Coherent X-rays reveal anomalous molecular diffusion and cage effects in crowded protein solutions

Domain: physics
Published: 2025-11-29
Editorial outcome: major_revision

Abstract
Understanding protein motion within the cell is crucial for predicting reaction rates and macromolecular transport in the cytoplasm. A key question is how crowded environments affect protein dynamics through hydrodynamic and direct interactions at molecular length scales. Using megahertz X-ray Photon Correlation Spectroscopy (MHz-XPCS) at the European X-ray Free Electron Laser (EuXFEL), we investigate ferritin diffusion at microsecond timescales. Our results reveal anomalous diffusion, indicated by the non-exponential decay of the intensity autocorrelation function g2(q,t) at high concentrations. This behavior is consistent with the presence of cage-trapping between the short- and long-time protein diffusion regimes. Modeling with the δγ-theory of hydrodynamically interacting colloidal spheres successfully reproduces the experimental data by including a scaling factor linked to the protein direct interactions. These findings offer insights into the complex molecular motion in crowded protein solutions, with potential applications for optimizing ferritin-based drug delivery, where protein diffusion is the rate-limiting step.

Introduction
In living organisms, proteins are embedded in crowded environments such as in the cytoplasm of cells, membranes, and lipid vesicles1. Understanding protein diffusion in these environments is essential, as it directly impacts several critical cellular processes, including metabolism (connected to reaction rates), self-assembly of supramolecular structures and signal transduction1,2.

However, predicting and studying protein diffusion in crowded conditions is challenging due to the complex interplay of multiple factors. One such factor is the hydrodynamic interactions, where the motion of each biomolecule is influenced by the flow field generated by its neighbors3,4. These hydrodynamic flows can affect the entire biomembrane and enhance the diffusive motion of passive particles within the cytoplasm5. Additionally, protein diffusion is affected by direct forces, including electrostatic interactions and nonspecific attractive interactions between molecules such as van der Waals forces6. Excluded volume effects can further complicate diffusion by reducing macromolecular mobility, although these effects alone do not fully account for the tenfold decrease in diffusion rates observed in vivo7,8. Moreover, transient protein clusters due to attractive forces can further diminish mobility6,9. A major challenge in understanding protein diffusion in crowded environments lies in deciphering the competition among these factors, all of which influence dynamics on similar timescales.

Crowding can also lead to deviations from simple Brownian motion, which is termed as anomalous diffusion10,11. For globular proteins and colloids, anomalous diffusion has been linked to cage effects12,13, where a protein is transiently confined in a “cage” formed by neighboring molecules. The rearrangement of this cage is what eventually allows the molecule to diffuse and leads to structural relaxation.

In this context, protein motion is typically characterized in terms of short- and long-time diffusion (see Fig. 1a). Short-time diffusion refers to the motion of proteins within the cage formed by its neighbors and occurs on timescales shorter than the interaction time τi, which is the travel time required for a protein to move a distance equal to its radius. The interaction time can be estimated as τi ≈ 6Rh2/Ds, where Rh is the hydrodynamic radius of the protein and Ds is the average self-diffusion coefficient. These two diffusion types are visible also by probing the mean square displacement (MSD) of the particles. They are characterized by a linear increase of MSD with time with one slope at t ≪ τi and a different one at t ≫ τi14.

Anomalous diffusion of globular proteins is most evident around the interaction time, t ∼ τi. At timescales much shorter than τi (t ≪ τi), the short-time diffusion is Brownian and is mainly affected by hydrodynamic interactions15. At much longer times (t ≫ τi), long-time diffusion is predominantly hindered by friction caused by interaction between the proteins15. Unlike short-time diffusion, there are no theoretical predictions for long-time diffusion due to the complex interplay of hydrodynamic and direct protein interactions. Therefore, it is essential to develop strategies to characterize the properties of the long-time diffusion, as well as the transition between the short- and long-time diffusion regimes.

To understand the role of hydrodynamic and direct interactions in long-time protein motion, it is necessary to measure the collective diffusion at length scales both larger and smaller than the protein intermolecular distances, which is a significant technical challenge. Single particle tracking, fluorescence correlation spectroscopy16, neutron backscattering, nuclear magnetic resonance (NMR) and gradient field NMR are limited to accessing the self-diffusion of proteins. While scattering techniques can probe collective diffusion, dynamic light scattering (DLS)17,18 provides insights at length scales significantly larger than the protein size, while neutron spin echo reflects the short-time diffusion dynamics, typically faster than ≈300 ns19–22. Thus, in addition to the theoretical challenges of describing long-time diffusion due to the complexity of inter-particle hydrodynamics and direct protein interaction, there is also an experimental research gap. Addressing this gap requires methods that can combine molecular-scale sensitivity with the capability to probe collective dynamics on microsecond timescales.

With the advent of high repetition rate X-ray free electron lasers (XFELs), such as the European XFEL (EuXFEL), it becomes possible to bridge this methodological gap. This capability is especially relevant for biological systems, enabling the study of relatively small proteins (on the order of several nanometers) at high volume fraction exceeding 20 vol%, which has been previously inaccessible. Measuring dynamics in the microsecond range opens the possibility to probe proteins around the interaction time τi (estimated in the order of few microseconds) and beyond. X-ray photon correlation spectroscopy (XPCS) enables direct access to collective protein diffusion on molecular length scales23,24. XPCS measurements have been demonstrated for measuring protein dynamics, both indirectly with microrheology25, and directly when related to Brownian diffusion26, cage relaxation27,28, liquid–liquid phase separation dynamics29–31, gelation processes32–34, and nanoscale fluctuations in hydration water35. The recent development of Megahertz XPCS (MHz-XPCS) at EuXFEL has further enhanced the ability to capture equilibrium protein diffusion before the onset of radiation effects using the “correlation before aggregation” approach36. This technique offers a powerful tool to address the experimental research gap, enabling detailed investigation of long-time protein diffusion at the molecular scale and microsecond timescales, which we exploit in the present study.

Here, we investigate the short- and long-time molecular diffusion and cage effects in crowded protein solutions using MHz-XPCS at the EuXFEL (see Fig. 1b). The central questions are (i) how the interplay between hydrodynamic interactions and direct interactions influence the long-time protein diffusion, (ii) how long-time protein diffusion depends on the momentum-transfer q approaching protein molecular length scales, (iii) what the characteristic signatures associated with the cage effects are in terms of time and length scales.

To address these questions, we utilize ferritin, a globular protein known for its non-toxic iron-storing capabilities, which is produced by nearly all living organisms. Ferritin has relevant applications in vaccine development37,38, drug delivery39 and nanomaterials40,41. Structurally, ferritin consists of a protein shell with 24 sub-units arranged in pairs to form a hollow nanocage, with a core enriched in iron ions. This robust and monodisperse system is ideal for dynamic measurements, providing a strong scattering signal as a result of the iron-rich core, which facilitates data interpretation. The expected interaction time τi for ferritin in water is τi ≈ 0.3 μs and τi ≈ 3 μs for ferritin in water-glycerol (with volume fraction νglyc = 0.55), calculated using the dilute limit diffusion coefficient as an approximation of Ds and the experimental Rh (see “Methods”).

Our experimental results indicate that as the protein concentration increases, there is a continuous transition from simple Brownian motion to anomalous diffusion. This transition is characterized by the deviation from single-exponential behavior in the intermediate scattering function, particularly evident at high protein concentrations, where two distinct decay processes become apparent. The hydrodynamic function, H(q), is determined experimentally and serves as a signature of many-body hydrodynamic and direct protein interactions. Modeling based on the δγ-theory of hydrodynamically interacting colloidal spheres in the short-time limit42–44 shows quantitative agreement with the experimental results only by accounting for both short- and long-time diffusion.

Results
Scattering intensity I(q) and structure factor S(q)
The experiments were conducted at the Material Imaging and Dynamics (MID) instrument of the EuXFEL45, where the protein solutions, contained in capillaries, were measured using a small-angle X-ray scattering (SAXS) geometry with photon energy 10 keV (see “Methods”). Figure 2a shows the azimuthally integrated intensity, I(q), for various protein concentrations, ranging from c = 9 mg/ml (volume fraction ϕ = 0.0042) to c = 730 mg/ml (ϕ = 0.34). A spherical form factor fit to the scattering intensity of the dilute solution (c = 9 mg/ml) yields a radius Rs ≈ 4.0 nm (see also Supplementary Section 1), corresponding to the size of the ferritin cavity46. This result suggests that the SAXS signal predominantly originates from the protein core, which exhibits a higher scattering cross-section at this photon energy compared to the protein shell, because of the high iron content.

For the more concentrated solutions (c = 70, 180, 400, and 730 mg/ml; see Table 1 for specific sample compositions), I(q) exhibits correlation peaks that are indicative to the formation of a structure factor S(q), as shown in Fig. 2b (see “Methods” for details on data analysis). The presence of the S(q) peak at q = q0 reflects interference between proteins, with a correlation length ξp = 2π/q0. The values of q0 and ξp are provided in Table 2. As the protein concentration increases, the peak shifts to higher q-values, indicating that the inter-protein correlation length, ξp, decreases due to crowding. Additionally, Fig. 2b demonstrates that the S(q) measured at the European Synchrotron Radiation Facility (ESRF), beamline TRUSAXS (ID02), is consistent with the S(q) obtained at the EuXFEL.

