Crowding-induced protein destabilization in the absence of soft attractions

Abstract

It is generally assumed that volume exclusion by macromolecular crowders universally stabilizes the native states of proteins and destabilization suggests soft attractions between crowders and protein. Here we show that proteins can be destabilized even by crowders that are purely repulsive. With a coarse-grained sequence-based model, we study the folding thermodynamics of two sequences with different native folds, a helical hairpin and a β-barrel, in a range of crowder volume fractions, ${\phi }_{\text{c}}$. We find that the native state, N, remains structurally unchanged under crowded conditions, while the size of the unfolded state, U, decreases monotonically with ${\phi }_{\text{c}}$. Hence, for all ${\phi }_{\text{c}}>0$, U is entropically disfavored relative to N. This entropy-centric view holds for the helical hairpin protein, which is stabilized under all crowded conditions as quantified by changes in either the folding midpoint temperature, $T_{\text{m}}$, or the free energy of folding. We find, however, that the β-barrel protein is destabilized under low-T, low-${\phi }_{\text{c}}$ conditions. This destabilization can be understood from two characteristics of its folding: 1) a relatively compact U at $T<T_{\text{m}}$, such that U is only weakly disfavored entropically by the crowders; and 2) a transient, compact, and relatively low-energy nonnative state that has a maximum population of only a few percent at ${\phi }_{\text{c}}=0$, but increasing monotonically with ${\phi }_{\text{c}}$. Overall, protein destabilization driven by hard-core effects appears possible when a compaction of U leads to even a modest population of compact nonnative states that are energetically competitive with N.

Significance

The thermodynamic and kinetic properties of proteins can be drastically altered by various macromolecular crowding effects. Here we show that, contrary to common assumption, the excluded volume effect can destabilize a protein’s native state if compact nonnative states with sufficiently low energies are formed during folding. Even sparsely populated nonnative states might be sufficient for crowding-induced destabilization under conditions for which the unfolded state is relatively compact, such as low temperatures. Because the excluded volume effect is always present under crowded conditions, our results have implications also for macromolecular crowders capable of attractive interactions.

Introduction

Many valuable advances in protein biophysics have come from experiments on proteins in dilute solutions. However, the native environment of proteins, i.e., the cell interior, is spatially inhomogeneous and highly crowded (1). A substantial fraction of this space is occupied by macromolecules. For instance, the average total concentration of protein and RNA in the Escherichia coli cytosol is in the range 300–400 g/L, corresponding to volume fractions of around 30%–40% (2). Macromolecular crowding effects have been shown to impact a range of protein processes, such as folding (3), aggregation (4), and liquid-liquid phase separation (5).

An unavoidable consequence of macromolecular crowders is that they reduce the volume available to other molecules in the solution (6). Minton was first to quantify the impact of this excluded volume effect on protein stability, predicting it to be universally stabilizing (7). Experiments using artificial polymer crowders, such as Ficoll, dextran, or polyethelene glycol, as excluded volume agents, indeed very often stabilize proteins as indicated by, e.g., an increase in the free energy of unfolding or the folding midpoint temperature (8, 9, 10, 11, 12, 13, 14, 15). Some exceptions have also been found (11,16, 17, 18). For example, the protein apoazurin exhibits a decreased unfolding free energy at low concentrations of Ficoll-70 (18). Computational studies (19, 20, 21, 22, 23, 24, 25, 26, 27, 28) have also examined the effect of volume exclusion on protein stability, showing results that range from negligible to robustly stabilizing depending on crowding conditions and protein studied. Most of these computational studies, although not all (20), relied on so-called structure-based or Gō-type (29) models for folding in which attractive nonnative interactions are typically ignored. To the best of our knowledge, no computational crowding study has reported a decrease in protein stability from purely repulsive crowders.

More recent experiments (30, 31, 32, 33, 34) and theory (34,35) have shown that the stabilizing effect of hard-core steric repulsions can be opposed by nonspecific soft attractions (also known as chemical interactions) between crowders and protein, by energetically favoring the unfolded state. These soft attractions can even dominate over hard-core effects leading to a net destabilization (33), which appears to be common when the crowder molecules are proteins (21,36,37). As shown by Zhou (35), soft attractions may lead to a crossover temperature, $T_{\text{cross}}$, at which crowding effects switch from destabilizing ($T<T_{\text{cross}}$) to stabilizing ($T>T_{\text{cross}}$). Examining the temperature dependence has provided additional insights by allowing the enthalpic and entropic components of the unfolding free energy to be determined. In the case of artificial polymer crowders, stabilization was in some cases found to be driven by enthalpy rather than entropy, contrary to the expectation of the excluded volume effect (10,31,34). These results have been interpreted in terms of a preferential hydration effect (31,38), akin to the protective mechanism of osmolytes (39), in which the crowders are preferentially excluded from the protein-water interface.

Here we revisit the issue of the excluded volume effect on protein folding and stability, focusing on the role of nonnative interactions in folding. To this end, we use a coarse-grained model (40) for folding with three amino acid types, and a potential energy function based on hydrogen bonding and effective hydrophobic attractions. In this model, sequences can be designed using basic design principles (41) to fold into thermally stable states with protein-like native structures. As a focus of our study, we take two 35-amino-acid sequences (42,43) (see Table 1) that adopt a helical hairpin or a five-stranded β barrel, as shown in Fig. 1 A and B. We study their folding in the presence of spherical crowders with purely repulsive interactions. Unexpectedly, we find that the β-barrel protein is destabilized at low temperatures, contrary to the expectation of the excluded volume effect. Our results can, however, be fully rationalized by a crowder-driven population shift toward more compact states, both native and nonnative.

Table 1.

The two 35-amino acid model sequences studied in this work

${\alpha }_{35}$	p(phpphhp)2ptttp(phpphhp)2p
${\beta }_{35}$	p(hp)3tt(ph)2(hp)3tt(ph)2tt(hp)3

The three types of amino acids, p (polar), h (hydrophobic), and t (turn), are described in section “materials and methods.” Subscripts are used to indicate repeats. For instance, (ph)₂ means phph.

Figure 1.

Folding curves and native structures. Representative low-energy conformations from Monte Carlo simulations of (A) ${\alpha }_{35}$ and (B) ${\beta }_{35}$ in the presence of excluded volume crowders (blue spheres). The protein structures are shown in ribbon representation and rainbow color scheme (from N terminus in blue to C terminus in red). The native state population, $P_{\text{nat}}$, as a function of temperature, for (C) ${\alpha }_{35}$ and (D) ${\beta }_{35}$ in the absence of crowders (open black squares) and in the presence of crowders with radius $R_{\text{c}}=12$ Å and volume fraction ${\phi }_{\text{c}}\approx 0.40$ (solid green squares). Solid curves are obtained by using the multistate Bennett acceptance ratio (MBAR) reweighting technique (44). Statistical errors are estimated from independent runs and are smaller than the plot symbols. To see this figure in color, go online.

