellular environment by using Brownian dynamics simulations and a theory of hard-sphere fluids. Our study suggests that the size heterogeneity of macromolecules or crowders is an important factor to be considered in analyzing crowding effects and relating in vitro studies using uniform crowders to in vivo situations. The analysis done here is clearly simplified. Shapes, chemical properties, interactions, and flexibility of macromolecules are ignored. Evaluating how these factors alter the crowding effects will be subject of future studies. Recent experimental and computational studies suggest that the crowding in some cases destabilize these structures in vitro and in vivo.10,56−61 Volume exclusion analyzed here always stabilizes compact native or complex states of macromolecules. Therefore, our study cannot directly explain such destabilization. However, the results and theory shown here would provide basic insights and controls for future studies on in vivo crowding effects.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b06243.

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Additional information on the cytoplasmic model and theoretical analysis for crowding effect (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Tel +81-48-462-1407 (Y.S.). 11863

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T.A.: Department of Applied Electronics, Tokyo University of Science, 6-3-1 Niijuku, Katsushika-ku, Tokyo 125-8585, Japan. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Huan-Xiang Zhou for providing us with the experimental data in Figure 6. Simulations and analysis were carried out using the RIKEN Integrated Cluster of Clusters (RICC) and RIKEN HOKUSAI supercomputer systems. This work was supported in part by RIKEN QBiC and iTHES (to Y.S.), a Grant-in-Aid for Scientific Research on Innovative Area “Novel measurement techniques for visualizing “live” protein molecules at work” (No. 26119006) (to Y.S.), a grant from JST CREST on “Structural Life Science and Advanced Core Technologies for Innovative Life Science Research” (to Y.S.), a Grant-in-Aid for Scientific Research (C) from MEXT (No. 25410025) (to I.Y.), and support from the U.S. National Institutes of Health (NIH, GM092949, GM084943) and the U.S. National Science Foundation (NSF, MCB 1330560) (to M.F.).

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