Intensity autocorrelation functions g2(q,t) and q-dependent diffusion coefficient D(q)
The intensity autocorrelation function, g2(q,t), is shown in Fig. 3a–d (see “Methods” for definition). The g2(q,t) was collected for varying concentrations (c = 70, 180, 400, and 730 mg/ml) and q-values (q = 0.225−0.625 nm−1), as indicated by the color-code. The solid lines represent the stretched exponential fits according to the equation

g2(q,t) = 1 + β(q)exp[−2(Γ~(q)t)α], (1)

where β(q) is the speckle contrast, modeled as in ref. 36, Γ~(q) is the decorrelation rate and α is the Kohlrausch–Williams–Watts (KWW) exponent47. As protein concentration increases, we observe an overall slowdown of dynamics and a systematic change of the g2(q,t) lineshape from an almost simple exponential decay (α = 1) at c = 70 mg/ml to a stretched exponential decay (α < 1) at higher concentrations. The condition α < 1 indicates heterogeneous protein motion at elevated concentrations. To account for the varying KWW exponents, we compute the average decorrelation rate48 Γ(q) = Γ~(q)α = Γ(1/α), where Γ(x) is the Γ-function.

The average decorrelation rate, Γ(q), is shown in Fig. 4a. The solid lines denote the expected relationship Γ(q) ∝ q2 for Brownian diffusion. However, the experimental data deviate from this behavior, as highlighted by the shaded regions. To quantify these deviations, we calculate the q-dependent diffusion coefficient, D(q) = Γ(q)/q2, shown in Fig. 4b. The diffusion coefficient D(q) decreases as a function of q, reaching a minimum at q = qD0, which coincides with q0 (see Table 2). This modulation in D(q) is due to the De Gennes narrowing49, observed in a variety of systems50–55, including apoferritin20,56,57. Thus, the observed D(q) indicates that collective protein motion is hindered (i.e., slowed down) by neighboring proteins, at length scales approaching the correlation length ξp = 2π/q0.

Hydrodynamic function H(q) and self-diffusion coefficient Ds
In the short-time limit, the diffusion coefficient D(q) can be linked to the hydrodynamic interactions by computing the hydrodynamic function H(q)15, defined as:

H(q) = D(q)/D0 * S(q), (2)

where D0 is the diffusion coefficient in the dilute limit and S(q) the structure factor. Since there is no theoretical description beyond the short-time limit, we use Eq. (2) to estimate the hydrodynamic function, even though the timescale probed exceeds the interaction time and thus can also be influenced by long-time diffusion.

The hydrodynamic function H(q) can be expressed as a sum of two contributions via43

H(q) = H~(q) + H0, (3)

where H~(q) is the q-dependent term determined by protein–protein hydrodynamic interactions and the spatial arrangement of the proteins in the solution, i.e., the S(q), while H0 is a q-independent term, related to the protein self-diffusion Ds by H0 = Ds/D0. At length scales much smaller than the typical protein–protein correlation length (q ≫ q0), the contribution of H~(q) is negligible and thus H(q ≫ q0) = Ds/D0.

Figure 5a shows the experimental D(q)S(q)/D0 for different ferritin concentrations (c = 70, 180, 400, and 730 mg/ml) using Eq. (2). Here, S(q) was determined from the SAXS data (shown in Fig. 2b) and D0 from the DLS data (see “Methods” and Supplementary Section 3). The value of D0 was corrected to account for beam-induced heating (approximately ΔT ≈ 1 K as detailed in Table S1). D(q)S(q)/D0 exhibits maxima at q = qH0 in agreement with the q0 values listed in Table 2. At increased protein concentrations, the overall magnitude of D(q)S(q)/D0 decreases due to the reduction in self-diffusion Ds and enhanced hydrodynamic interactions.

Modeling based on the δγ-theory
H(q) was modeled using the colloid δγ-theory15,42,43,58. This theory accounts for many-body hydrodynamic interactions of spherical particles in suspension in the short-time limit and uses S(q) as input, as described in the “Methods” section. Figure 5a shows the results of the model (solid lines), including a scaling factor that was fitted to the experimental D(q)S(q)/D0 data (see Supplementary Section 5). The model demonstrates good agreement with D(q)S(q)/D0 within the measured q-range. The extracted Ds/D0 as a function of the hydrodynamic volume fraction is shown in Fig. 5b (see also Fig. S7). The hydrodynamic volume fraction is ϕh = ϕ(Rh/Rp)3 ≈ 1.59ϕ, where Rh is the protein hydrodynamic radius (Rh = 7.3 nm obtained by DLS, see “Methods”) and Rp the protein radius (Rp = 6.25 nm based on the structure reported in ref. 59).

Alternatively, the hydrodynamic volume fraction ϕh dependence of Ds/D0 can be modeled independently60 by the expression

Dshort(ϕh)/D0 = 1 − 1.73ϕh + 0.88ϕh2, (4)

as shown in Fig. 5b (dashed line). This formula, which assumes that Ds refers to short-time self-diffusion, appears to overestimate Ds/D0 compared to the experimental values extracted by fitting D(q)S(q)/D0. Such deviations from colloid theory has been observed for various protein systems, including myoglobin19, hemoglobin21, and γB crystalline22. In the case of hemoglobin21, this discrepancy was partially explained by accounting for the hydrodynamic volume fraction. Here, implementing the ϕh correction is insufficient to explain the discrepancy seen for ferritin, where ϕh is included in Fig. 5. Another possible explanation for an overestimation of Ds by the theory is the presence of patchy interactions (non-isotropic protein-specific interactions)22, which is presumably not the case for ferritin given the repulsive interactions visible in the S(q) lineshape, characterized by the reduction in S(q = 0) with increasing concentration and the presence of a strong correlation peak17,18,61.

At the observed timescales, it is possible that Ds/D0 is influenced by a combination of short-time and long-time diffusion as previously observed by fluorescence recovery after photobleaching (FRAP) experiments62,63. To explore this hypothesis, the long-time self-diffusion coefficient is approximated by63,64

Dlong = Dshort * Ddirect, (5)

The short-time self-diffusion coefficient, Dshort, contains the hydrodynamic interactions, but not the direct protein interactions (i.e., electrostatic and nonspecific attractive interactions). The latter are included in Ddirect = D0/(1 + 2ϕhχ), where χ is the contact value of the pair-correlation function65, which was estimated approximating the protein interactions as hard spheres (where χ = (1 + 0.5ϕh)/(1 − ϕh)2). This model appears to better describe the volume fraction dependence of Ds/D0, as shown in Fig. 5b (solid line). Therefore, the experimental data indicate that the steep decrease of Ds/D0 with increasing ϕh is likely due to the Dlong component, which becomes dominant with increasing volume fraction.

Cage effects and long-time diffusion
To investigate the origin of the heterogeneous dynamics indicated by α < 1 for high protein concentrations, we calculate the intermediate scattering function, f(q,t), related to the g2(q,t) by the Siegert relation,

g2(q,t) = β(q)|f(q,t)|2 + 1. (6)

For Brownian diffusion in the dilute limit, f(q,t) is expected to be a simple exponential function, meaning that ln[f(q,t)] depends linearly on time, and ln[f(q,t)] = −〈r2(t)〉q2/6 with 〈r2(t)〉 the mean square displacement. Figure 6a shows f(q,t) for different protein concentrations, where the dotted line corresponds to a linear fit over the first 5 μs. We observe that for c = 70 mg/ml, the data follow closely the single exponential fit, while deviations from single exponential behavior become apparent at higher concentrations. Notably, at c = 730 mg/ml, a distinct slope is observed at timescales longer than 5 μs. This deviation is evident for all measured q values, as shown in Fig. 6b, and becomes more pronounced when ln[f(q,t)] is normalized by the slope of the linear fit over the first 2.5 μs, as seen in Fig. 6d. This observation suggests the presence of two components in the decay.

Using a double exponential fit, we extract the diffusion coefficients of the two components, by:

g2(q,t) = 1 + β(q){[1 − A(q)]exp[−Γ1(q)t] + A(q)exp[−Γ2(q)t]}2, (7)

where Γ1(q) and Γ2(q) correspond to fast and slow decorrelation rates and A(q) represents the relative amplitude of the slow component. Figure 6c shows the diffusion coefficients, Dshort(q) = Γ1(q)/q2 and Dlong(q) = Γ2(q)/q2, obtained from the decorrelation rates of the double exponential fit. Dshort(q) and Dlong(q) exhibit similar q-dependence and minima for q = q0 in agreement with the behavior of D(q) at lower concentrations, shown in Fig. 4. At c = 730 mg/ml, the ratio Dlong(q)/Dshort(q) is 0.09 ± 0.02.

One possibility is that the slow component is due to aggregates with size ≈ 11.1 times larger than a single protein (based on Dshort(q)/Dlong(q)), predicting a correlation peak at q = q0/11.1 = 0.050 nm−1. However, in the S(q) data (Fig. 2b) we do not see any enhancement at low q, which would indicate such a peak. Alternatively, the two decays represent short- and long-time diffusion. From Dshort(q), we estimate Dshort = Dshort(q → 1) = 2.35 nm2 μs−1 which yields an interaction time of τi = Rh2/(6Dshort) = 3.77 μs. This value corresponds to the cross-over time between the two exponential functions, supporting the hypothesis that the double exponential decay arises from short- and long-time diffusion. It is noteworthy, that estimating the interaction time based on the Stokes-Einstein equation in the dilute limit for ferritin in water yields τi ≈ 0.3 μs, whereas in crowded conditions τi is ≈ 12.6 times larger. This calculation implies a significantly slower reaction rates than those predicted from simple Stokes-Einstein estimations, even at molecular length scales.