Materials and methods

Excluded volume crowders

We model crowder-protein and crowder-crowder interactions using the pair potential suggested by Mittal and Best (25),

$$V\left(r\right)=k_{\text{cr}}{\left(\frac{\sigma }{r-\rho +\sigma }\right)}^{12},$$ (1)

where $r\ge \rho -\sigma$ is the center-to-center distance either between two crowders or between a crowder and a protein atom. For $r<\rho -\sigma$, V is +∞. The two parameters ρ and σ control respectively the range and softness of the interaction, as illustrated in Fig. S1. For crowder-crowder pairs, we set $\rho =2R_{\text{c}}$ and $\sigma =2{\sigma }_{\text{cr}}$, where $R_{\text{c}}$ is the crowder radius and ${\sigma }_{\text{cr}}=3$ Å is a parameter setting the softness of the crowders. For crowder-atom pairs, we set $\rho =R_{\text{c}}+{\sigma }_{\text{a}}$ and $\sigma ={\sigma }_{\text{cr}}+{\sigma }_{\text{a}}$, where ${\sigma }_{\text{a}}$ is the atom radius. The interaction strength is set to $k_{\text{cr}}=1.0$. As typical, we describe the concentration of crowders with the fraction of the total simulation volume V occupied by the crowders, ${\phi }_{\text{c}}=4\pi R_{\text{c}}^3{N}_{\text{cr}}/3V$. The number of crowding particles in our simulations range from $N_{\text{cr}}=6$ for ${\phi }_{\text{c}}=0.10$ and the largest crowder ($R_{\text{c}}=16$ Å) to $N_{\text{cr}}=186$ for ${\phi }_{\text{c}}=0.41$ and the smallest crowder ($R_{\text{c}}=8$ Å).

Coarse-grained model for protein folding in the presence of crowders

As a model for protein folding, we use the coarse-grained " ${\text{C}}_{\beta }$-model" developed in Ref. (40), which includes three types of amino acids: hydrophobic (h), polar (p), and turn (t). While this three-letter alphabet is limited, it suffices to construct sequences with amphipatic secondary structure character. For example, repeats of the αsegment phpphhp can produce amphipathic α-helices and repeats of the βsegment ph can produce amphipatic β-strands. Such segments can be further organized into higher-order, tertiary structures through hydrogen bonding and effective hydrophobic attractions. We study two sequences: ${\alpha }_{35}$, constructed from two identical 16-amino-acid α segments linked by ttt, and ${\beta }_{35}$, constructed from five β segments linked by tt segments. In our model, ${\alpha }_{35}$ folds spontaneously into a stable helical hairpin (43) and ${\beta }_{35}$ folds into a stable five-stranded β-barrel (42).

Geometrically, the ${\text{C}}_{\beta }$-model includes seven atoms per amino acid. All backbone atoms are explicitly represented (N, ${\text{C}}_{\alpha }$, ${\text{C}}^{\prime }$, H, ${\text{H}}_{\alpha 1}$, and O), while the sidechain is represented by a single large ${\text{C}}_{\beta }$ atom. In contrast to p and h, type t lacks a ${\text{C}}_{\beta }$ atom, which is replaced by an ${\text{H}}_{\alpha 2}$ atom. Hence, t strongly resembles a glycine residue. All bond lengths and angles, and some dihedral angles (e.g., the peptide plane angle $\omega =180°$), are held fixed at standard values. As a result of these constraints, the internal conformation of a chain with N amino acids is completely specified by the $2N$ Ramachandran angles ${\left\{{\phi }_{\text{i}},{\psi }_{\text{i}}\right\}}_{\text{i}=1}^N$.

To incorporate crowders we extend here the ${\text{C}}_{\beta }$-model energy function $E_{\text{pr}}$ (40) to include also terms for crowder-crowder ($E_{\text{cr}}$) and crowder-protein ($E_{\text{cr}-\text{pr}}$) interactions. The total energy function is thus $E=E_{\text{pr}}+E_{\text{cr}}+E_{\text{cr}-\text{pr}}$. Crowder interactions are assumed to be pairwise additive and modeled with the pair potential $V\left(r\right)$, as described in section “excluded volume crowders.” Hence, for a system with $N_{\text{cr}}$ crowders and $N_{\text{a}}$ protein atoms,

$$E_{\text{cr}}=\sum _{\text{i<j}}^{N_{\text{cr}}}V\left(r_{\text{ij}}\right)$$ (2)

and

$$E_{\text{cr}-\text{pr}}=\sum _{\text{i}}^{N_{\text{cr}}}\sum _{\text{j}}^{N_{\text{a}}}V\left(r_{\text{ij}}\right),$$ (3)

where $r_{\text{ij}}$ is the center-to-center distance between the crowder i and crowder j (Eq. 2) or between crowder i and protein atom j (Eq. 3).

The ${\text{C}}_{\beta }$-model energy function, $E_{\text{pr}}=E_{\text{hp}}+E_{\text{hbond}}+E_{\text{exvol}}+E_{\text{loc}}$, is described in the following (see also Ref. (40)). The first term,

$$E_{\text{hp}}=-k_{\text{hp}}\sum _{\text{ij}}{e}^{-{\left(r_{\text{ij}}-{\sigma }_{\text{hp}}\right)}^2/2},$$ (4)

describes effective hydrophobic interactions. The sum goes over all pairs of hydrophobic C_β atoms, excluding nearest and next-nearest amino acid neighbors along the chain. The strength of hydrophobicity is $k_{\text{hp}}=0.805$, and the energetically optimal C_β-C_β distance is ${\sigma }_{\text{hp}}=5$ Å. The second term represents hydrogen bonding between NH and C′O groups, and can be written

$$E_{\text{hbond}}=k_{\text{hbond}}\sum _{\text{ij}}{\gamma }_{\text{ij}}\left[5{\left(\frac{{\sigma }_{\text{hb}}}{r_{\text{ij}}}\right)}^{12}-6{\left(\frac{{\sigma }_{\text{hb}}}{r_{\text{ij}}}\right)}^{10}\right]\times {\left(\cos {\alpha }_{\text{ij}}\cos {\beta }_{\text{ij}}\right)}^{\frac{1}{2}},$$ (5)

