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Crowding-induced Elongated Conformation of Urea-unfolded Apoazurin: Investigating the Role of Crowder Shape In Silico

Here, we show by solution nuclear magnetic resonance measurements that the urea-unfolded protein apoazurin becomes elongated when the synthetic crowding agent dextran 20 is present, in contrast to the prediction from the macromolecular crowding effect based on the argument of volume exclusion. To explore the complex interactions beyond volume exclusion, we employed coarse-grained molecular dynamics simulations to explore the conformational ensemble of apoazurin in a box of monodisperse crowders under strong chemically denaturing conditions. The elongated conformation of unfolded apoazurin appears to result from the interplay of the effective attraction between the protein and crowders and the shape of the crowders. With a volume-conserving crowder model, we show that the crowder shape provides an anisotropic direction of the depletion force, in which a bundle of surrounding rod-like crowders stabilize an elongated conformation of unfolded apoazurin in the presence of effective attraction between the protein and crowders.


Abstract
Here, we show by solution nuclear magnetic resonance measurements that the urea-unfolded protein apoazurin becomes elongated when the synthetic crowding agent dextran 20 is present, in contrast to the prediction from the macromolecular crowding effect based on the argument of volume exclusion. To explore the complex interactions beyond volume exclusion, we employed coarse-grained molecular dynamics simulations to explore the conformational ensemble of apoazurin in a box of monodisperse crowders under strong chemically denaturing conditions. The elongated conformation of unfolded apoazurin appears to result from the interplay of the effective attraction between the protein and crowders and the shape of the crowders. With a volumeconserving crowder model, we show that the crowder shape provides an anisotropic direction of the depletion force, in which a bundle of surrounding rod-like crowders stabilize an elongated conformation of unfolded apoazurin in the presence of effective attraction between the protein and crowders.

I. INTRODUCTION
The cellular environment is highly crowded with different types of macromolecules that can occupy volumes up to 40 % of the total volume inside cells [1][2][3] . To properly understand how proteins fold and function in vivo, it is necessary to consider the volume exclusion from surrounding macromolecules, known as the macromolecular crowding effect 4 , which influences the structures and dynamics of a biopolymer in this restricted space. To test the crowding effect in vitro, sugar-based, branched polymers such as Ficoll and dextran 5 are typically utilized as synthetic crowder agents to investigate the macromolecular crowding effect because they are considered to be chemically inactive agents in solution. Under the assumption that the dominant interaction between a crowding agent and a biopolymer is steric repulsion, the macromolecular crowding is shown to stabilize the conformation of compact biopolymers as shown in proteins [6][7][8] , nucleic acids 9 , and chromatins 10 .
However, here we made the puzzling discovery by solution nuclear magnetic resonance (NMR) spectroscopy that our model protein, apoazurin, unfolded by a high urea concentration adopts more elongated conformations in the presence of dextran 20 than without a crowding agent (bulk). This is a deviation from a previous study from the Wittung-Stafshede group where they have shown that apoazurin increases in stability with synthetic crowders 11 . The thermodynamic stability of the two-state folding protein apoazurin increases with the increased volume fraction of crowders, and this effect was independent of the crowder size when various dextran sizes were analyzed, suggesting that a cylindrical representation from Minton's model 12 is appropriate to model dextran that interacts with apoazurin through steric repulsions 13 .
To find an explanation to this counterintuitive result, we considered in silico coarse-grained molecular dynamics simulations, allowing more complex modeling of crowders than a simple hard sphere model for volume exclusion. The question central to all computational modeling is how to choose an appropriate level of the coarse graining of crowders that captures essential features without a prohibitive computational cost. All-atom molecular dynamics simulations have shown an unprecedented resolution of protein and protein crowders that allows investigation of subtle chemical interactions affecting protein stability and kinetics [14][15] . These simulations demand massive computational resources and still require advanced sampling techniques to explore a wide range of phase space in a system for the computation of thermodynamic properties [16][17] . Coarsegrained models, although losing atomistic details, allow for greater sampling than in all-atom models and provide key polymeric insight into the distribution of protein conformations and the folding energy landscape under the macromolecular crowding effect [18][19][20][21] .
To investigate the structural ensemble of apoazurin in 6 M urea in a periodic box of dextran crowders with coarse-grained molecular simulations, we must consider the appropriate dextran model. It is challenging to determine the geometry and property of dextran in a coarse-grained model because of its branched topology and polydisperse spatial distribution in solution 22 . Early experiments based on filtration have characterized that the shapes of dextran crowders are asymmetrical by showing that dextrans pass through a pore much smaller than their presumed diameters 5 . Sasahara et al have further modeled the shape of Dextrans as rods to fit the data from experimental measurement of the protein stability in dextrans with circular dichroism 12 . Our previous work has adopted a spherocylindral model for dextrans to explain the features of folding from various protein mutants 23 . Nevertheless, these studies have assumed dextrans to interact with a protein through hard-core interactions. Interestingly, studies by Jiao et al. 24 and, recently, by Ebbinghaus et al. 25 showed that dextran stabilizes a protein through enthalpic interactions similar to its building block glucose, instead of stabilizing through volume exclusion. A suitable choice of dextran models again has become elusive from the past literature.
A mixture of dextran with destabilizing urea complicates our analysis of apoazuring folding. The net opposing impact from dextran and urea on protein stability can depend on the composition and become nonadditive 26 . Given only the hydrodynamic radius of dextran in 6.8 M urea from the NMR experiments, our modeling approach was to vary the shape of a volumeconserving crowder model for dextran while changing the effective attraction between ureaunfolded apoazurin and its surrounding dextran crowders.
We first showed that a naïve crowder model of hard-core repulsion between dextrans and apoazurin fails to capture the experimental observation. The addition of an effective attraction between dextrans and apoazurin is necessary to shift the protein ensemble towards elongated conformations. These effective attractions are unlikely the ones such as electrostatics and hydrogen bonding interactions found in protein crowders [27][28] . We are attracted to another possible explanation that compressible (physically soft) crowders 29 or chemically soft repulsion 30 produce effective attractions between a protein and a crowder because of the increased depletion between soft repulsive depletants. It is noteworthy that the soft repulsion from Zaccarelli's work 29 is due to deformation of polymers, which is an entropic effect. In Harries's work [30][31] , however, the soft repulsion from cosolute-solvent mixtures, resulting from changes in the hydrogen bonding in solvent near the first hydration shell, is represented by a step-like function in the interacting potential -an enthalpic effect. Our effective attractive potential can rise from either or both factors.
The geometry of an elongated rod model for dextrans further shifts the ensemble of apoazurin towards elongated conformations (favoring urea-unfolded conformations). The attraction between urea-unfolded apoazurin and dextrans facilitates dextrans to form aligned bundled conformations. The shape of the dextran breaks the symmetry of the depletion force, thus favoring an elongated geometry for the unfolded apoazurin. With coarse-grained modeling, we provide a plausible explanation to the puzzling discovery from the NMR experiments.