The short-time diffusion coefficient Dshort(q) is almost 15 times smaller than the dilute-regime diffusion coefficient. The reason for the discrepancy between Dshort(q) and the dotted line in Fig. 5b is due to the limit of the validity of Eq. (4) to volume fractions below 0.45 according to ref. 60. For ferritin in heavy water at ϕ = 0.153 and ϕ = 0.327 (corresponding to ϕh of 0.21 and 0.44), the values of the normalized short-time self-diffusion Ds/D0, extracted with simulations and experimental data employing neutron scattering20, are 0.76 and 0.48, respectively. These values are close to the expected short-time diffusion coefficient (dotted line in Fig. 5b). Hence, we conclude that the large difference seen between the samples in glycerol-water mixtures and water cannot be attributed to the different solvents, but rather to very elevated crowding conditions of the proteins in solution at 730 mg/ml.

The lineshape of Dshort(q) and Dlong(q) coincides with the prediction of the δγ-theory, as shown in Fig. 6c (black line). Since hydrodynamic forces influence both diffusion coefficients, while direct interactions are relevant only in the long-time diffusion, we conclude that hydrodynamic forces act similarly on the short- and long-time diffusion and the direct interactions do not provide additional q-dependent changes on the diffusion coefficient.

These results are also in agreement with previous studies performed on larger colloidal particles66–68, where the short- and long-time diffusion coefficients, Dshort(q) and Dlong(q), were shown to have the same lineshape at qR > 2.5. This finding indicates that for the investigated system at 1.5 < qR < 4, the direct interactions, relevant only for the long-time diffusion, do not influence the q-dependence of Dlong(q). Consequently, hydrodynamic functions can be effectively extracted from long-time diffusion, disentangling the contributions of direct forces and hydrodynamic interactions (see also Supplementary Section 5).

Additionally, the relative amplitude of the two decays, A(q), provides insight into protein motion. It is given by:

A(q) = A0 exp(−q2δ2/6), (8)

reflecting the loss of correlation due to a Debye-Waller-like movement around a central position, with average displacement δ and percentage A0 of particles involved69. For this system, A0 = 89 ± 3% and δ = 1.0 ± 0.3 nm indicating that the majority of protein molecules participate in cage formation, with an average displacement of 1.0 nm. This finding suggests that protein rattling within the cage occurs only within a fraction of their radius due to the limited available space at high concentrations. This result aligns with the expected reduction in cage size due to confinement with increasing volume fraction, approaching the glass transition12,66,70. Cage effects have been probed also in biological systems28,71 and inorganic colloids with different sizes66. Here, the main parameter that drives the cage formation is the excluded volume effects, which can be affected by the inter-particle interactions72.

Here, we have used two models to extract the diffusion coefficient: the stretched exponential and the double exponential models. In the stretched exponential the short- and long-time diffusion are effectively merged into a single time constant, and the exponent indirectly reflects the extent to which these two components are separated as a function of concentration. In the double exponential model, the long- and short-time diffusion are treated separately and the amplitude indicates the relative contributions of the two components. The latter indicates that 90% of the decay is from the long-time diffusion component. Since the contribution of long-time diffusion component is predominant, the extracted value of D(q) obtained from the stretched exponential model is effectively close to the value of Dlong(q). The solid black line shown in Fig. 4b and in Fig. 6c (it refers to the y-axis on right-hand side) is the same between the same panels, showing good agreement between D(q) and Dlong(q).

Methods
Sample preparation
Protein solutions of different concentrations were prepared in steps, starting from centrifuging the parent protein saline solution and followed by dilution with glycerol. Ferritin from equine spleen solutions (Sigma-Aldrich, F4503) was used as the parent solution, with an initial protein concentration of c = 71 mg/ml and NaCl content of 150 mM. This parent solution was centrifuged at 10,000 × g for 1 h using 10 kDa Millipore filters, resulting in a concentrated ferritin solution of c = 730 ± 20 mg/ml, as determined by UV-Vis spectroscopy at λ = 280 nm78, where the error bar corresponds to the standard error. This concentration was estimated using a calibration curve based on samples of known concentration.

The concentrated ferritin solution was subsequently diluted with glycerol and water to achieve protein concentrations c = 70, 180, and 400 mg/ml (with 55 vol% glycerol). As the protein concentration decreased, the NaCl concentration also decreased, as the diluting solvent did not contain NaCl. These solutions were filled in quartz capillaries with an outer diameter of ≈ 1 mm and a wall thickness of ≈ 20 μm. The volume fraction is calculated from the protein concentration, c, using the equation ϕ = cν, where ν = 0.463 ml/g is the specific volume of the protein79, which is the inverse of the protein density. A summary of the samples and their composition is shown in Table 1.

X-ray experimental parameters
The data were acquired at the MID instrument45 of the EuXFEL in small-angle X-ray scattering (SAXS) configuration using a pink beam with a photon energy of 10 keV. The X-ray pulses, with an intra-train repetition rate of 4.5 MHz (222.2 ns pulse spacing), were delivered in trains of up to 310 pulses, with an inter-train repetition rate of 10 Hz. The Adaptive Gain Integrating Pixel Detector (AGIPD)80,81 was positioned at a sample-detector distance of 7.687 m. The X-ray beam was focused to a diameter of 12 ± 1 μm and the X-ray bandwidth was ΔE/E = 7 × 10−3, determined by fitting the speckle contrast. This beam size, achieved using compound refractive lenses (CRL), was chosen to enhance both the speckle contrast and the signal-to-noise ratio (SNR) of the XPCS measurements82. X-ray intensity was adjusted by chemically vapor deposited (CVD) diamond attenuators of variable thickness to minimize beam-induced effects while maintaining high SNR (see Table S1).

To refresh the sample volume between trains, the sample was moved through the X-ray beam at a translation motor speed of 0.4 mm s−1, as described in ref. 36. Table 3 summarizes the experimental parameters for the EuXFEL measurements, along with the SAXS experimental parameters at ESRF (beamline ID02). The measurements were repeated to increase the SNR. This also ensured that the sample was measured in different positions along the capillary to take into account local variations, which were not seen. The total repetition number for the data shown is 6000 for each protein concentration for an X-ray transmission of 5.35 × 10−6.

Brownian diffusion coefficient D0 and hydrodynamic radius Rh
The protein diffusion coefficient, D0, was measured in dilute conditions (c = 9 mg/ml) using DLS (see Section 4 of the Supplementary Information). For ferritin in water-glycerol (55 vol% glycerol), we obtained D0 = 3.05 ± 0.01 nm2 μs−1. For ferritin in water, the diffusion coefficient used is D0 = 30.1 ± 0.2 nm2 μs−1. The hydrodynamic radius, Rh, was calculated using the Stokes-Einstein equation, D0 = kBT/(6πηRh), where η is the viscosity, T the temperature and kB the Boltzmann constant. Here, for water-glycerol by using η = 9.3 ± 0.6 mPas83 and T = 298 K, we estimated Rh = 7.3 ± 0.1 nm. This value is comparable to literature values of apoferritin (hollow shell without the core) in saline solutions obtained with DLS (Rh = 6.9 nm17, Rh = 6.3 nm18) and HYDROPRO (Rh = 6.4 nm)84. The slightly larger Rh obtained here could be due to the presence of the iron (ferritin vs apoferritin), which may influence the protein hydrodynamic radius as surrounding hydration layer. For ferritin solutions, the Rh = 6.85 ± 0.8 nm obtained by DLS85 is consistent with our estimate.

SAXS data analysis
The protein scattering intensity, I(q), was determined by subtracting the background scattering intensity, Ibkg (scattering from the capillary containing only the solvent), from the measured scattering intensity, Im(q):

I(q) = Im(q) − Ibkg(q). (9)

The scattering intensity, I(q), of mono-dispersed spherical proteins is modeled by

I(q) = cΔρ2P(q)S(q), (10)

where Δρ is the scattering contrast between the protein and the solvent, c the protein concentration, S(q) the structure factor, and P(q) the form factor. The P(q) was determined experimentally by measuring the scattering intensity in the dilute limit, I0(q), where S(q) ≈ 1. Here, I0(q) is estimated using low concentration sample (c = 9 mg/ml). The experimental S(q) was then calculated as follows:

S(q) = C * (Im(q) − Ibkg(q)) / (I0(q) − Ibkg(q)), (11)

where C is a normalization constant. More detailed are provided in section 1 of the Supplementary Information.

Intensity autocorrelation function g2 calculation and fit
The g2(q,t) function was calculated starting from the two time correlation function (TTCs), which represent the correlation between speckle images taken at times t1 and t2 at momentum transfer q:

c2(q,t1,t2) = 〈I(q,t1)I(q,t2)〉 / (〈I(q,t1)〉〈I(q,t2)〉), (12)

where 〈…〉 indicates averaging over all pixels with the same momentum transfer, q. The TTCs functions were extracted with the MID data processing pipeline embedded in the DAMNIT tool of the EuXFEL data analysis group (https://damnit.readthedocs.io/en/latest/), which includes a correction using train cross-correlation86. The average TTC was obtained by averaging the TTCs of the single trains, where the weights correspond to the inverse of the standard errors on the TTCs squared. The corresponding error bars δc2(q, t1, t2) were estimated from the standard error between the individual TTCs. A baseline subtraction on the average TTCs was performed at each q. The value of the baseline was calculated as weighted mean of the delay times above 46 μs which correspond to pulse numbers from 210 to 310. Finally, g2(t, q) was obtained by g2(q,t) = 〈c2(q,t1,t1+t)〉, with the weighted average, with weights corresponding to 1/δc2(q,t1,t1+t)2. The error bars on g2(t, q), were estimated from the error transposition of the weighted average.