where the sum is over all CO and NH pairs, but with the restriction that i and j are separated by at least two CO groups or two NH groups, and the strength is controlled by $k_{\text{hbond}}=3.1$. The strength is modified by a sequence-dependent scale factor taken to be ${\gamma }_{\text{ij}}=1$ for hh, hp, and pp pairs, and 0.75 for tt, th, and tp pairs. The reduced hydrogen-bonding capacity of t amino acids is meant to mimic the tendency for glycine residues to break secondary structure through weaker hydrogen bonds (45). For a hydrogen bond, the optimal HO distance is ${\sigma }_{\text{hb}}=2.0$ Å. A directional dependence is implemented via the factor ${\left(\text{cos}{\alpha }_{\text{ij}}\text{cos}{\beta }_{\text{ij}}\right)}^{\frac{1}{2}}$, where ${\alpha }_{\text{ij}}$ and ${\beta }_{\text{ij}}$ are the NHO and HOC′ angles, respectively. Additionally, for any bond with either ${\alpha }_{\text{ij}}<90°$ or ${\beta }_{\text{ij}}<90°$, the bond contribution is set to zero. The third term is the excluded volume energy, which can be expressed as

$$E_{\text{exvol}}=k_{\text{exvol}}\sum _{\text{i}<\text{j}}{\left(\frac{{\lambda }_{\text{ij}}{\sigma }_{\text{ij}}}{r_{\text{ij}}}\right)}^{12},$$ (6)

where $k_{\text{exvol}}=0.10$. The sum is taken over all pairs of atoms ij connected by more than two covalent bonds. The scale factor ${\lambda }_{\text{ij}}=1.00$ for atom pairs connected by three covalent bonds. With two exceptions, all other global atom pairs have ${\lambda }_{\text{ij}}=0.75$. The two exceptions are carboxyl OO pairs and amide HH pairs for which ${\lambda }_{\text{ij}}=1.00$ and 1.25, respectively. The reduction factor ${\lambda }_{\text{ij}}=0.75$ for most global pairs is meant to accommodate the reduced flexibility of a chain with fixed bond lengths and angles. The parameter ${\sigma }_{\text{ij}}={\sigma }_{\text{i}}+{\sigma }_{\text{j}}$ is the sum of i and j atom radii, taken to be 1.75, 1.42, 1.55, and 1.00 Å for carbon, oxygen, nitrogen, and hydrogen atoms, respectively. An exception is ${\text{C}}_{\beta }-{\text{C}}_{\beta }$ pairs, for which ${\sigma }_{\text{ij}}=5.0$ Å, thereby accounting for some of the bulkiness of sidechains. The last term captures local interactions between partial charges in adjacent peptide planes,

$$E_{\text{loc}}=k_{\text{local}}\sum _{\text{I}}\sum _{\text{i},\text{j}}\frac{q_{\text{i}}{q}_{\text{j}}}{r_{\text{ij}}/\text{Å}},$$ (7)

where the strength is $k_{\text{local}}=50$ and the partial charges $q_{\text{i}}$ are $-0.2$, $+0.2$, $+0.42$, and $-0.42$ for N, H, C′, and O, respectively. The outer sum is over all amino acids I and the inner sum over the N${\text{C}}^{\prime }$, NO, H${\text{C}}^{\prime }$, and HO atom pairs of amino acid I.

Simulated tempering Monte Carlo

To find the thermodynamic behavior of various protein-crowder systems, as determined by the amino acid sequence, number of crowders, and the energy function $E\left(r\right)$, we use simulated tempering Monte Carlo (MC) (46, 47, 48). In addition to a random walk in conformational space, as in basic MC, simulated tempering also carries out a random walk in temperature while keeping the simulation at equilibrium. This is achieved by defining a set of temperatures, ${\left\{T_j\right\}}_{j=1}^M$, and simulating the joint probability distribution

$$P\left(r,j\right)\propto e^{-{\beta }_{j}E\left(r\right)+g_j},$$ (8)

where ${\beta }_j=1/k_{\text{B}}{T}_j$, $k_{\text{B}}$ is Boltzmann’s constant, and j has been made a dynamic parameter. The $g_{\text{j}}$s are M simulation parameters that control the marginal distribution $P\left(j\right)$. Jumps between temperatures, $j\to j^{\prime }$, are accomplished as MC updates, with acceptance probability

$$P_{\text{acc}}\left(r,j\to j^{\prime }\right)=\text{min}\left[1,e^{-E\left(r\right)\left({\beta }_{j^{\prime }}-{\beta }_j\right)+g_{j^{\prime }}-g_j}\right].$$ (9)

A common choice for the $g_j$ parameters, which we follow here, is to select them such that $P\left(j\right)$ is roughly flat, ensuring that sampling of conformations takes place at each temperature $T_j$.

Simulation and analysis details

Protein and crowders are placed in a $V=100\times 100\times 100$ ų box with periodic boundary conditions. Crowder positions are updated using single-particle translational MC moves, with random direction and maximum distance 8.7 Å. Two different types of MC moves are used for the protein chain: pivot moves and Biased Gaussian steps (BGS) (49). In pivots, a single ${\psi }_{\text{i}}$ or ${\phi }_{\text{i}}$ angle is chosen and assigned a new random value. This rotates the chain around a ${\text{NC}}_{\alpha }$ bond or a ${\text{C}}_{\alpha }{\text{C}}^{\prime }$ bond, respectively. In BGS, eight consecutive ${\psi }_{\text{i}},{\phi }_{\text{i}}$ angles are changed in a coordinated way to provide a roughly local chain deformation. The frequency of different updates are chosen as follows. Updates are divided equally between crowder particles and protein. The relative frequency between pivots and BGS is chosen to be temperature dependent, such that pivot dominates at high T and BGS dominates at low T. Temperature updates are attempted every 100 MC steps.

Simulated tempering runs for protein-crowder systems are carried out using eight different temperatures in the range $k_{\text{B}}T=$ 0.40–0.70 for ${\alpha }_{35}$ and 0.50–0.70 for ${\beta }_{35}$. For each system, 20 independent runs are carried out of each $5\times {10}^9$ elementary MC steps. Initial conformations are created by picking a random protein conformation and random crowder positions, followed by a relaxation step in which hard-core steric clashes involving the crowders are removed.

Statistical analysis is carried out using the multistate Bennett acceptance ratio (MBAR) technique (44), which optimally combines statistics from different thermodynamic conditions. We apply MBAR to combine the statistics from the eight different $T_j$ values in our simulated tempering runs and use it to determine the thermodynamic averages of various observables at narrowly spaced temperatures in the range $k_{\text{B}}T=$ 0.40–0.70. Since for ${\beta }_{35}$ the range of simulated $k_{\text{B}}{T}_j$ values is 0.50–0.70, the MBAR analysis involves an extrapolation to the range $k_{\text{B}}T=$ 0.40–0.50. Separate test simulations of ${\beta }_{35}$, covering a wider temperature range, confirm that the MBAR extrapolation is valid. Statistical errors are estimated from the 20 independent runs.