II.1 Apoazurin
We here used the apo-form Pseudomonas aeruginosa azurin, termed apoazurin, to study the excluded-volume effects of dextran 20 on the urea-denatured protein in vitro and in silico. It is a bacterial protein that binds a copper ion that can undergo oxidation-reduction as part of electron transfer chains in bacteria. Apoazurin has 128 residues, and the folded state adopts a β-sheet Greekkey fold with a disulfide bond (Cys3-Cys26) 32 (see Figure 1). In vitro studies on apoazurin have advanced our knowledge on the folding mechanism of sandwich-like proteins 33 . Apoazurin has provided an excellent template to compare experimental measurement with the predictions from protein folding theories 34 and simulations 35 . The folded structure is illustrated in a cartoon representation and a side-chain Cα representation. These protein illustrations were produced using VMD 36 . The algorithm DSSP 37 was used to assign the secondary structures. The residues that form the disulfide bond are shown in yellow. (b) Illustrations of two models for dextran 20 with the same volume: a spherical model (DX20-S), and a spherocylinder model with an aspect ratio of 4 (DX20-AR4). As reference, the folded structure of apoazurin is shown. The reduced unit of length is s.

II.2 NMR measurements
All NMR experiments were conducted at a Bruker Avance III HD 850 MHz spectrometer equipped with a z-gradient cryogenic probe using20 mM NaP, pH 7.0 and 5 % D2O (v/v) as sample buffer. All one-dimensional data sets were processed and analysed by Topspin 3.2 whereas twodimensional data sets were processed with NMRPipe 38 and analysed in NMRView 39 . Water suppression was achieved by presaturation and WATERGATE.

II.2.1 Diffusion experiments.
NMR diffusion spectra were recorded using pulsed field bipolar gradient stimulated echo experiments 40 at T = 298 K. For each diffusion profile, 21 different gradient strengths G were used for 6 ms along the z axis followed by a 100 ms recovery delay. The diffusion of apoazurin was allowed to proceed for 100 ms (c d20 = 0 g/l, c d20 = 100 g/l, c d20 = 150 g/l, c d20 = 180 g/l, c d20 = 200 g/l and c urea = 0 M; c d20 = 0 g/l, c d20 = 100 g/l and c urea = 6.8 M), 150 ms (c d20 = 150 g/l and c urea = 6.8 M) or 200 ms (c d20 = 180 g/l and c urea = 6.8 M). The calibration of G was performed by a standard protocol 41 . For error estimation, four different gradient strengths were repeated (relative gradient strengths of 1, 10, 40 and 70 %). The measured 1 H NMR spectra were integrated within the aliphatic signal region I ( 0.5... 1.5 ppm) and fitted to Equation (1): (1) where g is the gyromagnetic ratio, d is the gradient length, D is the diffusion time and D is the calculated diffusion coefficient 40 .

II.3 Coarse-grained models
We have performed coarse-grained molecular simulations of apoazurin under several crowding conditions and under 0 M and 6 M of urea denaturant. 6 M urea is a concentration that guarantees the denaturation of the protein at room temperature.