The fits of the exponential functions to g2(q,t) were performed simultaneously in both q and t dimensions, to ensure better stability of the fits. The fit contains the q-dependence and the overall values of the contrast β(q) as described in Supplementary Section 6. The contrast β(q) was included to overcome the limitation of 222.2 ns for the shortest timedelay. The experimental parameters for the contrast estimation (X-ray bandwidth, sample-detector distance, photon energy, sample thickness, etc.) were fixed with the values reported in Table 3 except for the beam size. The latter was set as fit parameter to account for the inherent fluctuations of the XFEL beam (for values, see Supplementary Section 6). The value of α was set to be constant with q. The q-dependence of Γ~(q) was not fixed to be able to extract D(q).

Beam-induced effects assessment
Identifying the onset of X-ray beam-induced effects is crucial for MHz-XPCS. Beam-induced effects can manifest as fluence-dependent changes in the scattering intensity, I(q), and the diffusion coefficient, D(q). These changes may result from beam-induced heating or radicals formation in the solution, potentially leading to protein aggregation or denaturation36.

Supplementary Fig. S2 shows the I(q) for varying pulse accumulations at different concentrations, c, and transmissions, T0. In addition, the I(q) as a function of pulse number for q = 0.2 nm−1 is shown in Supplementary Fig. S3. Minor changes in the I(q) are observed across all T0. Specifically for data used for the figures presented, i.e., at T0 = 5.35 × 10−6, intensity changes are less than 1%.

The fluence-dependence of D(q) and TTCs are discussed in Supplementary Section 3. Within the error bars, changes in the KWW exponent α with dose are negligible. However, a slight increase in the diffusion coefficient is observed with increasing pulse number, likely due to beam-induced temperature rise36,87, estimated to be below 1 K. This approach allows us to identify the optimal parameters to mitigate beam-induced effects, such as optimal transmission T0, number of repetitions nrep, and the number of pulses-per-train npulses. Considering the minor changes in both I(q) and D(q) under these conditions, we conclude that protein aggregation is not detected. The intrinsic protein dynamics can be accurately accessed by including a correction for the temperature increase (in the order of 1 K) as reported in Table S2.

Hydrodynamic function modeling
The hydrodynamic function, H(q), was modeled according to the framework outlined in refs. 43,58 by:

H(q) = H~(q) + Ds(ϕ)/D0, (13)

The q-dependent part H~(q) is calculated using jscatter88 with the expression:

H~(q) = 2/3π ∫0∞ dR p k(Rpk) s2i [n12 + (RpϕSγ)(Rpk)] − 1 ∫1−1 dx(1−x2)[S(j|q−k|j)−1], (14)

where x = cos(q,k) is the angle between q and k vectors, ϕ is the protein volume fraction. The protein radius Rp = 6.25 nm is based on the structure reported in ref. 59, and Sγ(q) is the volume fraction dependent function given in ref. 58. The S(q) used for the calculation is obtained by interpolating the experimental S(q) (see Supplementary Section 1).

The normalized self-diffusion coefficient Ds(ϕ)/D0 was estimated via58:

Ds(ϕ)/D0 = 1 − 2π ∫0∞ dR p k(Rpk)2[1 + ϕpSγ(Rpk)] (15)

