Dependence of Protein Folding Stability and Dynamics on the Density and Composition of Macromolecular Crowders

Abstract

We investigate the effect of macromolecular crowding on protein folding, using purely repulsive crowding particles and a self-organizing polymer model of protein folding. We find that the variation in folding stability with crowder size for typical α-, β-, and α/β-proteins is well described by an adaptation of the scaled particle theory. The native state, the transition state, and the unfolded protein are treated as effective hard spheres, with the folded and transition state radii independent of the size and concentration of the crowders. Remarkably, we find that, as the effective unfolded state radius is very weakly dependent on the crowder concentration, it can also be approximated by a single size. The same model predicts the effect of crowding on the folding barrier and therefore refolding rates with no adjustable parameters. A simple extension of the scaled-particle theory model, assuming additivity, can also describe the behavior of mixtures of crowding particles.

Introduction

In contrast with conventional laboratory experiments conducted under dilute conditions, protein folding in a cell occurs in a dense environment consisting of various other macromolecules, commonly referred to as crowders (1). Although the detailed interactions of the protein with the crowders may be quite complex (e.g., direct protein-crowder attractions, water-mediated interactions (2)), the primary physical effect of macromolecular crowding on the protein folding reaction is to reduce the volume available to the protein by that occupied by the crowders (3–10). A complete theoretical understanding of the excluded volume effects will greatly enhance our ability to interpret experiments, as well as all-atom simulations, and to develop coarse-grained models (11,12), of concentrated protein solutions.

Several theories have been put forward to predict the effect of excluded volume crowders on the folding free energy. However, these theories provide strongly contrasting predictions for even the qualitative effects of crowding. We focus here on two theories which provide convenient closed-form expressions suitable for fitting to theory or simulation, although we note that other theories requiring numerical solution have been put forward (13). By treating the folded and unfolded proteins as effective hard spheres, Minton utilized scaled particle theory (SPT) to estimate the change in folding free energy as the difference between the insertion free energy for the folded and the unfolded states. The SPT free energy of inserting a hard sphere of radius R in a hard-sphere fluid of particle radius R_c is (14)

$$\beta F=\left(3y+3y^2+y^3\right)\rho +\left(9y^2/2+3y^3\right){\rho }^2+3y^3{\rho }^3-\ln \left(1-{\varphi }_{\text{c}}\right),$$

where β = 1/k_BT, T is the temperature, k_B is the Boltzmann's constant, y = R/R_c, ρ = ϕ_c/(1 − ϕ_c), and ϕ_c is the fluid volume fraction. This theory predicts a strong effect of macromolecular crowding on the folding free energy (13), with a monotonic increase in stability with increasing crowder packing fraction ϕ_c, in qualitative agreement with previous molecular simulation results (4). We note that the insertion free energy of any given protein conformation in a hard sphere solvent can also be calculated directly from its geometric properties using, e.g., a recently proposed morphometric approach (15). In this work, we concentrate on theories that do not require enumeration of the protein configurations, which cannot be done experimentally.

An alternative theory by Zhou (16) also uses SPT for the effect of crowders on the folded protein, but the free energy of the unfolded protein is calculated using an elegant model of the unfolded chain as a random walk in the presence of a spherical trap. For a Gaussian chain of radius of gyration R_g, with z = R_g/R_c, the change in free energy is (17)

$$\beta F=3{\varphi }_{\text{c}}{z}^2+6{\pi }^{-1/2}{\varphi }_{\text{c}}z-\ln \left(1-{\varphi }_{\text{c}}\right)\text{.}$$

This model predicts a much weaker effect of crowding on the unfolded state free energy, as the unfolded polypeptide can access the voids between the crowders. In addition, a stability maximum is predicted as the crowder packing fraction is increased. The extension of this theory to binary mixtures of different size crowders leads to an intriguing conclusion that there will be an optimum mixing ratio of the two components to achieve maximum protein stability (18). One may expect that different crowding theories will work better under certain conditions of crowder packing fraction and size. However, the boundaries for the validity of these theories in parameter space are relatively unknown.