II.3.1
Coarse-grained apoazurin model. Our model protein apoazurin (PDB ID: 1E65; Figure 1) is studied employing a coarse-grained model. We coarse-grained each residue into a side-chain and Cα bead with exception to glycine, which does not contain a side-chain 44 . The Cα bead is located at the position of the α-carbon, and the side-chain bead is located at the center of mass of the side-chain atoms. We have employed a structure-based Hamiltonian for our protein model, i.e., the energy landscape of the protein follows the principle of minimal frustration [45][46][47] , by following previous work 44,48 . The dimensions of the protein are expressed in units of s (s = 3.8 Å), which is the average distance between two consecutive Cα beads in a protein. This reduced unit of length is also used for the dimensions of the systems with crowders.

II.3.2 Coarse
-grained Dextran 20 model. We employed twelve volume-conserving models on dextran 20, a glucose-based polydisperse polymer of 20 kDa molar mass, to investigate the effects of crowder shapes and effective attraction with a protein (Figure 2). The radius of rH d20 = 32.4 Å was taken from the current experimental measurement. The rod-like shapes used by the models are spherocylinders 21 with various aspect ratios (AR). A spherocylindral model is created by connecting several overlapping spheres where the center-to-center distance between adjacent spheres is the radius of a sphere. The AR of a spherocylinder is defined as the ratio L/D, where L is the distance between the centers of the two spheres located at the ends of the spherocylinder, and D is the diameter of a sphere. For hard-core crowder models, the AR varies from 0 (DX20-S), to 1 (DX20-AR1), 2 (DX20-AR2), 4 (DX20-AR4), 8 (DX20-AR8), and 16 (DX20-AR16), while keeping the volume the same as DX20-S with the radius of rH d20 = 32.4 Å which is equivalent to 8.3 s. This value corresponds to the hydrodynamics radius of dextran 20 from our in vitro experiment, and it is approximately twice the radius of gyration Rg of the folded state (Rg N ≈ 4 σ).
The crowders with DX20-series interact with a protein through repulsion, and the crowders with DX20A-series interact with the protein through attractive interactions. Table 1 shows the number of beads and radii of those beads needed to create the various ARs.
DX20A-series are crowders with attractive interactions with a protein.
An apoazurin protein is placed in the center of the box of crowders of the same type. The number of the crowders in a box depends on the volume fraction ϕc under study. The volume fractions studied are ϕc = 0 % (or bulk), 5 %, 10 %, and 20 %.

II.3.3. The energy function of apoazurin under urea interference in a periodic cubic box of crowders.
The Hamiltonian of the system (Es) is the sum of the potential energy of the protein (Ep), the interaction between protein and crowders (Epc), and the interaction between crowders (Ecc),  i.e., Es = Ep+Ecc+Epc. Ep adopts a variant of structure-based potentials where only the contacts found in the native states are attractive according to the amino acid types, while the non-native contacts are repulsive. We have incorporated the chemical interference by adopting the "Urea-SC-DIH" model from a previous study 48 where the urea solvent-mediated interactions were included in the interactions between side-chain beads and the dihedral-potential that corresponds to four successive Cα beads. The expression of Ep can be found in the supplementary information.
For Ecc the interactions between crowders are repulsive and inversely related to the distance between the two crowders to the 12 th power [Equation (S3)]. We set the interactions between protein and crowders, Epc, to be repulsive or attractive. The expression of repulsion between hard spheres of crowders labeled as DX20-series in Table 1 follows the same expression as Ecc. The expression of non-specific attraction between a crowder labeled as DX20A-series in Table 1 and a bead from the apoazurin model follows a Lennard-Jones potential with the depth well of εs. The relative strength of the attractive interaction is denoted by λ = εs/ε, where = 0.6 kcal/mol is the solvent-mediated interaction between Thr and Thr residues at 0 M. We used λ = 0.83 and 2.50 for this study.

II.4. Molecular simulations
We used an in-house version of the Amber 10 molecular dynamics package 49 to integrate the Langevin equation of motion in the low friction limit 50 . The integration time step was set to 10 -4 # , where # = %( ) ⁄ ), is the mass of a Cα bead, is the solvent-mediated interaction, and is the van der Waals radius of a Cα bead. The replica exchange method (REM) [51][52]  where = 1 D ⁄ , D is the Boltzmann constant, T is the temperature, and (r) is the potential energy of the system. Twenty replicas were employed where the temperature range used in a REM simulation varies for each condition. For each replica, more than 100 000 statistically significant conformations were collected.
For each condition of the simulations with crowders, there are ten sets of randomly assigned crowder configurations in a periodic boundary cubic box (PBC) as initial conditions. The size of PBC is 120×120×120 σ 3 for the crowders with AR from 0 to 8, and it is 185×185×185 σ 3 for the crowders with AR = 16. The canonical simulations were performed at a temperature that corresponds to the in silico room temperature Tu. Tu is obtained by transferring the difference between the melting temperature (Tm) and the room temperature (298 K) for an in vitro experiment to the in silico computational study as suggested in a prior study 35 . The in silico room temperature is given by Where Tf is the in silico folding temperature. In our case the in silico room temperature at 0 M urea is kBTu/ε = 1.14.