More information on the fit is provided in Supplementary Section 5.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript explores protein diffusion dynamics in a crowded environment using MHz XPCS. I believe the experiment described presents a compelling scientific case for XPCS at FELs, particularly with the use of EuXFEL pulse trains. However, there are several key aspects that need to be addressed before the paper can be considered for publication.\n1. XFELs are known for the excellent transverse coherence, and this measurement was conducted at a very small angle. However, the observed contrast is only a few percent. What could be the reason for this? Please perform single-shot speckle visibility analysis to verify the contrast at each Q. Additionally, could you provide details on the theoretical calculations for the expected contrast values based on the measurement geometry?\n2. In reference to question 1, Fig. 4 shows different starting contrast values for various concentrations at the same Q, while the starting contrast is a very important parameter in the fitting. It is hard to argue this discrepancy is due to the background. For example, at the smallest Q, the 70 mg/mL sample exhibits the highest scattering intensity, as shown in Fig. 1a, yet its contrast is not the highest. Why is that? Furthermore, since the contrast begins to decrease at the shortest separation (~220 ns), it becomes challenging to achieve a convincing fit without having the first data point at time zero from visibility extraction. For instance, the DLS measurement in Fig. S6a includes numerous short time g2 data points, which is what makes the fit reliable and accurate.\n3. For figure 2, there should be error bars for each g2 values.\n4. Regarding the error bars, the data points in Fig. 4b show significantly larger deviations from the theoretical model than suggested by the error bars, making it difficult to argue that the model is still valid.\n5. Line 151 and Line 157, is the pre-factor (1.59) the same for hemoglobin and ferritin?\n6. The authors should double check the numbers for consistency throughout the paper. From Fig. S6b, D0 is about 3 nm^2 μs^(-1), which differs from the value stated on line 277 (page 17) of the main paper. Moreover, using this D0, and the viscosity η = 92 mPa s, I cannot get the correct R_h. D0 of ferritin in water and the hydrodynamic radius makes sense on the other hand.\n7. How is D_s/D_0 in Fig. 5b estimated? According to line 136 of the paper, H (Q >> q_0) = D_s/D_0. However, when looking at Fig. 5a, the fit extending to q = 0.8 shows an increasing trend, suggesting that H(q=0.8) does not represent the small length scale limit. Yet, the D_s/D_0 value in Fig. 5b appears to match H (q = 0.8). It should also come with an error bar.\n8. In this study, only the 730 mg/ml is studied in water, we see a considerate slow down considering the difference in τ_i (0.3 microsecond from stokes Einstein equation vs the measured interaction time of 4.25 microsecond). However, if using 55% vol fraction glycerol, is there a slow down? (D1 seems to be within a factor of 2 as compared with D0) What causes the difference in the magnitude of the change in interaction rate between these two environments? The data for 400mg/mL in water is needed as a comparison.", "reviewer_2": "The manuscript describes a comprehensive study on the diffusion of biomolecules in 'crowded' environment using megahertz X-ray photon correlation spectroscopy. Using ferritin as the model system, the authors revealed the cage-trapping phenomenon. This particular observation is relevant to understanding the diffusivity of ferritin, particularly - as mentioned in the manuscript - for its use as delivery system in biomedical applications.\nThe work was carried out comprehensively and shed lights on the understanding of how protein particles diffuse in solution.\nIt will be interesting for the authors to comment on the cage-trapping behavior on other particles. Have they been observed on inorganic nanoparticles of similar size? What would be the parameters or surface properties that may drive such trapping?\nWhile the reviewer appreciates the implications on biomedicine, ferritin has a unique property that each particle contains iron core that has been shown to be paramagnetic. This magnetic behavior is attractive for its application in MRI which the authors may want to include. How does the iron core affect the measurement? How does it play a role in the diffusivity of the ferritin?", "reviewer_3": "In a paper titled “Coherent X-rays reveal anomalous molecular diffusion and cage effects in crowded protein solutions”, Girelli et al. investigate translational diffusion of ferritin in self-crowded systems using X-ray scattering. In particular, authors study the interplay of direct and hydrodynamic interactions, and timescales involved in the caging effects. The key result is a similar wavevector dependence of long- and short-time ferritin diffusivity, resulting in transferability of hydrodynamic function between these two regimes. The topic is very interesting and important for the biophysics and soft condensed matter community, and the findings are of high importance. Moreover, the methodology applied by the authors probes the microsecond timescale which was previously relatively unexplored.\nThe strength of this paper lies in the data which beyond any doubts is comprehensive (multiple techniques, multiple concentrations, multiple wavevectors) and accurate (small error bars, good agreement between model fits and experimental points). However, at the same time, multiple techniques and analysis approaches make it hard to follow and relate between the results of various subsections. Specifically, connection between various diffusion coefficients (Dshort, Dlong, D1, D2) introduced in the paper and the temporal dynamics of mean squared displacement or concentration is not clearly expressed, which makes it hard to relate the findings with the problem of diffusive transport mentioned in the abstract. I would suggest improving these aspects. Below I list specific things that should be addressed.\n1. Page 3, line 40: short-time diffusion should not be affected by the excluded volume, but by hydrodynamic interactions only.\n2. Figure 4 is titled “Decorrelation rate (…)”, but the symbol corresponds to average decorrelation rate introduced earlier. I believe this distinction is important and consistency should be kept.\n3. In the caption of Figure 5b, after word “self-” the term “diffusion coefficient” is misplaced, or something is lacking.\n4. The estimate of short-time diffusivity from two-component exponential fit to the intermediate scattering function is almost 15 times smaller than the dilute-regime diffusion coefficient, leading to drastically different estimates of transition time between the two different diffusive regimes. Authors show that theory predicts much smaller short-time diffusivity slowdown (dotted line in Fig. 5b). What is the reason for this discrepancy, and more general: what is the relation between the estimates made via this two-component exponential fit and the previous estimates made based on wavevector-dependent diffusivities?\n5. Page 17, line 279: authors write viscosity = 92 mPa*S, which is 92 cP. It seems too high both for water and for water-glycerol mixture.\n6. The major application of the results of this study mentioned by authors in the abstract is predicting diffusive transport of ferritin-based drugs. It is not clear to me how the various diffusion coefficients obtained in the paper translate into temporal dynamics, either mean squared displacement curve with its distinct regimes or temporal evolution of concentration field. Authors should shed some light on that.\n7. Page 18, line 298: ”correspond to standard errors”"}, "author_response": "Rebuttal letter for ”Coherent X-rays reveal anomalous\nmolecular diffusion and cage effects in crowded protein\nsolutions” (NCOMMS-24-55663)\nWe appreciate the time and effort of each of the three reviewers dedicated to providing insightful\nfeedback on ways to strengthen our paper and the overall positive evaluation of our work. Here,\nwe\nprovide point-by-point responses to the reviewers’ comments. The text is color-coded: The comments\nofthereviewersarecoloredblack, ourresponsesarecoloredgreen, andthemodificationsincorporated\ninto the manuscript or the supporting information are colored blue.\nReviewer 1\nThis manuscript explores protein diffusion dynamics in a crowded environment using MHz XPCS. I\nbelieve the experiment described presents a compelling scientific case for XPCS at FELs, particularly\nwiththeuseofEuXFELpulsetrains. However,thereareseveralkeyaspectsthatneedtobeaddressed\nbefore the paper can be considered for publication.\nWethankthereviewerforprovidinginterestinganddetailedremarksaboutourwork. Wehaverevised\nourmanuscripttoaddressthecommentsofthereviewer,whichhelpedsignificantlyimproveourarticle.\nPoint 1 XFELs are known for the excellent transverse coherence, and this measurement was con-\nductedataverysmallangle. However,theobservedcontrastisonlyafewpercent. Whatcouldbethe\nreason for this? Please perform single-shot speckle visibility analysis to verify the contrast at each q.\nAdditionally, could you provide details on the theoretical calculations for the expected contrast values\nbased on the measurement geometry?\nWe have added a single-shot speckle visibility analysis suggested by the referee. We note here that\nthe values of the contrast are consistent with previous studies performed at the EuXFEL with similar\nexperimental parameters [1–3]. As the referee points out, the transverse coherence is very high but\nthe longitudinal is limited by the wavelength bandwidth, in accordance with contrast estimations [4].\nIn addition, in order to reduce the x-ray dose on the sample, the beam size is relatively large, which\ncausesafurtherdecreaseofthespecklecontrast. Furthermore,thecontrastislimitedbytherelatively\nlarge size of the pixel of AGIPD (200×200µm2), which is necessary to reach MHz resolution. We have\nadded the following section ”Contrast estimation” in the supporting information with details of the\ncontrast estimation:\nContrast evaluation\nThe values of the contrast β(q) were estimated by using the equation [1, 2]:\nβ(q)=β (q)β (1)\nl t\nThe value of the transverse coherence was found to be β ≈ 0.5 in various XFELs including the\nt\nEuXFEL [1, 2, 5]. The longitudinal coherence, β (q), describes the loss of coherence due to the\nl\nfinite bandwidth ∆E/E, the geometry and speckle shape factor. β (q) = (M M )−1 contains two\nl rad det\ncontributions M and M , which are:\nrad det\n(cid:115)\nq2 (cid:0)∆E(cid:1)2(cid:2) b2cos2(θ)+t2sin2(θ) (cid:3)\nM = 1+ E s (2)\nrad 4π2\n1\n(cid:115)\np4b2 (cid:2) b2cos2(2θ)+t2sin2(2θ) (cid:3)\nM = 1+ s s . (3)\ndet λ4L4M2\nrad\nWith the constant b being the beam size, ∆E/E being the bandwidth, 2θ being the scattering\ns\nangle, λ being the wavelength of the x-ray beam, L being the sample-detector distance, p being the\npixel size, t being the sample thickness. The experimental parameters in the fit were fixed, except for\nthe beam size b which effectively accounts for small changes in the experimental parameter between\ns\nmeasurements. The value of the beam size was optimized for each sample in the fit g function. The\n2\ncorresponding extracted values of the contrast with are reported in Fig.S9 and their respective b in\ns\nTable 4.