Model and Methods

In this article, we investigate the quantitative theoretical description of purely repulsive crowders on folding, using molecular simulations of a coarse-grained folding model in a bath of crowders. We consider two-state proteins from the three main protein structural classes: all-α (prb (19)), all-β (TNfn3 (20)) and α/β (protein G (21)), which are described using a self-organizing polymer model (or Gō-like model) (22,23). In our model, each residue is represented by a single particle and a standard procedure is used to build the potential from the experimental native-state structure (24). Interactions between the contacts present in the native state are treated as attractive and all others as repulsive, an approximation motivated by the funneled nature of the folding free energy landscape (25). Previous studies have shown that this type of simplified model can indeed capture relevant features of protein folding (26), such as mechanism (27) and kinetics (28,29). Repulsive protein-crowder and crowder-crowder interactions are given by the pair potential

$$V\left(r\right)=\varepsilon {\left[{\sigma }_{\text{ref}}/\left(r-\sigma +{\sigma }_{\text{ref}}\right)\right]}^{12},$$

where r is the distance between particle centers, ɛ = 1 kcal/mol sets the energy scale, σ is the hard core overlap distance, and σ_ref = 6 Å is a reference diameter: for σ = σ_ref, V(r) reduces to a more familiar form. For the pair (i, j), we define σ = R_i + R_j, where R_i, R_j are the radii of crowder particles R_c, or of the various protein residues R_p. We use Langevin dynamics simulations with a time step of 10 fs and a friction coefficient of 0.2 ps⁻¹ using the BBK integrator (30) in the CHARMM simulation package (31). Cubic periodic boundary conditions with a primary cell size of 100 Å were employed. To speed up equilibration at a given temperature, we use replica exchange moves every 30 ps between 12 replicas which are each biased using an umbrella potential of the form

$$V_{\text{i}}\left(Q\right)=0.5\kappa {\left(Q-Q_{\text{i}}\right)}^2,$$

where Q is the fraction of native contacts, 0 ≤ Q_i ≤ 1 is the target Q value for replica i, and κ = 300 kcal/mol is the force constant. We define

$$Q\left(\mathbf{R}\right)=N_{\text{ij}}^{-1}{\sum }_{\left(\text{i,j}\right)}{\left(1+\text{exp}\left(\mathit{\beta }\left(r_{\text{ij}}-\lambda r_{\text{ij}}^{\text{0}}\right)\right)\right)}^{-1},$$

where the sum runs over N_ij native contact pairs (i, j), which are separated by distances $r_{\text{ij}}^{\text{0}}$ in the native structure and by r_ij in configuration R; β = 5 Å⁻¹; λ = 1.2. The required thermodynamic information is extracted from simulations with different umbrella potentials and temperatures using the weighted histogram method (32,33).

Results and Discussion

For the model proteins considered here, Q is nearly an optimal coordinate for identifying transition states and capturing the dynamics of protein folding with a diffusive Markovian model (29,34,35). Here, we project folding in our systems onto Q, and use it to identify transition states as well as unfolded and folded free energy minima. To estimate the crowding-induced changes on the folding free energy surface, we construct the potential of mean force along Q, defined as

$$\beta F\left(Q\right)=-\ln \left[P\left(Q\right)/\Delta Q\right],$$

where P(Q) is the equilibrium probability of observing configurations between Q and Q + ΔQ. Fig. 1 shows representative free energy profiles in bulk (ϕ_c = 0) and under crowded conditions (ϕ_c > 0) for protein G. The destabilization due to the presence of crowders is clearly visible from an upward shift in the curves near the unfolded basin (Q ≈ 0.1), qualitatively consistent with earlier predictions (3). This is due to the greater cost of inserting the larger unfolded state into a bath of crowders. A secondary effect is that the average size of the unfolded state is reduced with increasing confinement, because the more expanded conformers are disfavored. At the highest packing fractions, we find a reduction of 15% in the average radius of gyration of the unfolded state. This compaction is qualitatively consistent with neutron scattering measurements on random-coil polymers, which have suggested a compaction of up to 30% (36). The quantitative difference from the experimental data may be due to differences between the compact unfolded state of the Gō model, due to residual nativelike interactions, and random-coil chains. For high crowding packing fractions (ϕ_c > 0.10), there is also a slight destabilization of the transition state (Q ≈ 0.5) with respect to the folded basin (Q ≈ 0.9). The increase in folding temperature T_f (the maximum in heat capacity C_v; see Fig. 1 inset) with increasing ϕ_c, demonstrates the stabilizing effect of the crowders as predicted by approximate theoretical models (3) and observed in molecular simulations previously (4).