II.4.1 Geometrical properties of a protein.
We measured the geometry of apoazurin by using the radius of gyration (Rg) and asphericity parameter (Δ) 53 [see Equation (S7)]. Rg measures the size of the protein, and Δ measures the degree of anisotropy of the structure of the protein. Δ has the following range: 0 ≤ Δ ≤ 1, where a perfect sphere has Δ = 0. When Δ approaches 1, it is an indication that the protein is fully extended.

II.4.2 The geometrical properties of a crowder void.
The crowder void is defined as the space between the crowders in which the protein lies. To find this void, we partition the entire simulation box through Voronoi tessellation 54 . From the tessellation, a 3D polygon is created around each bead (both crowders and protein residues). We select the crowder beads whose polygon is in contact with the polygon of the protein beads, and the positions of these selected crowder beads are used to define the vertices of the crowder void. We characterize the crowder void by computing two geometrical descriptors: the radius of gyration Rg and the asphericity parameter Δ.

III.1.1. Structural changes
It has been established that high-resolution NMR spectroscopy enables the profound molecular characterization of the structural 55 and dynamic properties 56 of a protein ensemble as it is present in solution. We have made use of that by applying two-dimensional heteronuclear 1 H-15 N NMR experiments on apoazurin present under native and denatured conditions in a macromolecular crowding environment. Moreover, we have used Pulse-field gradient (PFG)-NMR techniques to decode the potential impact of macromolecular crowding on the hydrodynamic dimension of apoazurin present under both folded and denatured conditions.
The addition of dextran 20 to apoazurin up to a concentration of c d20 = 200 g/l neither changes the chemical shifts nor the signal intensities of apoazurin as probed by one-dimensional proton ( Figure S1) as well as by two-dimensional 1 H-15 N NMR HSQC spectroscopy ( Figure S2). This indicates that the addition of dextran 20 does not modify the structural properties of folded apoazurin at secondary and tertiary levels.
The maximum change in 1 H, 15 N weighted chemical shifts, Δωmax, is 0.025 ppm with an average Δω mean of (0.012±0.004) ppm. Note that chemical shifts for aliphatic protons of apoazurin are highly conserved as well by comparing diluted and c d20 = 200 g/l conditions for the folded state. Next, dextran 20 of up to c d20 = 180 g/l was additionally supplemented to apoazurin that has been chemically unfolded using c = 6.8 M of urea ( Figure S3 and S4).
Again, changes in 1 H, 15 N weighted chemical shifts are small, with Δωmax = 0.025 ppm and Δω mean = (0.009±0.006) ppm. Chemical shifts for aliphatic protons are conserved in the presence of c urea = 6.8 M by comparing diluted and c d20 = 180 g/l conditions as observed for folded apoazurin.
Taken together, 200 g/l of dextran 20 does not change the structural integrity of the folded state and 180 g/l of dextran 20 does not perturb the chemical shift pattern seen for chemically unfolded apoazurin. Consequently, we conclude that there is no significant solute-protein interaction between dextran 20 and apoazurin.

III.1.2. Molecular dimension
NMR diffusion methodology has been successfully employed for the profound characterization of the molecular dimensions of both folded and highly denatured proteins in solution 42 . The initial idea with these experiments was to probe the expected (based on excluded volume theory) compaction of unfolded apoazurin when macromolecular crowding agents have been added.  show the hydrodynamic radii that are theoretically expected for a globular folded protein (colored in red) and a denatured polypeptide chain of that size (colored in blue) according to 42 . All the data were recorded at T = 298 K. Error bars have been included (and seen). The calculation has been done in the same way as for we did Figure 3.  [Dex20] / % = 6.8 M. Again, D decreases with an increasing concentration of dextran 20 from D = (5.22±0.08)e-11 m 2 s -1 (c d20 = 0 g/l) to D = (0.62±0.06)e-11 m 2 s -1 (c d20 = 180 g/l). However, the decrease in D was not compensated by the increase in h as found under native conditions. Instead, surprisingly, the hydrodynamic radius for denatured apoazurin increases from rH = 35.8±0.6 Å (c d20 = 0 g/l) to rH = 46.6±2.5 Å (c d20 = 180 g/l) ( Figure 4). Notably, rH, as determined here for denatured apoazurin under dilute conditions, agrees with the molecular dimensions expected for a denatured polypeptide chain of apoazurin's size 42 , rH(N aa =128)=(35±18 Å). Because excluded volume theory predicts that unfolded proteins become compacted in the presence of macromolecular crowding, more factors must be at play. To find a molecular-mechanistic explanation to this in vitro observation, we turned to in silico computations.

III.2. In silico analysis of apoazurin unfolding
To compare our simulation results to the experimental measurements of apoazurin's hydrodynamic radii rH under different conditions, we used two order parameters: radius of gyration Rg and asphericity ∆. Rg is not an exact comparison to rH because the shape of the object also matters in determining rH. For a perfect sphere, rH can be approximated by Rg; however, as the protein becomes less spherical (or an increase in ∆), rH and Rg diverge (see Figure S5). rH/Rg can exceed 3 when ∆ approaches 1 (a rod-like shape). This is due to the diffusion of spherical molecules being different than that of other elongated molecules. Therefore, rH is in fact measuring a combination of the size and shape of a molecule.