Observables

The number of native contacts is defined by $Q_{\text{nat}}={\sum }_{\text{i}<\text{j}-3}{\text{Δ}}_{\text{ij}}{C}_{\text{ij}}$, where the sum goes over pairs of residues ij, and ${\text{Δ}}_{\text{ij}}=1$ if ij has formed a contact and 0 otherwise. A contact between amino acids i and j is considered formed if the distance between their ${\text{C}}_{\beta }$ atoms is $<7$ Å (positions with a t amino acid type, which lacks a ${\text{C}}_{\beta }$ atom, do not contribute toward contact counts). The native contact set, C, is defined so that $C_{\text{ij}}=1$ if ij is a native contact, and otherwise 0. We use the two native contact sets taken from our previous study (42). We define the native state as $Q_{\text{nat}}\ge Q_{\text{cut}}$, where $Q_{\text{cut}}=54$ and 68 for ${\alpha }_{35}$ and ${\beta }_{35}$, respectively. For ${\beta }_{35}$, $Q_{\text{cut}}$ was selected based on the peak in the free-energy barrier along the order parameter $Q_{\text{nat}}$, as shown in Ref. (42). For ${\alpha }_{35}$, which does not exhibit a clear folding barrier, $Q_{\text{cut}}$ is selected such that the folding midpoint temperature, $T_{\text{m}}^0$, roughly coincides with the temperature $T_{C_{\text{v},\text{max}}}$ at the maximum in the heat capacity curve (see Fig. S2). For $Q_{\text{cut}}=54$, we obtain $T_{\text{m}}^0/T_{C_{\text{v},\text{max}}}=0.985$). The number of nonnative contacts is determined from $Q_{\text{nonnat}}=Q-Q_{\text{nat}}$, where $Q={\sum }_{\text{i}<\text{j}-3}{\text{Δ}}_{\text{ij}}$ is the total number of contacts. The root-mean-square deviation (RMSD) is determined over all ${\text{C}}_{\alpha }$ atoms. As reference (native) structures for ${\alpha }_{35}$ and ${\beta }_{35}$, we pick the lowest energy conformations found from simulations with no crowders. These reference structures are similar to those shown in Fig. 1.

Results

Native structures are not changed by crowders

Using the model for protein folding and the MC sampling techniques described in section “materials and methods,” we determine the thermodynamic behavior of the two model sequences, ${\alpha }_{35}$ and ${\beta }_{35}$, given in Table 1, in the presence of crowders with radii $R_{\text{c}}=8$, 12, and 16 Å and volume fractions in the range $0\le {\phi }_{\text{c}}\le 0.41$. The crowders are comparable in size with the ${\alpha }_{35}$ and ${\beta }_{35}$ chains, which have radii of gyration ranging from $\approx$ 8–9 Å for the native state to $\approx$ 14–15 Å for the unfolded state (shown below).

As a first step in our analysis, we examine the temperature dependence of the native state population, $P_{\text{nat}}$, in the absence of crowders (${\phi }_{\text{c}}=0$), as shown in Fig. 1 C and D. For both proteins, these equilibrium folding curves are well described by a two-state equation, with only two free fit parameters (see Fig. S3). One of the free parameters is the folding midpoint temperature, $T_{\text{m}}^0$. We find (in model units) $k_{\text{B}}{T}_{\text{m}}^0=0.534$ and 0.515 for ${\alpha }_{35}$ and ${\beta }_{35}$, respectively, indicating similar native state stabilities. In calculating $P_{\text{nat}}$, we define the native state, N, as $Q\ge Q_{\text{cut}}$, where Q is the number of native contacts, and $Q_{\text{cut}}$ is chosen as described in section “materials and methods.” The folding curves are robust with respect to the definition of N. For example, they remain similar if N is instead defined using the RMSD (see Fig. S4).

For simulations at ${\phi }_{\text{c}}>0$, the protein chain must avoid overlapping with the hard cores of the crowding particles. Conversely, of course, the crowders must similarly avoid the protein chain. Because the volume available to the crowders is reduced when the protein is expanded, compact conformations will be entropically favored under crowded conditions. In our model, atoms on the protein chain can penetrate the soft shell of the crowder particles, at an energetic cost. The thickness of this soft shell is controlled by a softness parameter, which we hold fixed at ${\sigma }_{\text{cr}}=3.0$ Å (see section “materials and methods”). At this ${\sigma }_{\text{cr}}$, the total protein-crowder and crowder-crowder repulsive energy, $E_{\text{cr}-\text{pr}}+E_{\text{cr}-\text{cr}}$, turns out to be small. For example, for $R_{\text{c}}=12$ Å and ${\phi }_{\text{c}}=0.20$, this repulsive energy per crowding particle is $\approx 0.1k_{\text{B}}T$. We therefore expect our crowders to behave roughly as hard spheres. We confirm this by carrying out test simulations of ${\beta }_{35}$ in the presence of exact hard sphere crowders (i.e., with the soft shell removed), showing results similar to the ${\sigma }_{\text{cr}}=3.0$ Å case (see Fig. S5). For this reason, we refer to our crowders as excluded volume crowders.

We find that the folding curves for the ${\phi }_{\text{c}}=0$ and ${\phi }_{\text{c}}>0$ cases are generally different, as illustrated in Fig. 1 C and D (the folding curves for all ${\phi }_{\text{c}}$ values are given in Fig. S6). For all studied ${\phi }_{\text{c}}$, $P_{\text{nat}}\approx 1$ at very low T. We conclude that for ${\phi }_{\text{c}}\le 0.41$ the ${\alpha }_{35}$ and ${\beta }_{35}$ native structures are not substantially perturbed by the crowders.

Nonnative interactions are promoted under crowded conditions

Fig. 2 shows the number of nonnative contacts, $Q_{\text{nonnat}}$, as a function of temperature across different ${\phi }_{\text{c}}$. It is clear that $Q_{\text{nonnat}}$ increases monotonically with ${\phi }_{\text{c}}$ except at very low T. Interestingly, for both ${\alpha }_{35}$ and ${\beta }_{35}$, the $Q_{\text{nonnat}}$ curve exhibits a peak at intermediate T, which can be understood in the non-crowding case in the following way. At high T, the chain is in an entropy dominated state with contacts formed and unformed basically at random, leading to a mix of native and nonnative contacts. The total number of contacts, $Q_{\text{nat}}+Q_{\text{nonnat}}$, is not maximal at this T, however, because the chain is expanded (see section “impact of crowding on the unfolded state”). As T decreases, $Q_{\text{nonnat}}$ initially increases because the chain becomes more compact but then abruptly decreases when T approaches $T_{\text{m}}^0$ due to folding (and $Q_{\text{nat}}$ abruptly increases). The net result is a peak in $Q_{\text{nonnat}}$ at $T\approx 1.05T_{\text{m}}$ for both ${\alpha }_{35}$ and ${\beta }_{35}$. As it turns out, the $Q_{\text{nonnat}}$ peak remains under crowding conditions. Overall, we find that the excluded volume crowders generally promote formation of nonnative interactions, except at very low T where the native state is thermodynamically dominant. We note that the increase in $Q_{\text{nonnat}}$ with ${\phi }_{\text{c}}$ is more than linear, as shown in Fig. 2 (insets).