Figure 1.

Effect of crowding on folding free energy surface of protein G. The potential of mean force along the coordinate Q, the fraction of native contacts, is shown for bulk conditions and with different crowder packing fractions ϕ_c for crowder size R_c = 8 Å at the bulk folding temperature T_f^bulk = 310 K. The curves have been shifted by an arbitrary constant to match the free energy of the folded state. (Inset) Heat capacity C_v(T) from weighted histogram method analysis.

Fig. 2, A–C, shows the effect of crowding on the free energy of folding,

$$\Delta \Delta F_{\text{N}-\text{U}}=\Delta F_{\text{N}-\text{U}}\left({\varphi }_{\text{c}}\right)-\Delta F_{\text{N}-\text{U}}^{\mathrm{bulk}}$$

for each protein, where

$$\Delta F_{\text{N}-\text{U}}=-k_{\text{B}}T\text{ln}\left({\int }_{Q_{\mathrm{‡}}}^1{e}^{-\mathit{\beta }F\left(Q\right)}dQ/{\int }_0^{Q_{\mathrm{‡}}}{e}^{-\mathit{\beta }F\left(Q\right)}dQ\right)$$

is the difference between the native F_N and unfolded F_U free energy and Q_‡ is the location of the transition state along Q. We find a monotonic increase in stability with the packing fraction ϕ_c for all proteins and crowder sizes. For a given ϕ_c, smaller crowders are more strongly stabilizing because there are effectively fewer voids of the size of the protein. For a given size and concentration of crowders, we observe more stabilization for a longer protein chain because the size ratio between the unfolded and the folded state increases with chain length (35,37). Chain lengths and $R_{\text{g}}^{\text{U}}/R_{\text{g}}^{\text{N}}$ are 47 (1.67), 56 (1.89), and 90 (2.2) for prb, protein G, and TNfn3, respectively. We compute radii of gyration $R_{\text{g}}^{\text{N}}$ and $R_{\text{g}}^{\text{U}}$, for the folded and unfolded proteins, respectively, from the simulations and define the folded protein radius a_N = $R_{\text{g}}^{\text{N}}$ (Fig. 2 legend). The Zhou theory is in excellent agreement with the simulation data in the limit ϕ_c → 0 (Fig. 2). For smaller crowders and higher packing fractions, however, the theory predicts too small a stabilization—most likely due to the neglect of excluded volume in the unfolded chain.

Figure 2.

Effect of crowding on protein stability. The change in folding free energy with crowder size and packing fraction is given for (A) protein G (T = 320 K), (B) prb (T = 320 K), and (C) TNfn3 (T = 300 K); the corresponding protein structure is shown next to each plot. Symbols: simulations for R_c = 20 Å (circle), 16 Å (square), 12 Å (triangle up), and 8 Å (triangle right); (dashed lines) Zhou theory (16); (dot-dash lines) Minton theory; and (solid line) effective hard sphere radius for the unfolded state motivated by the form of the U-C g(r). Radius of gyration for the native state R_g^N = 9.5 (prb), 10.6 (protein G), and 13.0 (TNfn3). (D) Pair distribution functions g(r) between crowders (C) and the native (N) and unfolded (U) protein (C-C [U] and C-C [N] are the C-C g(r) in the presence of U and N protein, respectively), for prb with R_c = 8 Å, ϕ_c = 0.05. Arrows indicate corresponding sums of hard sphere radii. (E) Dependence of effective unfolded state radius a_U on crowder size R_c; each axis is given relative to the unfolded radius of gyration R_g^U.