\nFigure S9: a) Speckle contrast β versus momentum transfer q, obtained after the optimization of the\nbeam size in the fit of g (q,t) from Fig. 3 of the main manuscript. The shaded area represents the\n2\nstandard error due to the uncertainties of the fit. b) Contrast estimation with three different methods\nfor the data of protein concentration 180 mg/ml. The three methods correspond to: the fit results\nfromtheg (q,t)functions(greencirclesandshadedareatoindicatethestandarderror),theestimator\n2\nof eq. 7 , the fit of the probability of two photon events given in Eq. 6 (orange squares).\nc (mg/ml) b (µm)\ns\n70 13.4 ± 1.3\n180 12.4 ± 0.3\n400 12.8 ± 0.3\n730 13.2 ± 0.4\nTable 4: Beam size obtained from the fit of g (q,t) from Fig. 3 of the main manuscript.\nrayspecklevisibilityanalysis(XSVS).Here,thecontrastβ canbeobtainedbyanalysingtheproperties\nof scattering intensity distribution P(k,k¯,M) of a single frame [6]. The analytical form of P(k,k¯,M)\nis:\nΓ(k+M) (cid:18) M (cid:19)−k(cid:18) k¯ (cid:19)−M\nP(k,k¯,M)= 1+ 1+ (4)\nΓ(M)Γ(k+1) k¯ M\nwhere k is the photon count of each pixel per shot, k¯ is the average number of photons per pixel\nper shot and M is the number of modes, which is related to the speckle contrast by β = 1/M. The\ncontrast was estimated with two different methods: from the fit of the probability of finding a one-\nand two-photon events on a pixel p = P(k = 2,k¯,M) and from a contrast estimator formula. The\n2\n2\nanalytical form for p and p can be derived from Eq. 4, and is:\n1 2\n(cid:18) M (cid:19)−1(cid:18) k¯ (cid:19)−M\np =P(k =1,k¯,M)=M 1+ 1+ , (5)\n1 k¯ M\nM(M +1) (cid:18) M (cid:19)−2(cid:18) k¯ (cid:19)−M\np =P(k =2,k¯,M)= 1+ 1+ . (6)\n2 2 k¯ M\nFrom the analytical expression of p and p , and defining R as R =p /p , β can be obtained\n1 2 12 12 2 1\nwith the following estimator[7]:\n2·R −k¯\nβ = 12 (7)\nk¯(1−2·R )\nsame bunch trains used for the calculation of g (q,t). We observe that the contrast values obtained\n2\nwith different methods are overall consistent with each other. The minor discrepancies between the\ncontrast obtained from the fit of the TTCs and from the XSVS analysis can be due to several reasons:\nFirst, in the XSVS analysis it is not straightforward to apply a cross-correlation correction [8] which\ncorrects for the spurious correlations present due to the noise in the detector. Second, in the single\nimage XSVS analysis a baseline correction is not trivial as it is for the two-time correlation functions,\nwhere the correlation of images at large time difference can be used as baseline. For XSVS, the large\nerror bars are due to the very low counts which makes the baseline estimation challenging [9].\nPoint 2 In reference to question 1, Fig. 4 shows different starting contrast values for various con-\ncentrations at the same q, while the starting contrast is a very important parameter in the fitting. It\nis hard to argue this discrepancy is due to the background. For example, at the smallest q, the 70\nmg/mLsampleexhibitsthehighestscatteringintensity,asshowninFig. 1a,yetitscontrastisnotthe\nhighest. Why is that? Furthermore, since the contrast begins to decrease at the shortest separation\n(220ns),itbecomeschallengingtoachieveaconvincingfitwithouthavingthefirstdatapointattime\nzero from visibility extraction. For instance, the DLS measurement in Fig. S6a includes numerous\nshort time g data points, which is what makes the fit reliable and accurate.\nalthoughthisismainlyduetothechangesoftheXFELbeamproperties,andnotduetothescattering\nsignal of the sample. To account for the variation in the contrast, and the limitation to 222.2 ns for\nthe shortest time delay, a theoretical estimation of the contrast has been included in the fit of the\ng (q,t). The model takes into account the q dependence of the contrast based on the experimental\n2\nparameters (X-ray bandwidth, sample-detector distance, photon energy, sample thickness etc.) based\non Refs.[1, 2]. In order to take into account changes in contrast, the beam size is fitted along with the\ndecorrelationrateΓ˜. TheadditionalbeamparametersusedfortheestimationarereportedinTable3.\nWehaveaddedthefollowingsectionintheMethodssection,under”Intensityautocorrelationfunction\ng calculation and fit”:\n2\ndimensions, to ensure better stability of the fits. The fit contains the q-dependence and the overall\nvalues of the contrast β(q) as described in SI in section ”Contrast estimation”. The contrast β(q)\nwas included to overcome the limitation of 222.2 ns for the shortest time delay. The experimental\nparameters for the contrast estimation (X-ray bandwidth, sample-detector distance, photon energy,\nsample thickness etc.) were fixed with the values reported in Table 3 except for the beam size. The\nlatterwassetasfitparametertoaccountfortheinherentfluctuationsoftheXFELbeam(forvalues,see\ntheSIsection”Contrastestimation”). Thevalueofαwassettobeconstantwithq. Theq-dependence\nof Γ˜(q) was not fixed to be able to extract D(q).\nPoint 3 For figure 2, there should be error bars for each g values.\n2\nsize. Fig.3 of the main manuscript was changed.\ntained at EuXFEL. The intensity autocorrelation function, g (q,t), for different protein concentra-\n2\ntions (a) c = 70 mg/ml, (b) c = 180 mg/ml, (c) c = 400 mg/ml and (d) c = 730 mg/ml. Data in\npanels(a-c)wheremeasuredinwater-glycerol(withglycerolvolumefractionν =0.55),whilepanel\nglyc\n(d) presents data measured in water to reach the desired protein concentration (c = 730 mg/ml).\nThe different colors represent different q-values, changing from lighter to darker green for q increasing\nfrom q = 0.225 nm−1 to q = 0.625 nm−1 with equal spacing of dq = 0.05 nm−1. Solid lines represent\nstretched exponential fits, with the corresponding Kohlrausch-Williams-Watts (KWW) exponent α\nshown in the legend. The error bars shown correspond to the standard error, estimated as described\nin the SI.\nPoint 4 Regarding the error bars, the data points in Fig. 4b show significantly larger deviations\nfromthetheoreticalmodelthansuggestedbytheerrorbars,makingitdifficulttoarguethatthemodel\nis still valid.\nTo improve the stability of the fit, we have introduced a baseline correction in the TTCs. It is true\nthat, as the referee points out, we observe deviations from the theoretical model, but we note here\nthat our data provide a significant improvement from previous measurements. [10–12]. In Methods,\ninsection”Intensityautocorrelationfunctiong calculationandfit”thefollowingsentencewasadded:\ncalculated as weighted mean of the delay times correspondent to pulses 210 to 310, corresponding to\ndelay times above 46µs.\nPoint 5 Line 151 and Line 157, is the pre-factor (1.59) the same for hemoglobin and ferritin?\nhemoglobin the pre-factor is 1.36. We have changed it to:\nIn the case of hemoglobin [12], this discrepancy was partially explained by accounting for the\nhydrodynamic volume fraction.\nPoint 6 The authors should double check the numbers for consistency throughout the paper. From\nFig. S6b, D is about 3nmµs−2, which differs from the value stated on line 277 (page 17) of the main\n0\npaper. Moreover, using this D , and the viscosity η = 92 mPas, I cannot get the correct R . D of\n0 h 0\nferritin in water and the hydrodynamic radius makes sense on the other hand.\nWethankthereviewerfornoticingthediscrepancyinthevalues. Theyarenowcorrected. InMethods,\nthe section ”Brownian diffusion coefficient D and hydrodynamic radius R ” now reads:\n0 h\nThe protein diffusion coefficient, D , was measured in dilute conditions (c = 9 mg/ml) using\nerol), we obtained D = 3.05 ± 0.01nm2µs−1. For ferritin in water the diffusion coefficient used is\n0 h\nequation, D = k T/(6πηR ), where η is the viscosity, T the temperature and k the Boltzmann\nconstant. Here, for water-glycerol by using η = 9.3±0.6 mPas [13] and T = 298 K, we estimated\nR =7.3±0.1 nm .\nh\nPoint 7 How is D /D in Fig. 5b estimated? According to line 136 of the paper, H(Q >> q ) =\ns 0 0\nD /D . However, when looking at Fig. 5a, the fit extending to q = 0.8 shows an increasing trend,\ns 0\nsuggesting that H(q = 0.8) does not represent the small length-scale limit. Yet, the D /D value in\ns 0\nFig. 5b appears to match H(q =0.8). It should also come with an error bar.\nWe have added Figure S7 in the SI in section ”H(q) model fit” which shows the extended q-range, as\nwell as explanation in the main text to clarify this point:\nThe extracted D /D , estimated as the average value D(q)S(q)/D in a q range between 0.8 and\ns 0 0\n1.3 nm−1, as a function of the hydrodynamic volume fraction, is shown in Fig. 5b (see also Fig. S7).\nFigure S7: The experimental D(q)S(q)/D as a function of momentum transfer q (empty symbols)\n0\nand H(q) model results using the δγ-theory (solid lines). The data points and lines are the same as in\nFig. 5a, but the q range was extended to show the oscillations of the hydrodynamic function around\nthe estimated D /D . The figure is shown in a) linear scale and in b) log scale to be able to see the\ns 0\ndata points for the highest protein concentration.\nIn the caption of Fig.5 we added: The error bars correspond to error propagation from the fit param-\neters and, if not visible, they are smaller than the symbol.\nPoint 8 In this study, only the 730 mg/ml is studied in water, we see a considerate slow down con-\nsideringthedifferenceinτ (0.3microsecondfromstokesEinsteinequationvsthemeasuredinteraction\ni\n5\ntime of 4.25 microsecond). However, if using 55% vol fraction glycerol, is there a slow down? (D1\nseems to be within a factor of 2 as compared with D ) What causes the difference in the magnitude\n0\nof the change in interaction rate between these two environments? The data for 400 mg/mL in water\nis needed as a comparison.\nTherefereeraisesanimportantpoint. Adirectcomparisonofferritininwateratlowandintermediate\nconcentration is not possible with MHz-XPCS in the current setup, where the shortest time delay is\n222.2 ns. This can be extrapolated from the current data we collected on ferritin at 400 mg/ml in\nwater-glycerol (Fig. 3). Taking into account the difference in viscosity (factor of ∼ 10), most of the\ndecay of g (q,t) for ferritin in water is outside the experimental window. However, the diffusion of\n2\nferritininheavywateratintermediateandlowconcentrationwaspreviouslyextractedwithsimulations\nandneutronscatteringexperiments[10]. Wehaveaddedthefollowingsectiontoclarifythisissueunder\n”Cage effects and long-time diffusion”:\nThe short-time diffusion coefficient Dshort(q) is almost 15 times smaller than the dilute-regime\ndiffusion coefficient. The reason for the discrepancy between Dshort(q) and the dotted line in Fig. 5b\nis due to the limit of the validity of Eq. 4 to volume fractions below 0.45 according to Ref.