III.2.1. Structural changes for the repulsive crowder model and chemical denaturation
We have investigated the ensemble conformations of apoazurin in various crowding contents at a chemically denatured state at 6 M urea and at an in silico room temperature of kBTu/ε = 1.14 (see Methods) with coarse-grained molecular simulations.
First, we have assumed that crowders and apoazurin interact through a repulsive potential that accounts for the excluded volume effects at ϕc = 5 %, 10 %, and 20 %. While conserving the volume of the crowders, we tested the shape effect by varying the aspect ratio of the crowders on apoazurin unfolding. Figure 5(a) shows the average asphericity (Δ) normalized by the asphericity (ΔB) under the bulk condition for DX20-S, DX20-AR4 and DX20-AR8 against ϕc. Δ/ΔB are all less than 1 over a wide range of ϕc, implying that all dextran 20 models with repulsive interactions with apoazurin do not contribute to the elongation of apoazurin. For most of the conditions, the macromolecular crowding effect leads to compaction of apoazurin with Rg/Rg,B less than 1 [ Figure  5(b)]. The crowders with a rod-like geometry show greater compaction than the spherical crowder over all ϕc because rod-like crowders render a larger covolume than spherical crowders. Because the effects found upon including only volume exclusion cannot explain the experimental results of dextran 20-induced elongation of unfolded apoazurin, we next investigated a dextran 20 model that also provides effective attractive interactions.

III.2.2. Structural changes of apoazurin with which crowders interact attractively and under chemical denaturation
We hypothesize the existence of non-specific attractive interactions between the unfolded protein and crowders. For this, we tested two types of non-specific attractive interactions with λ = 0.83 and λ = 2.50 (see methods). λ measures the attraction between a protein and crowder relative to the attraction between two Thr residues at 0 M urea. λ = 0.83 represents a weak attraction, and λ = 2.50 represents a strong attraction.
We noticed that a box of spherical crowders with a weak attraction (DX20A-S at λ = 0.83) cannot produce elongated ensembles of apoazurin in silico at room temperature with kBTu/ε = 1.14 in Figure 6. The distribution of D for apoazurin, centered at D ≈ 0.16, is significantly narrower than the bulk condition over a wide range of ϕc. We note that, with a strong attraction at λ = 2.5, there was a slight increase in Δ/ΔB or Rg/Rg,B along ϕc ( Figure S6). However, Rg/Rg,B is still less than 1 when the aspect ratio of the crowder is less than 4.
This result leads us to a working hypothesis that, in addition to weak attractions, the crowders must be rod-like to stabilize an extended unfolded conformation of apoazurin. We showed that, in the system of weakly attractive crowders with a rod-like shape (DX20A-AR4), the distribution of D for the protein apoazurin under the chemically denatured condition becomes broad and comparable to that under the bulk condition in Figure 6. Interestingly, the distribution of D for apoazurin shifts toward D = 1 and forms a large peak at D ≈ 0.84 at ϕc = 5 % for DX20A-AR8 (Figure 7). This finding implies that more elongated conformations of apoazurin in chemically unfolded states are sampled in a system with the crowder model DX20A-AR8 than with the crowder model DX20A-AR4. However, the height and position of the peak shifts toward D = 0 with increasing ϕc. At a high ϕc of 20 %, apoazurin adopts a compact configuration again ( Figure 8). This feature was not observed for spherical crowders (DX20A-S) with a weakly attractive interaction with a protein at λ = 0.83 ( Figure S6).   These ratios are shown as a function of the aspect ratio of crowders, AR. The concentration of urea is 6 M, and the temperature is kBTu/ε = 1.14 for all conditions. Error bars are included.

III.2.3. Both the rod-like shape of a crowder and attractive interaction with a protein contribute to the structural changes of apoazurin under chemically denatured conditions.
We evaluated Δ/ΔB as well as Rg/Rg,B of chemically denatured apoazurin at varying aspect ratios (AR) of crowders for a wide range of ϕc in Figure 8. Overall, there is an increase in Δ/ΔB when AR increases [ Figure 8(a)]. Δ/ΔB exceeds 1 at AR greater than 8. Interestingly, Rg/Rg,B also exceeds 1 at the most extreme case of AR = 16 [ Figure 8(b)] at ϕc = 5 %. However, we noted that the increasing trends in Δ/ΔB or Rg/Rg,B are not uniform with ϕcs. When AR is less than 4, the protein has the largest Δ and Rg values at ϕc = 20 % compared with the other ϕc conditions. When AR is greater than 4, the largest Δ and Rg values occur at ϕc = 5 %.
Such a crossover behavior at AR = 4 is contributed by the shape of the protein-crowder assembly (Figure 9). The unfolded ensemble of apoazurin conformations adopts a shape according to the available void formed by surrounding crowders [ Figure 9(a)]. A rod-like crowder with nonspecific attractive interactions provides an elongated surface where several regions of the protein can stick [ Figure 9(b)]. This particular geometry of rod-like crowder favors the formation of extended unfolded conformations of apoazurin. As several crowders stick to a protein, they tend to align themselves and restrict the available space for the protein. It appears that, at high ϕc, there is insufficient space to accommodate the volume of a large assembly of protein crowders. Therefore, the uptick of Δ/ΔB or Rg/Rg,B at high ϕc is weaker than that at low ϕc.