By contrast, the Minton theory predicts a stabilization much greater than found in the simulations over the full range of ϕ_c. This theory treats the unfolded protein (with radius of gyration $R_{\text{g}}^{\text{U}}$) as an equivalent hard sphere with radius $\sqrt{5/3}{R}_{\text{g}}^{\text{U}}$. An alternative theory by Minton (13), using the so-called Gaussian cloud model for the unfolded state, works slightly better but still does not explain our simulation data over the whole range (A. P. Minton, personal communication, 2009). The use of an effective hard sphere for the unfolded chain is justified by the highly successful Asakura-Oosawa theory (38) of polymer-colloid mixtures and the statistical-mechanical description of such mixtures by Shaw and Thirumalai (39). However, as indicated by the pair distribution function g(r) between the unfolded protein and the crowders (Fig. 2), the unfolded protein is quite soft, so that a smaller choice of hard sphere radius may be more appropriate (40). We find that the ΔΔF_N–U data are well described by SPT with the folded protein radius a_N = $R_{\text{g}}^{\text{N}}$, and an unfolded radius a_U < $R_{\text{g}}^{\text{U}}$. The optimal value of a_U is approximately independent of ϕ_c as evident from the fit using a single a_U for each R_c (Fig. 2, A–C). In fact a_U is also only weakly dependent on R_c (Fig. 2 E) and a similar quality fit can be obtained using a single a_U = 0.8$R_{\text{g}}^{\text{U}}$ (data not shown). The slight reduction in a_U with decreasing R_c results from greater penetration of the unfolded protein by the smaller crowders. This weak dependence on crowder size can be related to the analogous mapping of polymer-hard spheres mixtures onto the Asakura-Oosawa model (41), where the effective hard sphere radius of the polymer scales as (R_g/R_c)^−1/3; the results of this mapping for simulations of hard spheres in a bath of ideal chains (42) are superimposed on Fig. 2 E.

We have also studied mixed macromolecular crowding with binary A:B and ternary A:B:C mixtures for the α-helical protein prb. For A:B mixtures, we calculated the change in folding free energy with the packing fraction of the type-A crowders (radius 8 Å) fixed at ${\varphi }_c^A=0.05$, and the fraction of type-B crowders (radius 12 Å) varying (Fig. 3). To test whether the effect of mixed crowding is simply additive and can be easily estimated from the pure crowder simulations, we also calculate the change in folding free energy from the following additivity Ansatz,

$$\Delta \Delta F_{\text{add}}\left({\mathit{\varphi }}_{\text{c}}^{\text{1}},{\mathit{\varphi }}_{\text{c}}^{\text{2}},\dots {\mathit{\varphi }}_{\text{c}}^{\text{N}}\right)={\sum }_{\text{i}}{x}_{\text{i}}\Delta \Delta F_{\text{i}}\left({\sum }_{\text{i}}{\mathit{\varphi }}_{\text{c}}^{\text{i}}\right),$$

where index i runs over N different types of crowding particles,

$$x_{\text{i}}={\varphi }_{\text{c}}^{\text{i}}/{\sum }_{\text{j}}{\varphi }_{\text{c}}^{\text{j}}$$

is the fraction of crowder type i in the mixture, and ${\varphi }_c^i$ is the volume fraction of i. We note that our expression differs from a previously proposed model in which (43)

$$\Delta \Delta F_{\text{add}}\left({\varphi }_{\text{c}}^{\text{1}},{\varphi }_{\text{c}}^{\text{2}},\mathrm{..}{\varphi }_{\text{c}}^{\text{N}}\right)={\sum }_{\text{i}}\Delta \Delta F_{\text{i}}\left({\mathit{\varphi }}_{\text{c}}^{\text{i}}\right)\text{.}$$

Figure 3.

SPT prediction for binary mixtures of 8 Å and 12 Å crowders. With the packing fraction of particle type A fixed to ${\varphi }_c^A=0.05$ and the fraction of particle type B, ${\varphi }_c^B=0.05\text{–}0.20$, the change in folding free energy of prb from simulations ΔΔF_sim is shown as a function of change in folding free energy ΔΔF_add from the additivity Ansatz (circle), as discussed in the text. (Inset) Simulation data (circle) along with SPT model predictions (line) using the additive model.