[14]. For\nferritin in heavy water at ϕ=0.153 and ϕ=0.327 (corresponding to ϕ of 0.21 and 0.44), the values\nh\nof the normalized short-time self-diffusion D /D , extracted with simulations and experimental data\ns 0\nemployingneutronscattering[10],are0.76and0.48respectively. Thesevaluesareclosetotheexpected\nshort-time diffusion coefficient (dotted line in Fig.5b). Hence, we conclude that the large difference\nseen between the samples in glycerol-water mixtures and water cannot be attributed to the different\nsolvents, but rather to very elevated crowding conditions of the proteins in solution at 730 mg/ml.\nReviewer 2\nThe manuscript describes a comprehensive study on the diffusion of biomolecules in ’crowded’ envi-\nronment using megahertz X-ray photon correlation spectroscopy. Using ferritin as the model system,\nthe authors revealed the cage-trapping phenomenon. This particular observation is relevant to un-\nderstanding the diffusivity of ferritin, particularly - as mentioned in the manuscript - for its use as\ndeliverysysteminbiomedicalapplications. Theworkwascarriedoutcomprehensivelyandshedlights\non the understanding of how protein particles diffuse in solution.\nWethankthereviewerforprovidingremarksaboutthepossibleapplicationsofourwork. Wehave\nrevised our manuscript to address the comments of the reviewer, which helped to deepen the clarity\nof our article and connect it more with applications and previous studies.\nPoint 1 It will be interesting for the authors to comment on the cage-trapping behavior on other\nparticles. Have they been observed on inorganic nanoparticles of similar size? What would be the\nparameters or surface properties that may drive such trapping?\nWe thank the referee for raising these questions. In section ”Cage effects and long-time diffusion” we\nhave added the following sentences:\nThis result aligns with the expected reduction in cage size due to confinement with increasing\nvolume fraction, approaching the glass transition [15–17]. Cage effects have been probed also in\nbiologicalsystems[18,19]andinorganiccolloidswithdifferentsizes[17,20]. Here,themainparameter\nthatdrivesthecageformationistheexcludedvolumeeffects,whichcanbeeffectedbytheinter-particle\ninteractions [21].\nPoint 2 While the reviewer appreciates the implications on biomedicine, ferritin has a unique prop-\nerty that each particle contains iron core that has been shown to be paramagnetic. This magnetic\nbehavior is attractive for its application in MRI which the authors may want to include. How does\nthe iron core affect the measurement? How does it play a role in the diffusivity of the ferritin?\nWe thank the reviewer for pointing out the important application in MRI, we have included this\npossible application in the text. In our experiment the diffusion coefficient is specifically related to\nthe probed length scale associated with the average protein-protein distance, which is larger than the\ncoreitself. However,therecanbesmallerindirectcontributionsgiventhatthehydrodynamicradiusof\nferritin and apoferritin is different, as indicated previously [11]. We have added the following section\nin the main manuscript:\ntiesofitscore[22],andthepossibilityofexchangingthecontentofthecore[23]. Inthiscontext,ferritin\naggregation may be a useful strategy to produce a functional reporter gene for magnetic resonance\nimaging [24].\nReviewer 3\nIn a paper titled “Coherent X-rays reveal anomalous molecular diffusion and cage effects in crowded\nprotein solutions”, Girelli et al. investigate translational diffusion of ferritin in self-crowded systems\nusing X-ray scattering. In particular, authors study the interplay of direct and hydrodynamic interac-\ntions,andtimescalesinvolvedinthecagingeffects. Thekeyresultisasimilarwavevectordependenceof\nlong- and short-time ferritin diffusivity, resulting in transferability of hydrodynamic function between\nthese two regimes. The topic is very interesting and important for the biophysics and soft condensed\nmatter community, and the findings are of high importance. Moreover, the methodology applied by\nthe authors probes the microsecond timescale which was previously relatively unexplored.\nThe strength of this paper lies in the data which beyond any doubts is comprehensive (multiple\ntechniques, multipleconcentrations, multiplewavevectors)andaccurate(smallerrorbars, goodagree-\nment between model fits and experimental points). However, at the same time, multiple techniques\nand analysis approaches make it hard to follow and relate between the results of various subsections.\nSpecifically, connection between various diffusion coefficients (Dshort, Dlong, D1, D2) introduced in\nthe paper and the temporal dynamics of mean squared displacement or concentration is not clearly\nexpressed,whichmakesithardtorelatethefindingswiththeproblemofdiffusivetransportmentioned\nin the abstract. I would suggest improving these aspects. Below I list specific things that should be\naddressed.\nWethankthereviewerfortheattentivereadinganddetailedremarksaboutourwork. Wehaverevised\nour manuscript to improve the readability and added more explanations about the interpretation and\nunderstanding of our data. We have changed the names from D and D to Dshort and Dlong to\n1 2\nimprovetheclarityofthetext. Wehavealsoprovidedmoredetailsontheconnectionofthemeasured\nquantities with mean squared displacement in the SI as well as in the text (see point 7).\nPoint 1 Page 3, line 40: short-time diffusion should not be affected by the excluded volume, but by\nhydrodynamic interactions only.\nWe thank the reviewer for pointing this out, we have changed the wording. The sentence was changed\nto ”by hydrodynamic interactions.”\nPoint 2 Figure 4 is titled “Decorrelation rate (...)”, but the symbol corresponds to average decor-\nrelation rate introduced earlier. I believe this distinction is important and consistency should be kept.\nWe thank the reviewer for pointing out the inconsistency, it is now correct.\nPoint 3 In the caption of Figure 5b, after word “self-” the term “diffusion coefficient” is misplaced,\nor something is lacking.\nWe thank the reviewer for pointing out the inconsistency, we have corrected it.\nPoint 4 Theestimateofshort-timediffusivityfromtwo-componentexponentialfittotheintermedi-\nate scattering function is almost 15 times smaller than the dilute-regime diffusion coefficient, leading\nto drastically different estimates of transition time between the two different diffusive regimes. Au-\nthors show that theory predicts much smaller short-time diffusivity slowdown (dotted line in Fig. 5b).\nWhat is the reason for this discrepancy, and more general: what is the relation between the estimates\nmade via this two-component exponential fit and the previous estimates made based on wavevector-\ndependent diffusivities?\nWe thank the referee for pointing out the lack of clarity of these two points. We have answered the\nquestions and addressed the points by adding the two following paragraphs in section ”Cage effects\nand long-time diffusion” :\ndiffusion coefficient. The reason for the discrepancy between Dshort(q) and the dotted line in Fig. 5b\nis due to the limit of the validity of Eq. 4 to volume fractions below 0.45 according to Ref. [14].\nHere, we have used two models to extract the diffusion coefficient: the stretched exponential and\nthe double exponential models. In the stretched exponential the short- and long-time diffusion are\neffectively merged into a single time constant, and the exponent indirectly reflects the extent to which\nthese two components are separated as a function of concentration. In the double exponential model,\nthe long- and short-time diffusion are treated separately and the amplitude indicates the relative\ncontributionsofthetwocomponents. Thelatterindicatesthat90%ofthedecayisfromthelong-time\ndiffusion component. Since the contribution of long-time diffusion component is predominant, the\nextracted value of D(q) obtained from the stretched exponential model is effectively close to the value\nof Dlong(q). The solid black line shown in Fig. 4b and in Fig. 6c (it refers to the y-axis on right-hand\nside) is the same in the two panels, showing good agreement between D(q) and Dlong(q).\nPoint 5 Page 17, line 279: authors write viscosity = 92 mPa s, which is 92 cP. It seems too high\nboth for water and for water-glycerol mixture.\nWe thank the reviewer for pointing out this discrepancy, we have corrected this (see also response to\nreviewer 1 Point 6).\nPoint 6 The major application of the results of this study mentioned by authors in the abstract is\npredicting diffusive transport of ferritin-based drugs. It is not clear to me how the various diffusion\ncoefficientsobtainedinthepapertranslateintotemporaldynamics,eithermeansquareddisplacement\ncurvewithitsdistinctregimesortemporalevolutionofconcentrationfield. Authorsshouldshedsome\nlight on that.\nWe have added the section ”Mean square displacement and intermediate scattering function” in the\nsupporting information:\nTheintensityautocorrelationfunctionandtheintermediatescatteringfunctionarerelatedtoboth\nthe mean square displacement of the particle and the time dependent concentration fluctuations. The\nwidth function, defined as\nln[f(q,t)]\nW(q,t)=− , (8)\nq2\ncorresponds to the mean square displacement ⟨r2(t)⟩=W(q,t) in the case of a dilute system. In case\nof interacting particles this relation does not hold, as the mean-square displacement refers only to the\nself-partoftheintermediatescatteringfunction. Inotherwords,⟨r2(t)⟩includesonlyself-diffusionand\nnot the contribution of collective diffusion. In case of interacting particles, only in the limit of q ≫q\n0\nthe intermediate scattering function does not contain contributions from the collective diffusion, and\ntherefore Eq. 8 holds.\nForlowproteinconcentrationtherelationbetweentheself-diffusioncoefficientandthemeansquare\ndisplacement is\n⟨r2(t)⟩=6D t. (9)\ns\nThepresenceoftwoexponentialdecaysindicatesthatinthemeansquaredisplacementisnotshowing\na simple relation as seen in Eq. 9. Instead, two different slopes are present, one Dshort corresponding\ns\nto time t≪τ and the other one Dlong at t≫τ .\ni s i\nIn the introduction we have added the following sentence:\nThese two diffusion types are visible also by probing the mean square displacement (MSD) of the\nparticles. They are characterized by a linear increase of MSD with time with one slope at t≪τ and\ni\na different one t≫τ [25]\ni\nIn the section ”Cage effects and long-time diffusion” the following sentence was added:\nFor Brownian diffusion in the dilute limit, f(q,t) is expected to be a simple exponential function,\nmeaning that ln[f(q,t)] depends linearly on time, and ln[f(q,t)] = −⟨r2(t)⟩q2 with ⟨r2(t)⟩ the mean\nsquare displacement.