III.2.3. The geometry of crowder-induced voids is modulated by the protein-crowder interactions
Next, we investigated the geometry of voids, in terms of shape and size, formed by crowders of the various models modulated by the protein-crowder interactions. The voids are the depletion zones due to non-overlapping volume exclusion between a protein and crowders. The distribution of the asphericity of the void D void is virtually unchanged between spherical and rod-like crowders without attractive interactions [ Figure 10(a)]. Additionally, as ϕc increases, the peaks of the distributions are unperturbed. These results indicate that, without attractive interactions, the variation in the shape effect from rod-like crowders has been averaged in the ensemble. However, when attractive interactions are added to the crowders, there is a clear shift rightward (towards higher asphericity) of the peak. The spread of the distributions increases with AR when the shape of a crowder becomes elongated [ Figure 10(b)]. These effects are greatest for ϕc = 5 %.
We show that the distribution of the size of a void (Rg void ) depends on the volume fraction of crowders, the shape of a crowder, and interaction between the protein and crowder. To make a fair comparison between spherical and rod-like crowders, we plotted the distribution of (Rg void -R)/σ in Figure 10(c) and Figure 10(d), where R is the radius of a sphere in a crowder model. The overall size of (Rg void -R)/σ is smaller for rod-like crowders (DX20-AR8) than spherical crowders (DX20-S) by roughly a third at each ϕc when a protein and crowders interact through hard-core interaction [ Figure 10(c)]. However, with a weak attraction between a protein and crowders (λ = 0.83), (Rg void -R)/σ is reduced by half across all crowder types [ Figure 10(d)] compared to those in Figure 10(c). The peaks in the distribution of (Rg void -R)/σ for rod-like crowders (DX20A-AR8) at all ϕc nearly overlap at the same position around 10. This feature is consistent with the finding of a protein-crowder assembly in Figure 9.