Figure 2.

Formation of nonnative interactions under crowded and non-crowded conditions. Number of nonnative contacts, $Q_{\text{nonnat}}$, as a function of temperature for (A) ${\alpha }_{35}$ and (B) ${\beta }_{35}$, at different crowder volume fractions ${\phi }_{\text{c}}$. Insets: $Q_{\text{nonnat}}$ as a function of ${\phi }_{\text{c}}$ (solid circles) and fits to $Q_{\text{nonnat}}\propto {\phi }_{\text{c}}^{\gamma }$, giving $\gamma =1.8$ (dotted curves) taken at $k_{\text{B}}T=0.599$ for ${\alpha }_{35}$ and $k_{\text{B}}T=0.567$ for ${\beta }_{35}$, in both corresponding to a temperature $T\approx 1.10T_{\text{m}}^0$. All results are for $R_{\text{c}}=12$ Å. To see this figure in color, go online.

Excluded volume crowders can both increase and decrease native state stability

We turn now to the effect of crowders on the stability of ${\alpha }_{35}$ and ${\beta }_{35}$. Because crowding effects can be strongly dependent on solution conditions (11,35), we examine stability changes at two different temperatures, above ($T^{+}=1.05T_{\text{m}}^0$) and below ($T^{-}=0.95T_{\text{m}}^0$), the folding midpoint temperature in the absence of crowders ($T_{\text{m}}^0$). As a direct measure of stability, we use the free energy of folding,

$$\text{Δ}F=F_{\text{N}}-F_{\text{U}}=-k_{\text{B}}T\ln \frac{P_{\text{nat}}}{1-P_{\text{nat}}},$$ (10)

where $F_{\text{N}}$ and $F_{\text{U}}$ are the free energies of the native and unfolded states, respectively, and we have assumed that the unfolded state population is $1-P_{\text{nat}}$. Fig. 3 A and B show the crowding-induced change in the free energy, $\text{ΔΔ}F\left({\phi }_{\text{c}}\right)=\text{Δ}F\left({\phi }_{\text{c}}\right)-\text{Δ}{F}^0$, where $\text{Δ}{F}^0=\text{Δ}F({\phi }_{\text{c}}=0)$, as function of ${\phi }_{\text{c}}$. The stability of ${\alpha }_{35}$ increases with ${\phi }_{\text{c}}$ at both $T^{-}$ and $T^{+}$, as indicated by a negative $\text{ΔΔ}F$ and an increase in $T_{\text{m}}$ (see Fig. 3 C). By contrast, for ${\beta }_{35}$, both $\text{ΔΔ}F$ and $T_{\text{m}}$ depend nonmonotonically on ${\phi }_{\text{c}}$. Specifically, $\text{ΔΔ}F$ initially increases until a minimal stability is reached at some packing fraction ${\phi }_c^{\ast }$, whereafter $\text{ΔΔ}F$ decreases. The minimal stability occur at quite different packing fractions for the two temperatures: ${\phi }_c^{\ast }\approx 0.10$ at $T^{+}$ and ${\phi }_c^{\ast }\approx 0.30$ at $T^{-}$, indicating a rather strong T dependence on $\text{ΔΔ}F$. Strikingly, at $T^{-}$, the ${\beta }_{35}$ protein is destabilized over the entire range $0<{\phi }_{\text{c}}\le 0.39$ relative to the no-crowder case.

Figure 3.

Effect of crowding on native state stability and folding cooperativity. Change in the free energy of folding, $\Delta \Delta F=\Delta F-\Delta F^0$, as a function of ${\phi }_{\text{c}}$, at temperatures (A) $T^{-}=0.95T_{\text{m}}^0$ and (B) $T^{+}=1.05T_{\text{m}}^0$. $\Delta \Delta F<0$ indicates stabilization. (C) Midpoint folding temperature, $T_{\text{m}}$, and (D) maximum heat capacity, $C_{\text{v}}^{\text{max}}$, as functions of ${\phi }_{\text{c}}$, where the heat capacity is determined from $C_{\text{v}}=\left(⟨E^2⟩-{⟨E⟩}^2\right)/k_{\text{B}}{T}^2$. Results are shown for ${\alpha }_{35}$ (filled circles) and ${\beta }_{35}$ (open squares). $\Delta F^0$, $T_{\text{m}}^0$, and $C_{\text{v}}^{\text{max},0}$ are the ${\phi }_{\text{c}}=0$ values of $\text{Δ}F$, $T_{\text{m}}$, and $C_{\text{v}}^{\text{max}}$, respectively. The crowder radius $R_{\text{c}}=12$ Å. Error bars in parts A and B indicate statistical errors estimated from independent runs. Dashed lines between points are drawn to guide the eye.

Because our crowders are purely repulsive, one might expect any change in stability $\text{Δ}F$ to originate entirely from the entropic component ($\text{Δ}S$) while the energetic component ($\text{Δ}E$) is left unchanged. However, this is not guaranteed because interactions within the protein include a mix of energy driven attractions and steric repulsions, and this mix of interactions can be affected by the crowders. To examine this issue, we dissect the folding free energy into its energy and entropy components, $\text{Δ}F=\text{Δ}E-T\text{Δ}S$, by measuring $\text{Δ}E=E_{\text{N}}-E_{\text{U}}$, where $E_{\text{N}}$ and $E_{\text{U}}$ are the average energies of the native and unfolded ensembles, respectively, and applying Eq. 10 to find $\text{Δ}S=\left(\text{Δ}E-\text{Δ}F\right)/T$. $\text{Δ}E$ and $\text{Δ}S$ are strongly negative for both ${\alpha }_{35}$ and ${\beta }_{35}$ due to the energy-entropy compensation of folding (50); however, they become less negative with increasing ${\phi }_{\text{c}}$ (see Fig. S7). For example, for ${\beta }_{35}$ at $T^{+}$, we find that $\text{Δ}E/k_{\text{B}}T=-72.1$ and $\text{Δ}S/k_{\text{B}}=-74.9$ at ${\phi }_{\text{c}}=0$ while, at ${\phi }_{\text{c}}=0.39$, $\text{Δ}E/k_{\text{B}}T=-56.4$ and $\text{Δ}S/k_{\text{B}}=-58.6$. The decrease in the energy gap $\left|\text{Δ}E\right|$ between native and unfolded states is in line with the reduced folding cooperativity we observe under crowded conditions, as quantified by the peak in heat capacity (see Fig. 3 D). It is likely that these crowding-induced changes in folding thermodynamics primarily arise from a compaction of the unfolded state, because $E_{\text{N}}$ and $S_{\text{N}}$ are left relatively unchanged. Benton et al. found that CI2 was enthalpically stabilized by both Ficoll and its monomer unit sucrose, and coupled this stabilization to a preferential hydration mechanism (10). Our analysis highlights that a re-modeling of the unfolded state ensemble, which can be accomplished by hard-core effects, may lead to changes in both the energetic component and the entropic component of the folding free energy.