\nWe thank the reviewer for pointing out typo. It is corrected now.\nReferences\n1. Lehmku¨hler,F.etal.EmergenceofAnomalousDynamicsinSoftMatterProbedattheEuropean\nXFEL. Proc. Natl. Acad. Sci. U.S.A. 117, 24110–24116 (2020).\n2. Reiser, M. et al. Resolving molecular diffusion and aggregation of antibody proteins with mega-\nhertz X-ray free-electron laser pulses. Nat. Commun. 13, 5528 (2022).\n3. Dallari, F. et al. Coherence properties from speckle contrast analysis at the European XFEL. J.\nPhys. Conf. Ser. 2380, 012085 (2022).\n4. Perakis, F. & Gutt, C. Towards molecular movies with X-ray photon correlation spectroscopy.\nPhys. Chem. Chem. Phys. 22, 19443–19453 (2020).\n5. Madsen, A. et al. Materials Imaging and Dynamics (MID) instrument at the European X-ray\nFree-Electron Laser Facility. J. Synchrotron Radiat. 28, 637–649 (2021).\n6. Hruszkewycz, S. O. et al. High Contrast X-ray Speckle from Atomic-Scale Order in Liquids and\nGlasses. Phys. Rev. Lett. 109, 185502 (2012).\n7. Perakis,F.et al.CoherentX-raysrevealtheinfluenceofcageeffectsonultrafastwaterdynamics.\nen. Nat. Commun. 9, 1917 (2018).\n8. Dallari, F. et al. Analysis Strategies for MHz XPCS at the European XFEL. Appl. Sci. 11, 8037\n(2021).\n9. M¨oller, J. et al. Using low dose X-ray Speckle Visibility Spectroscopy to study dynamics of soft\nmatter samples. New J. Phys. 23, 093041 (2021).\n10. Gapinski, J. et al. Diffusion and microstructural properties of solutions charged nanosized pro-\nteins: experiment versus theory. J. Chem. Phys. 123, 054708 (2005).\n11. H¨aussler,W.Neutronspinechostudiesonferritin:free-particlediffusionandinteractingsolutions.\nEur. Biophys. J. 37, 563–571 (2008).\n12. Doster,W.&Longeville,S.MicroscopicDiffusionandHydrodynamicInteractionsofHemoglobin\nin Red Blood Cells. Biophys. J. 93, 1360–1368 (2007).\n13. Cheng, N.-S. Formula for the Viscosity of a GlycerolWater Mixture. Ind. Eng. Chem. Res. 47,\n3285–3288 (2008).\n14. Beenakker, C. Self-diffusion of spheres in a concentrated suspension. Physica A 120, 388–410\n(1983).\n15. Weeks, E. R. & Weitz, D. A. Properties of Cage Rearrangements Observed near the Colloidal\nGlass Transition. Phys. Rev. Lett. 89, 095704 (2002).\n16. Weeks, E. R., Crocker, J. C., Levitt, A. C., Schofield, A. & Weitz, D. A. Three-Dimensional\nDirect Imaging of Structural Relaxation Near the Colloidal Glass Transition. Science 287, 627–\n631 (2000).\n17. Segr`e,P.N.&Pusey,P.N.ScalingoftheDynamicScatteringFunctionofConcentratedColloidal\nSuspensions. Phys. Rev. Lett. 77, 771–774 (1996).\n18. Chushkin, Y. et al. Probing cage relaxation in concentrated protein solutions by XPCS. Phys.\nRev. Lett. 129, 238001 (2022).\n19. Goiko,M.,deBruyn,J.R.&Heit,B.Short-LivedCagesRestrictProteinDiffusioninthePlasma\nMembrane. Sci. Rep. 6, 34987 (2016).\n20. Kwa´sniewskiPawel(cid:32)andFluerasu,A.&Madsen,A.Anomalousdynamicsatthehard-sphereglass\ntransition. Soft Matter 10, 8698–8704 (43 2014).\n21. Kompella,V.P.S.,Romano,M.C.,Stansfield,I.&Mancera,R.L.Whatdeterminessub-diffusive\nbehavior in crowded protein solutions? Biophys. J 123, 134–146 (2024).\n9\n22. Gossuin, Y., Roch, A., Muller, R. N., Gillis, P. & Lo Bue, F. Anomalous nuclear magnetic re-\nlaxation of aqueous solutions of ferritin: An unprecedented first-order mechanism. Magn. Reson.\nMed. 48, 959–964 (2002).\n23. Ruggiero, M. R., Alberti, D., Bitonto, V. & Geninatti Crich, S. Ferritin: A Platform for MRI\nContrast Agents Delivery. Inorganics 7 (2019).\n24. Bennett,K.M.,Shapiro,E.M.,Sotak,C.H.&Koretsky,A.P.ControlledAggregationofFerritin\nto Modulate MRI Relaxivity. Biophys. J. 95, 342–351 (2008).\n25. H¨ofling, F. & Franosch, T. Anomalous transport in the crowded world of biological cells. Rep.\nProg. Phys 76, 046602 (2013).\n10"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I would like to thank the authors for their efforts in addressing my comments and suggestions.\nI still have a following up question on point 1 and 2. The details are attached.", "reviewer_2": "The authors have address all comments satisfactorily."}, "author_response": "Follow-up question on Manuscript ”Coherent\nX-rays reveal anomalous molecular diffusion and\ncage effects in crowded protein solutions”\nIwouldliketothanktheauthorsfortheireffortsinaddressingmycomments\nand suggestions.\nIstillhaveafollowingupquestiononpoint1and2. Understoodthattheg2\nanalysis utilizes the cross correlation as described in Ref 8, the excessive corre-\nlation due to detector artifacts as extracted from the cross-correlation analysis\nin principle can be directly used for correcting the visibility extraction. Below\nI briefly wrote my reasoning.\nSay the intensity from pulse 1 and pulse 2 is I and I (pulse 1 and pulse 2\n1 2\ncan be from different trains and sample is decorrelated), the detector artifact is\nI , the cross correlation gives\na\n⟨(I +I )(I +I )⟩ ⟨I2⟩\nC = 1 a 2 a = a . (1)\n⟨I 1 +I a ⟩⟨I 2 +I a ⟩ ⟨I 1 +I a ⟩2\nMyunderstandingisthatthisishowthecrosscorrelationmethodworks. Then\nin principle, for single beam contrast\n⟨(I +I )(I +I )⟩ ⟨I2⟩ ⟨I2⟩\nβ = 1 a 1 a −1= 1 + a −1 (2)\n0,extracted ⟨I +I ⟩2 ⟨I +I ⟩2 ⟨I +I ⟩2\n1 a 1 a 1 a\nI guess it might be just a simple subtraction, which also agrees with the fact\nthat the current contrast estimator is overestimating.\nWe thank the reviewer for their comment. Their explana6on would indeed be accurate if the\nphoton counts were sufficiently high. However, in our case, the probability of finding a pixel\nwith an intensity greater than 0 in a single pulse is approximately 0.01 (depending on the\nsample and the q range see Fig.1). Therefore, it is not possible to calculate the contrast using\nthe formula suggested.\nFig. 1: The probability to find one and two photon events per pixel at different q values for a Ferri>n\nconcentra>on of 180 mg/ml for the q values indicated above the panels.\nFurthermore, I is not a Gaussian or a Poissonian random variable, and it depends\na\nspecifically which depends on storage cell in the detector and 6me (we cannot increase the\nsta6s6cs by probing for longer 6me to apply the central limit theorem and treat it as a\ngaussian variable). Hence, including its presence in Eq. S5 and consequently S7 and S8 is not\ntrivial. This is visible also from the fit of p which is not perfectly describing the data.\nWe thank the reviewer for their comment. Their explana6on would indeed be accurate if the\nphoton counts were sufficiently high. However, in our case, the probability of finding a pixel\nwith an intensity greater than 0 in a single pulse is approximately 0.01 (depending on the\nsample and the q range see Fig.1). Therefore, it is not possible to calculate the contrast using\nthe formula suggested.\nFig. 1: The probability to find one and two photon events per pixel at different q values for a Ferri>n\nconcentra>on of 180 mg/ml for the q values indicated above the panels.\nFurthermore, I is not a Gaussian or a Poissonian random variable, and it depends\na\nspecifically which depends on storage cell in the detector and 6me (we cannot increase the\nsta6s6cs by probing for longer 6me to apply the central limit theorem and treat it as a\ngaussian variable). Hence, including its presence in Eq. S5 and consequently S7 and S8 is not\ntrivial. This is visible also from the fit of p which is not perfectly describing the data.\nsuggested.\nFirst, in this shot noise dominated regime, the statistics are well described by the negative\nbinomial distribution, from which one can get the extracted contrast 𝛽 from a certain\n!,#$%&'(%#)\nnumber of pixels and detector frames.\nSecond, the equation (2) in my previous comment only relies on the assumption that the\ndetector artifact is uncorrelated with the shot noise, thus equation (2) will work if we can\ncharacterize <𝐼*>.\n'\nSince g₂ analysis can be corrected using cross-correlation between pulse trains, this implies\nthat the artifact appears consistently at certain pulse numbers within each train, likely due to\nthe AGIPD readout scheme. I believe such a behavior, if it drifts, also takes place on second or\nlonger timescales to allow this correction.\nThis suggests that the same cross-correlation value (1 + Δ) used in g₂ correction can be applied\nto correct the additional term in the speckle visibility. Therefore, subtracting Δ from\n𝛽 may provide a more accurate estimate of the true contrast. One important thing is\n!,#$%&'(%#)\nthat the pulse IDs (within a train) used to extract Δ from cross-correlation must match those\nused in the contrast analysis.\nOn the other hand, I believe this excess term does not affect the outcome of the g₂ fitting, which\nwas the original motivation behind my comment. Given that including or excluding the “time\nzero” point does not lead to a significant change in the fit, I find the scientific conclusions of\nthe manuscript to be well supported and coherent. I therefore agree that the manuscript is\nsuitable for acceptance."}, {"round": 3, "reviewer_comments": {"reviewer_1": "Referee 1\nI thank the reviewer for answer. However, there might be a misunderstanding in what I\nsuggested.\nFirst, in this shot noise dominated regime, the statistics are well described by the negative\nbinomial distribution, from which one can get the extracted contrast 𝛽0,extracted from a certain\nnumber of pixels and detector frames.\nSecond, the equation (2) in my previous comment only relies on the assumption that the\ndetector artifact is uncorrelated with the shot noise, thus equation (2) will work if we can\ncharacterize. Since g₂ analysis can be corrected using cross-correlation between pulse trains,\nthis implies that the artifact appears consistently at certain pulse numbers within each train,\nlikely due to the AGIPD readout scheme. I believe such a behavior, if it drifts, also takes\nplace on second or longer timescales to allow this correction.\nThis suggests that the same cross-correlation value (1 + Δ) used in g₂ correction can be\napplied to correct the additional term in the speckle visibility. Therefore, subtracting Δ from\nit may provide a more accurate estimate of the true contrast. One important thing is that the\npulse IDs (within a train) used to extract Δ from cross-correlation must match those\nused in the contrast analysis.\nOn the other hand, I believe this excess term does not affect the outcome of the g₂ fitting,\nwhich was the original motivation behind my comment. Given that including or excluding the\n“time zero” point does not lead to a significant change in the fit, I find the scientific\nconclusions of the manuscript to be well supported and coherent. I therefore agree that the\nmanuscript is suitable for acceptance."}, "author_response": "Comments to referee 1\nWe thank the referee for their comments. We agree with them, including or excluding the\n“9mezero” point does not lead to a significant change in the fit results with the fi>ng\nrou9ne we used since the contrast is fi?ed with the majority of the experimental\nparameters fixed, as described in the SI."}]

Decision letter