IV. DISCUSSION
This in silico investigation was initiated by the anomalous observation from NMR experiments that the addition of 180 g/l of dextran 20 alters the ensemble of urea-unfolded apoazurin in the direction of an extended conformation. Initially, this observation contrasts the physical intuition of the excluded volume theory that proteins generally favor a collapsed, compact conformation in the presence of macromolecular crowders 57 . This intuition is built on a hard-sphere crowder (or depletant) that exerts entropic depletion interactions, or the macromolecular crowding effect 4 , on the protein of interest, that effectively makes the unfolded expanded states of a protein entropically unfavorable, thus favoring a collapsed, compact conformation over an unfolded, extended one 18 . The strength of the macromolecular crowding effect grows with the volume fraction of crowders and is often strongest when the radius of the crowders is smaller than the Rg of the test protein 58 . However, recent studies have shown that the sequence-dependent properties of a protein from protein crowders can offset the volume exclusion of protein crowders 24,59 . The sequence dependent interactions between a protein and protein crowder include chemically soft interactions such as hydrogen bonding, electrostatic interactions, or dispersion. In some cases, these chemical attractions that possibly stick a protein to the surface of crowders further form quinary structures and destabilize the native folded state 28 60-61 .
Chemically inert crowders such as Ficoll 70, PEG, dextrans do not share the same chemical surfaces as protein crowders. However, a spherocylinder model for dextrans that lacks atomistic details and interacts with a protein through hard-core potentials fails to explain the anomaly from experiments, unless we consider the effective attractions between a protein and dextrans. A recent study by Zhou et al. also suggests that synthetic crowders may attain strong attractions with intrinsically disordered proteins (IDPs) such that IDP polypeptides populate expanded configurations under crowded conditions 62 . The underlying reasons may include soft repulsion between the protein and synthetic crowders, deepening the depletion in the potential of mean force [29][30] . This physically soft aspect of an interaction (in contrast to chemically soft as mentioned above) is due to the compressibility of soft-matter materials that does not exist in non-overlapping hard spheres. It is possible that, in the mixture of urea and dextran, dextran becomes physically soft. This speculation is supported by a recent essay from Ferreira et al. 26 that, in the mixture of osmolyte and a macromolecular crowder, the solvent properties may not always be additive, depending on the composition of the mixture. We speculate that, at 6 M urea, not only the solventmediated interactions between amino acids in the protein are weakened, leading to unfolding, but dextrans also become softened to trap unfolded apoazurin conformations. This scenario can be imagined as spaghetti trapped by a fork. This "fork-spaghetti" mechanism may be facilitated by the non-uniform length of the branches of dextran 63 . The geometric feature of branched dextrans is at odds with the dimensions of a linear dextran model 12 , especially when the molecular weight increases 22 . Noteworthily, the behavior of branched polymers (dextran) is distinct from those of linear polymers (PEG) with a similar molecular weight shown by Soranno et al. 64 who investigates the macromolecular crowding effect on the compaction of intrinsically disorder peptides (IDPs). When large PEG polymers reach a semi dilute region, they unwind and overlap one another at a moderate volume fraction of 10%. The monomers of overlapping PEG polymers still exert volume exclusion effect on the compaction of intrinsically disorder peptides (IDPs). However, the strength of volume exclusion from PEG monomers are not as strong as non-overlapping PEG polymers as predicted from the scaled particle theory 57 .
Notably, we showed that the effective attraction alone is not sufficient to bias the ensemble of unfolded apoazurin towards an elongated one as the shape of the crowder must be cylindrical with a sufficiently large aspect ratio. With spherical crowders containing either weak or strong effective attraction, the enhanced depletion interactions still favor a compact unfolded apoazurin configuration [ Figure 8(b)]. This contrasts the findings from Zhou's work in which peptide model samples with both extended and compact conformations interacted strongly with spherical crowders 62 . We speculate that the biphasic distribution might change if the length of the simulation is increased or if a more efficient sampling scheme is introduced. Because both the aspherical shape of a crowder and its effective attraction with a protein increase the correlation time in the simulations, we ensured that our conclusions are independent of the sampling scheme in our simulations. Our results at a volume fraction of 5 % for the DX20A-AR8 model at λ = 0.83 do not depend on the initial unfolded conformation of apoazurin. The temporal evolution of D is shown in Figure 11 at the early stage of two canonical simulations: one starting from a compact unfolded conformation and the other from an extended unfolded conformation. In the first case, D fluctuates and rearranges initially, and then it increases as time increases. In the second case, the conformation becomes compact from a stretched one, and then it rearranges to a have a less extended shape. Thus, the initial conformation is not the reason why the simulations favor the formation of extended conformations with crowders and non-specific attractive interactions. Figure 11. Initial stage of two canonical simulations for the model DX20A-AR8 at kBTu/ε = 1.14: one starting from (a) a compact configuration and the other from (b) an extended configuration. The snapshots of the protein depict the temporal structural changes.
To explain our in silico results, the elongated shape of a crowder must break the symmetry of the isotropic direction of the depletion force, creating an anisotropic force that sandwiches the backbone of the protein. We found that the elongated conformation of apoazurin is stabilized by a bundle of rod-like crowders in the presence of effective attraction. The rod-like crowders are still free to rotate, and there is no evidence of an isotropic-nematic transition in our system. In the absence of effective attraction, the repulsive cylinders do not self-assemble into nematic states. To reach a nematic state for rod-like crowders with an aspect ratio of 5, ϕc typically must exceed 40 % 65 . In fact, hard spherocylinders contribute to greater compactness of apoazurin because of an increased covolume compared with a volume-conserving hard sphere 4 . The difference here is the effective attraction of apoazurin and several rod-like crowders that collectively form a bundle. The bundles formed around elongated apoazurin maximize the volume accessible to the other crowders (not in a bundle), thus maximizing the total entropy of the system. We showed that elongated apoazurin appears at low ϕc = 5 %, but not at high ϕc = 20 %; the latter was observed in in vitro experiments. This difference may be due to the increase in excluded volume interactions between hard spherocylinders instead of the spherocylinder-apoazurin interactions, as the volume fraction of spherocylinders increases. To induce the formation of significant bundles at high ϕc, the concentration of apoazurin in a binary mixture of spherocylinder and apoazurin should be increased.

V. CONCLUSION
New understanding in science generally begins when theoretical expectations and new experimental findings do not agree. Here, the intriguing NMR discovery showing elongated ureaunfolded states in the presence of dextran goes against intuition from conventional macromolecular crowding theory and previous experimental findings. With very little clues from experimental data, we have employed coarse-grained modeling to investigate the underlying mechanism. Even though we cannot definitively exclude other possible explanations for the trends found in the experiments, our work denotes the importance of both crowder shapes and solventmediated interactions in manipulating the geometry of a protein. These two properties are pivotal to understand the full complexity of proteins in the cytoplasm. Furthermore, this newfound understanding can potentially be developed into a design principle for probes to accurately measure crowding volume fraction, shapes, and interactions locally in a subcellular environment.

Supporting Information
Hamiltonian of the coarse-grained protein model in the presence of urea, NMR measurements ( Figure S1-S4), definition of the asphericity ∆, ratio of rH and Rg as a function of ∆ ( Figure S5), changes in the geometry of the protein as a function of volume fraction of crowders ( Figure S6), and protein stability at different concentrations of urea ( Figure S7) are presented in the supporting information (SI).

The Hamiltonian of a protein at varying concentration of urea
We adopted the "Urea-SC-DIH" model from a previous study 1 where solvent-mediated interactions ε α ij for a particular concentration of urea (α) were used in the native side-chain interactions between the side-chain beads i and j and in the dihedral angle formed by four successive Cα beads. The potential energy of the protein (Ep) is the sum of a structural term (Estructural-urea) and several non-bonded terms. The list of native pair contacts between the ith and jth side-chain beads was established by the CSU software 2 .