Crossover temperature

The temperature-dependent crowding effect for ${\beta }_{35}$ (cf. Fig. 3 A and B) suggests there is a temperature, $T_{\text{cross}}$, such that crowders enhance stability at $T>T_{\text{cross}}$ but reduce stability at $T<T_{\text{cross}}$. The ${\beta }_{35}$ sequence does indeed exhibit such a crossover temperature, as seen in Fig. 4 B. $T_{\text{cross}}$ is slightly below $T_{\text{m}}^0$ at ${\phi }_{\text{c}}=0.20$. Moreover, Fig. 4 C and D show that $T_{\text{cross}}$ decreases with ${\phi }_{\text{c}}$ for fixed crowder radius $R_{\text{c}}$, and $T_{\text{cross}}$ increases with $R_{\text{c}}$ for fixed ${\phi }_{\text{c}}$. The ${\alpha }_{35}$ sequence does not have a crossover temperature because it is always stabilized, even though the magnitude of the stabilization is still temperature dependent (see Fig. 4 A). The situation for ${\beta }_{35}$ is qualitatively similar to that of ubiquitin, which was studied in the presence of synthetic (PVP or Ficoll) and protein (BSA or lysozyme) crowders (11). Crossover temperatures were later determined for these systems (35). For example, $T_{\text{cross}}$ for ubiquitin in the presence of Ficoll at concentration 100 g/L was estimated to be 301 K, much lower than the folding midpoint temperature of this protein (370 K). The existence of $T_{\text{cross}}$ was proposed to originate from “soft” attractive interactions between protein and crowders (35). That such soft attractions can occur in the case of protein crowders is by now well established (1,20,37). It is much less clear, however, if they occur for all synthetic crowders. Our results demonstrate that destabilization at low T can occur even in the absence of protein-crowder attractions.

Figure 4.

Existence of a crossover temperature, $T_{\text{cross}}$, in a system with only excluded volume crowders. $\Delta \Delta F$ as a function of temperature for (A) ${\alpha }_{35}$ and (B) ${\beta }_{35}$, under fixed crowding conditions (${\phi }_{\text{c}}=0.20$ and $R_{\text{c}}=12$ Å). A crossover temperature, defined by $\Delta \Delta F=0$, exists for ${\beta }_{35}$ but not for ${\alpha }_{35}$. (C) $T_{\text{cross}}$ as a function of ${\phi }_{\text{c}}$ for fixed $R_{\text{c}}=12$ Å. (D) $T_{\text{cross}}$ as a function of $R_{\text{c}}$ for fixed ${\phi }_{\text{c}}=0.20$.

Impact of crowding on the unfolded state

In order to understand the distinct responses of ${\alpha }_{35}$ and ${\beta }_{35}$ to crowded conditions, we examine more closely the character of the folding transition (see Fig. 5). Both proteins undergo a collapse at low T, as seen Fig. 5 A and B (solid curves). At the highest studied T, the (average) radius of gyration $R_{\text{g}}\approx$ 14–15 Å, which can be compared with the value $\approx 15.6$ Å obtained from the scaling law $R_{\text{g}}=R_0{N}^{\nu }$ where $R_0$ is a constant and $\nu =0.588$, which holds for fully chemically denatured proteins (51). Hence, ${\alpha }_{35}$ and ${\beta }_{35}$ transition from a random coil at high T to the compact folded state at low T.

Figure 5.

Effect of crowders on the size of the native and unfolded states. The average radius of gyration, $R_{\text{g}}$, of (A) ${\alpha }_{35}$ and (B) ${\beta }_{35}$, as a function of temperature (solid curves). Shown are also $R_{\text{g}}$ determined over the unfolded ($R_{\text{g}}^{\text{u}}$, dashed) and native ($R_{\text{g}}^{\text{n}}$, dotted) ensembles. Estimates of $R_{\text{g}}^{\text{n}}$ become unreliable at $k_{\text{B}}T\gtrsim 0.6$ due to the small native state population at these Ts, especially for ${\beta }_{35}$. The change in unfolded state radius of gyration, $\Delta R_{\text{g}}^{\text{u}}\left({\phi }_{\text{c}}\right)=R_{\text{g}}^{\text{u}}\left({\phi }_{\text{c}}\right)-R_{\text{g}}^{\text{u}}({\phi }_{\text{c}}=0)$, as function of ${\phi }_{\text{c}}$, for (C) ${\alpha }_{35}$ and (D) ${\beta }_{35}$, taken at the highest ($k_{\text{B}}T=0.70$, circles) and lowest ($k_{\text{B}}T=0.40$, triangles) studied temperatures. Statistical errors are estimated for parts C and D and are small.

It is instructive to consider also the size of the folded ($R_{\text{g}}^{\text{n}}$) and unfolded ($R_{\text{g}}^{\text{u}}$) state ensembles. In particular, the size (and shape) of the unfolded state is important for how volume exclusion affects protein stability (52). In contrast to the native state size, which is basically independent of T, the unfolded state is very sensitive to temperature changes (see Fig. 5 A and B). Above $T_{\text{m}}$, both the ${\alpha }_{35}$ and ${\beta }_{35}$ unfolded states become increasingly compact as conditions become more stabilizing. Below $T_{\text{m}}$, the ${\beta }_{35}$ unfolded state remains compact following the chain collapse, while a re-expansion occurs for ${\alpha }_{35}$. Visual inspection of low-T unfolded structures provides some insight into the dramatic difference between the proteins. Unfolded ${\beta }_{35}$ is characterized by a partial loss of the β-barrel organization, with one or more strands detached. These strands remain close to the remaining part of the barrel through hydrophobic attractions, which are strong at low T in our model, leaving $R_{\text{g}}^{\text{u}}$ small. Unfolded ${\alpha }_{35}$ is characterized by a partial or complete opening of the helical hairpin. As T decreases, the two α-helices become increasingly stable on their own and thus stiffer. At very low T, these stiff helices are unable to accommodate a hydrophobically collapsed unfolded state, but must instead dissociate while remaining well formed. As a result, $R_{\text{g}}^{\text{u}}$ increases with decreasing T.