The predictions of our additive model ΔΔF_add for A:B mixtures are in extremely good agreement with the simulation results ΔΔF_sim as shown in Fig. 3. Moreover, for an A:B:C mixture of crowder radii 8, 12, and 16 Å with volume fraction ${\varphi }_c^i$ = 0.05 of each component, the agreement between the simulation result (0.55 kcal/mol) and the additive model (0.54 kcal/mol) is remarkably good. Applying additivity to our SPT model predictions for single crowders also provides reasonable estimates for ΔΔF (Fig. 3, inset). Our results and the additive model for mixed crowding do not predict an optimal mixing ratio as expected from a previous theory (18). Indeed, for a mixture of crowders at given ϕ_c, the greatest stabilization will occur when all the crowders are of the more stabilizing (smaller) type.

From our explicit dynamical model of folding, we are able to estimate folding rates directly from mean first-passage time calculations. Starting from at least 400 different initial coordinates of prb drawn from an equilibrium unfolded ensemble (Q ≈ Q_U ≡ 0.2) at a given packing fraction, we calculate the average time τ_f = 1/k_f taken to reach the folded state (Q > 0.9). We estimate the change in apparent barrier height from the folding rates as

$$\Delta \Delta F_{\mathrm{‡}-\text{U}}=\Delta F_{\mathrm{‡}-\text{U}}\left({\varphi }_c\right)-\Delta F_{\mathrm{‡}-\text{U}}^{\mathrm{bulk}}\approx k_{\text{B}}T\ln \left(k_{\text{f}}^{\text{bulk}}/k_{\text{f}}\right),$$

which is justified if the position of the folding barrier and the diffusion coefficient along the reaction coordinate are unchanged (Fig. 4) (44). We also estimate directly the contribution of the barrier to the rate from the free energy surface F(Q) using

$$\Delta F_{\mathrm{‡}-\text{U}}=k_{\text{B}}T\text{ln}\left({\int }_{\mathrm{‡}}{\text{e}}^{\mathit{\beta }F\left(Q\right)}dQ{\int }_{\text{U}}{\text{e}}^{-\mathit{\beta }F\left(Q\right)}dQ\right),$$

with integration bounds Q_‡ ± 0.1 and Q_U ± 0.1 for ‡ and U, respectively. With this definition, the change of barrier height from bulk to crowded should match that calculated from the rates, if the dynamics along Q is diffusive with a constant, position-independent diffusion coefficient (44).

Figure 4.

Influence of crowding on folding kinetics. The change in barrier height ΔΔF_‡–U is estimated using k_BT ln(k_f^bulk/k_f) (solid symbols) and from F(Q) (open symbols) for crowders of radius 8 Å (circle), 12 Å (square), and 16 Å (triangle up); lines are the SPT predictions using transition state $R_g^{‡}$ = 11.4 Å (prb). (Inset) Folding rates with binary crowder mixtures (A, 8 Å radius; B, 12 Å radius; ${\varphi }_c^A=0.05$). Results from direct simulations (circle) are compared with the predictions based on simulations with a single crowder size (square) and from SPT (line).

The folding barrier height decreases monotonically with increasing ϕ_c in qualitative agreement with previous theoretical models (3). Indeed, it is possible to predict the change in rates using SPT with no adjustable parameters: we calculate the transition state radius a_‡ from the Q umbrella simulations (Q = 0.55), and the change in barrier height from SPT, using a_‡ in place of a_N. The agreement between the change in barrier height estimated from the rates and from the free energy surface confirms that the barrier height rather than variations in diffusion coefficient or dynamics is the dominant contribution to kinetics. The effect of mixed macromolecular crowding on kinetics can also be obtained from single crowder simulations by assuming additivity (Fig. 4, inset).

In summary, we find that scaled particle theory provides an accurate description of the effect of macromolecular crowding on both folding stability and rates. The effect of purely repulsive crowders can be well approximated over a wide range of crowder sizes and packing fractions by treating the unfolded state as a hard sphere of fixed radius a_U. In all the cases considered, we find $a_{\mathrm{U}}\approx 0.8R_{\mathrm{g}}^{\mathrm{U}}$. Our results have several important consequences. For crowders of a single size, folding rate, and stability will increase with increasing packing fraction monotonically under the relevant physiological conditions. When considering mixtures of purely repulsive crowders of different sizes the effects of crowding are additive, and the most stabilizing composition will consist completely of the smallest crowder. Therefore, any maximum in stabilization as a function of the ratio of various components at a fixed total packing fraction implies the existence of attractive interactions with at least one of the crowders (45).