S2
The structural potential term (Estructural-urea) includes a bond-length potential, bond-angle potential, dihedral-angle potential modulated by the concentration of urea, and a chiral potential term that considers the L-form of every side-chain 3 . The equilibrium values, used to estimate the potential terms, were taken from the crystal structure of apoazurin (PDB ID: 1E65) 4 . The disulfide bond between the side-chain beads of Cys3 and Cys26 was modeled by a bond-length potential.
The non-bonded potential term considers the native and non-native interactions among residues.
The potential term for the native pairs consists of the side-chain-side-chain potential term ( ./

23+'&,)
) and the backbone hydrogen-bonding potential term ( ./ The influence of urea at a certain concentration was introduced in the Hamiltonian by using two terms as indicated by the protocol "Urea-SC-DIH " from our prior study 1 : the side-chain-side-chain potential term and the dihedral-angle potential term.

23+'&,)
) for a specific concentration of urea (α) is given by a Lennard-Jones potential in Equation (S4) where the solvent-mediated interactions ( ./ M ) were derived from all-atom molecular dynamics simulations using a Boltzmann inversion approach 6 . α represents a solution with a concentration of urea of 0M, 6M, or 8M. In the absence of urea (i.e. 0M), the solventmediated interactions ( ./ M ) for the non-bonded interactions were obtained from the Betancourt-Thirumalai potential 7 , in which case the solvent-mediated interaction between Thr and Thr residues was used to obtain the solvent-mediated interactions of other residue pairs. The distance between the side-chain beads from two different residues is represented by ./ , and the equilibrium position between two beads is given by The interaction between crowder and crowder (Ecc) of the spherical model of dextran 20 (DX20-S or DX20A-S) is given by a repulsive term as described by Equation (S3). Ecc of a spherocylinder crowder (DX20-series or DX20A-series) is composed of the potential energy of crowders and the repulsive interaction between crowders. The potential term for the interaction between crowder and crowder is also described by Equation (S3), and the intra-potential energy of a spherocylinder crowder consists of a bondlength potential, bond-angle potential, and a dihedral-angle potential, which equilibrium values were set to the radius a crowder bead, 180°, and 0, respectively. Figure S1. High-field chemical shifts for one-dimensional 1 H NMR spectra for native apoazurin in absence (colored in black) and in presence of c d20 = 200 g/l (colored in red). Both spectra were recorded at T = 298 K and B0 = 20 T.  Figure S2. Two-dimensional 1 H-15 N NMR HSQC spectra for native apoazurin at different concentrations c of dextran 20 (c d20 = 0 g/l colored in black, c d20 = 100 g/l colored in red and c d20 = 200 g/l colored in blue). All spectra were recorded for at T = 298 K and B0 = 20 T. The assignment of signals is found in 8 . Figure S3. Two-dimensional 1 H-15 N NMR HSQC spectra for denatured apoazurin (c urea = 6.8 M) at different concentrations c of dextran 20 (c d20 = 0 g/l colored in black, c d20 = 100 g/l colored in red and c d20 = 180 g/l colored in blue). All spectra were recorded at T = 298 K and B0 = 20 T. Figure S4. High-field chemical shifts for one-dimensional 1 H NMR spectra for denatured apoazurin (c urea = 6.8 M) in absence (colored in black) and in presence of c d20 = 180 g/l (colored in red). Both spectra were recorded at T = 298 K and B0 = 20 T.

The asphericity parameter ∆
The asphericity parameter Δ is defined as 9 where . is an eigenvalue of the inertia tensor T and ̅ is the average eigenvalue given by tr is the trace of T. The elements of T are obtained from the following expression, where α and β represent the axes x, y, and z. N is the number of beads of the coarse-grained protein model, and .M is the component of the position of the bead i in the α-direction.

Analysis of rH and Rg
Hydrodynamic radius rH is an effective radius, which assumes the diffusing protein is spherical. However, as the asphericity ∆ of the protein increases, rH and radius of gyration Rg diverge. For pedagogical purposes, we use an extreme case to see how both ∆ and Rg are needed to describe rH. This example assumes that as the unfolded protein becomes less spherical, it becomes more like a cylindrical rod with increasing aspect ratio AR. We can write an expression z = < (Δ) by using the Stokes-Einstein relation where Dsphere is the diffusion coefficient of a sphere, … is the Boltzmann constant, is temperature, and is the viscosity, and also using the modified relation for a rod 10 and letting = < , we can show how z / < diverges as ∆ of the protein increases (see Figure S5). Since AR is the ratio of z and x principle components of gyration tensor and since ∆ is a function of the principle components, we can obtain (Δ) by assuming x and y principle components are equal (as in a cylinder).

Figure S5.
Hydrodynamic radius rH over radius of gyration Rg of a protein with respect to asphericity ∆. Figure S6. (a) Average asphericity (Δ) of protein apoazurin in varying shapes of crowders normalized by the average asphericity (ΔB) of protein apoazurin at the bulk condition against the volume fractions of crowders (ϕc). (b) Average radius of gyration (Rg) of apoazurin normalized by the average radius of gyration under the bulk condition against ϕc. The interactions between the protein apoazurin and crowders are attractive. The concentration of urea is 6 M. The temperature is kBTu/ε=1.14, corresponding to a room temperature (see Method), for all conditions. Error bars are included.