In the presence of excluded volume crowders, an expanded state will typically become more compact. The ${\beta }_{35}$ unfolded state, however, is already quite compact at low T and therefore basically unaffected by the crowders (see Fig. 5 D). This explains, in particular, the weak dependence of $T_{\text{m}}$ on ${\phi }_{\text{c}}$ for this protein (see Fig. 3 C). Although ${\beta }_{35}$ is stabilized at high T (i.e., $P_{\text{nat}}$ increases), a strong shift in $T_{\text{m}}$ does not occur unless $P_{\text{nat}}$ also increases at $T\lesssim T_{\text{m}}^0$, which is not the case for ${\beta }_{35}$. Stabilization at both low and high T occurs for ${\alpha }_{35}$, and this sequence indeed exhibits a strong shift in $T_{\text{m}}$ (see Fig. 3 C). The above observations highlight the need for measuring crowding-induced stability changes over a range of temperatures (11).

Compact nonnative states leads to destabilization

Crowding-induced protein destabilization must arise from a shift in the protein’s conformational ensemble such that the population of the native state decreases relative to other states. For crowders with soft attractions, such a population shift away from the native state can be driven by favorable crowder-protein interactions that energetically stabilize the unfolded state. In order to rationalize the destabilization of ${\beta }_{35}$ due to entirely repulsive crowders, we look for nonnative states stabilized by interactions within the protein. Fig. 6 A shows the free-energy surface $F\left(E,\text{RMSD}\right)$ at $T\approx T_{\text{m}}^0$. Although $F\left(E,\text{RMSD}\right)$ exhibits an overall funnel shape toward the low-E/low-RMSD native state, there is a small but non-negligible population with relatively low E and $\text{RMSD}\gtrsim$ 7Å. We define a state based on the criterion $E<0$ and 7Å $<\text{RMSD}<$ 9Å (see dashed box in Fig. 6 A), with the limit in E chosen because the distribution of energies in the native state has an upper bound of $E\approx 0$ (see Fig. S8 A). This way, included conformations will be energetically competitive with the native state. We find that the state defined this way includes both native and nonnative contacts, with the ratio $Q_{\text{nat}}/Q_{\text{nonnat}}\approx 3$. Structurally, it is compact, rich in β-sheet structure, and exhibits a nonnative organization of its β strands.

Figure 6.

Nonnative transient state in the folding of ${\beta }_{35}$. (A) Free-energy surface $F\left(E,\text{RMSD}\right)=-k_{\text{B}}T\text{ln}P\left(E,\text{RMSD}\right)$, where the joint probability distribution $P\left(E,\text{RMSD}\right)$ is taken at $T\approx T_{\text{m}}$, E is the total energy and RMSD is taken with respect to the representative native structure of ${\beta }_{35}$ (see Fig. 1B). The nonnative state is defined by the boundaries $E<0$ and 7 Å $<\text{RMSD}<$ 9 Å (dashed lines), and a representative structure is shown in cartoon representation. (B) $P_{\text{transient}}$ as a function of temperature at different packing fractions ($R_{\text{c}}=12$ Å), where $P_{\text{transient}}$ is the population of the low-E, nonnative state defined in (A). This nonnative state is transient in the sense that $P_{\text{transient}}$ is sharply peaked close to the folding midpoint. To see this figure in color, go online.

This nonnative state is also transient in the sense that it reaches a maximum population, $P_{\text{transient}}\approx 2\text{\%}$, at around $0.9T_{\text{m}}^0$, as shown in Fig. 6 B. Upon the addition of crowders, $P_{\text{transient}}$ increases monotonically with ${\phi }_{\text{c}}$. However, $P_{\text{transient}}$ remains very small at low Ts, such that the native state is structurally unaltered ($P_{\text{transient}}<0.3$% at $k_{\text{B}}T=0.40$). The crowding-induced increase in $P_{\text{transient}}$ results in a decrease in $P_{\text{nat}}$, and hence a destabilization, because of two factors: 1) the excluded volume effect favors compact states, both native and nonnative, relative to more expanded states; and 2) the energy of the transient state, although higher on average than the native state, is low enough that it can compete with the native state. With increasing ${\phi }_{\text{c}}$, the population of the two states will increasingly be determined by their relative sizes and less by their energy difference. As a result, as ${\phi }_{\text{c}}$ increases, the nonnative state gains population relative to the native state, at least in a temperature range around $T_{\text{m}}^0$.

To determine if the general shift of the unfolded state toward more compact states also contributes to the destabilization of ${\beta }_{35}$, we consider the full ensemble of low-E, nonnative (i.e., $E<0$ and $Q<Q_{\text{cut}}$) conformations. This ensemble, which includes the transient state, can be thought of as the low-E “tail” of the unfolded state. We find that its population peaks at $\approx 9\text{\%}$ at ${\phi }_{\text{c}}=0$, increasing to $\approx 13\text{\%}$ at ${\phi }_{\text{c}}=0.39$ (see Fig. S8 B). Therefore, it is possible that the shift of the unfolded state to more compact and lower-energy conformations also contributes to the destabilization. The relative population increase of the transient state, by approximately a factor two from the no crowder case to ${\phi }_{\text{c}}=0.39$, is, however, much more pronounced than for conformations found generally in the low-E tail of the unfolded state. This suggests that the transient state is energetically more competitive than the native state, and therefore has a greater impact on the native state population.

Apparent stabilization effect is observable dependent

Finally, we examine the folding progress in variables other than $P_{\text{nat}}$. Fig. 7 shows the ${\phi }_{\text{c}}$ dependence of the end-to-end distance, $R_{\text{ee}}$, and secondary structure contents. We note especially that the destabilization of ${\beta }_{35}$ is not apparent in $R_{\text{ee}}$, which follows a trend closely related to that of $R_{\text{g}}$. The β-structure content is rather insensitive to an increase in ${\phi }_{\text{c}}$, showing no detectable change or a small increase, in contrast to the decrease in $P_{\text{nat}}$ seen under low-T, low-${\phi }_{\text{c}}$ conditions (cf. Figs. 3 A and 7 D).

Figure 7.

Impact of crowding on secondary structure contents and end-to-end distance. Average of (A and B) end-to-end distance, $R_{\text{ee}}$; (C) number of helical amino acids, $N_{\alpha }$; and (D) number of β sheet amino acids, $N_{\beta }$, as functions of temperature, shown for ${\alpha }_{35}$ and/or ${\beta }_{35}$ and different ${\phi }_{\text{c}}$ . In determining $N_{\alpha }$ and $N_{\beta }$, a residue position i is classified in the following way: a helical state if $-90°<{\phi }_{\text{i}}<-30°$ and $-77°<{\psi }_{\text{i}}<-17°$, and a β sheet state if $-160°<{\phi }_{\text{i}}<-50°$ and $100°<{\psi }_{\text{i}}<160°$, where the ${\phi }_{\text{i}}$ and ${\psi }_{\text{i}}$ are the Ramachandran angles of residue i. $R_{\text{ee}}$ is the ${\text{C}}_{\alpha }$-${\text{C}}_{\alpha }$ distance between the terminal amino acids. To see this figure in color, go online.

Discussion

It was first realized on theoretical grounds that the native state of proteins should be stabilized by the presence of surrounding macromolecules, if these macromolecules are inert and simply occupy space (7). The reason is that the unfolded state, on account of its conformationally expanded character, will leave a smaller volume for the crowder molecules to occupy than the volume left by the more compact native conformation. As a result, the native state will be entropically favored relatively to the unfolded state, which should stabilize the protein. Indeed, there is wide support for at least a moderately stabilizing effect from experiments (8, 9, 10, 11, 12, 13, 14, 15, 16), and from theory (19, 20, 21, 22, 23, 24, 25, 26, 27).

However, as noted by Minton (53), the above theoretical argument can be applied not just to the native state but to any compact nonnative state, which will also be stabilized relative to more expanded conformations. Indeed, compaction of the unfolded state ensemble under crowding has been observed for several proteins (13,54,55), although an exception was recently found (56). Intrinsically disordered proteins also tend to become more compact under crowded conditions (57). For a strict two-state protein, the excluded volume effect is expected to be generally stabilizing even with a collapsed unfolded state because, in such an ideal system, even compact nonnative conformations will have energies much higher than the native state. Indeed, computer simulations of structure-based models for protein folding, which do not permit energetically favorable nonnative interactions within the protein, consistently lead to an enhanced stability along with a compaction of the unfolded state (24, 25, 26, 27).

The perspective provided by our model is partly different in that it takes into account attractive nonnative (intra-protein) interactions. We have found that such nonnative interactions during folding become increasingly prevalent with increasing crowder concentration, even when the native structure is left structurally unchanged. Moreover, these nonnative interactions can turn the excluded volume effect from being stabilizing to destabilizing, under some conditions (see Fig. 3). The destabilization we observe for ${\beta }_{35}$ occurs mainly at low temperatures and low crowder concentrations, through a combination of two factors. First, this protein has a relatively compact unfolded state. As a result, it is only weakly disfavored entropically by the crowders. Second, the presence of a compact, nonnative state with an energy that is low enough that its population can increase relative to the native state under crowded conditions. Importantly, the exclude volume-induced destabilization of ${\beta }_{35}$ occurs even though this protein exhibits an unfolding curve that is well described by a two-state equation.

The maximum population of the compact nonnative state in ${\beta }_{35}$ is around 1%–2% in the absence of crowders. Such low populations are unlikely to be detected by standard biophysical characterization techniques, such as circular dichroism. However, sparsely populated nonnative states have been detected for several small globular proteins (32,58, 59, 60), e.g., using nuclear magnetic resonance (NMR) spectroscopy (59), hydrogen-deuterium exchange (32), and other methods (60), and may be more common than previously thought (61). For example, using relaxation dispersion NMR, Neudecker et al. (58) showed that the Fyn SH3 domain with a five-stranded β-barrel native fold exhibits a compact near-native intermediate state with an $\approx$2% population stabilized by both native and nonnative interactions.

Although artificial polymer crowders, such as Ficoll or dextran, are typically stabilizing to proteins (summarized in recent reviews (62,63)), exceptions have been seen (11,16, 17, 18). For example, weak destabilization of ubiquitin (11) and apoazurin (18) were observed with Ficoll-70 as a crowder. Crowding-induced destabilization is often interpreted as evidence for soft attractive (or chemical) interactions between the unfolded protein and crowders. Because of the presence of an energetic component, soft attractions will lead to a crossover temperature below which crowding becomes destabilizing (35). Here we have shown that nonnative states during folding can lead to a destabilization of the native state, as well as a crossover temperature, without soft attractions. This idea could be tested through crowding experiments on proteins for which sparsely populated transient (or intermediate) states have already been characterized.

Both our model proteins exhibit a compaction of the unfolded state with decreasing temperature, i.e., as conditions increasingly promote folding, which is in line with data from single-molecule FRET (smFRET) and small-angle X-ray scattering (SAXS) experiments (64). Interestingly, Radford et al., using smFRET, additionally detected an expansion of the unfolded state of the α-helical protein Im9 at very low denaturant concentrations (65). This behavior mirrors the re-expansion at low T we observe for ${\alpha }_{35}$, which does not occur for ${\beta }_{35}$ (see Fig. 5). The stabilization of ${\alpha }_{35}$ at both high and low Ts results in a clear increase in $T_{\text{m}}$ with ${\phi }_{\text{c}}$. In contrast, because ${\beta }_{35}$ is stabilized at high T but destabilized at low T, the change in $T_{\text{m}}$ is very limited (at most $\approx$ 0.5%). Indeed, capturing small changes in stability may require stability measurements across a range of temperatures, as was pointed out previously (11,66).

We have also found that the crowding-induced change in folding free energy $\text{Δ}F$ in general includes both entropic and energetic components, i.e., both $\text{Δ}E$ and $\text{Δ}S$ change with crowder concentration. The possibility of energetically driven depletion forces in macromolecular association and folding was demonstrated by Harries et al. (38,39). It was shown that such forces can arise from favorable soft interactions between crowders (cosolutes) and solvent molecules, such that the crowders are preferentially excluded from the macromolecular surface. The underlying mechanism we observe here is entirely different. In our work, the crowding-induced change in $\text{Δ}F$ is not entirely entropic, despite purely repulsive crowders, because the crowders re-model the unfolded state ensemble and thereby reduce the energy gap $\left|\text{Δ}E\right|$ between native and unfolded states. Interestingly, there is potentially an interplay between these two mechanisms because the preferential hydration of crowder particles may affect their ability to reduce the size of the unfolded state ensemble of the protein.

Conclusions

In summary, we have used a coarse-grained sequence-based model to study the folding and stability of two different sequences in the presence of excluded volume crowders over a range of sizes and concentrations. We find that during folding, nonnative interactions are generally promoted by the crowders. Moreover, under low-T, low-${\phi }_{\text{c}}$ conditions, the excluded volume effect of crowders can lead to a destabilization of the protein even when its native structure remains unchanged and the population of nonnative conformations is relatively small, as observed in our model. Such destabilization may, however, not be apparent in observables reporting on the total content of secondary structure or overall chain size. The results suggested by our model may be tested experimentally; for example, using an artificial crowder molecule, such as Ficoll, on proteins for which a low population of compact nonnative conformations have been detected (58,67,68).

Author contributions

S.W. designed the study. S.W. and S.B. performed research, carried out analysis, and wrote the manuscript.

Editor: Robert Best.