# The protein dynamical transition is independent of hydration

Johanna Kölbel  Department of Chemical Engineering, University of Cambridge,
Cambridge, CB3 0AS, UK  Current affiliation: School of Engineering, Brown
University, 184 Hope Street, Providence, RI 02912, USA  Moritz L. Anuschek
Department of Pharmacy – Center for Drug Research, Pharmaceutical Technology
and Biopharmaceutics, Butenandtstrasse 5, 81377 Munich, Germany  Current
affiliation: Novo Nordisk, Novo Nordisk Park 1, 2760 Måløv, Denmark  Ivonne
Stelzl  Department of Pharmacy – Center for Drug Research, Pharmaceutical
Technology and Biopharmaceutics, Butenandtstrasse 5, 81377 Munich, Germany
Current affiliation: Coriolis Pharma Research GmbH, Fraunhoferstraße 18b,
82152 Martinsried, Germany  Supawan Santitewagun  Department of Chemical
Engineering, University of Cambridge, Cambridge, CB3 0AS, UK  Wolfgang Frieß
Department of Pharmacy – Center for Drug Research, Pharmaceutical Technology
and Biopharmaceutics, Butenandtstrasse 5, 81377 Munich, Germany  J. Axel
Zeitler<EMAIL_ADDRESS>Department of Chemical Engineering, University of
Cambridge, Cambridge, CB3 0AS, UK

###### Abstract

Terahertz time-domain spectroscopy and differential scanning calorimetry were
used to study the role of the dynamics of biomolecules decoupled from solvent
effects. Lyophilised sucrose exhibited steadily increasing absorption with
temperature as anharmonic excitations commence as the system emerges from a
deep minimum of the potential energy landscape where harmonic vibrations
dominate. The polypeptide bacitracin and two globular proteins, lysozyme and
human serum albumin, showed a more complex temperature dependence. Further
analysis focused on the spectral signature below and above the boson peak. We
found evidence for the onset of anharmonic motions that are characteristic for
partial unfolding and molecular jamming in the dry biomolecules. The
activation of modes of the protein molecules at temperatures comparable to the
protein dynamical transition temperature was observed in the absence of
hydration. No evidence for Fröhlich coherence, postulated to facilitate
biological function, was found in our experiments.

##  I Introduction

The number of protein drugs in therapy is steadily increasing. These
biopharmaceuticals typically need to be administered by injection. Due to the
intrinsically limited stability of the molecules in aqueous solution, the
protein-drug products are frequently freeze-dried into an amorphous solid
matrix for storage and reconstituted immediately before use.

The molecules surrounding the protein molecules in solution constitute their
solvation shell (sometimes also called hydration shell); and the molecular
mobility of this shell affects the rates of conformational change, catalysis,
and protein/DNA-protein interactions [1, 2]. The misfolding propensity and the
pathways of protein aggregation depend on the protein’s local environment,
which is influenced by the solvent, water, as well as sugars, salts, metal
ions, and lipids that are part of the physiological environment or the
reconstituted formulation. Molecular crowding at high protein concentration,
pH and buffer also play a role [3, 4, 5, 6, 7, 8]. The predominant
intermolecular interactions that proteins form with their surrounding
environment are hydrogen bonds with water.

A recent paper highlighted the importance that the mobility of water plays in
protein dynamics and, ultimately, aggregation. Simulations in combination with
various experimental techniques focusing on the intrinsically disordered model
protein $\alpha$-synuclein (aSyn) showed that water mobility and aSyn mobility
are inextricably linked. Enhancing the water mobility reduces the propensity
of aSyn to aggregate [9].

The timescales of solvent motions and conformational changes in proteins
differ significantly. Solvent motions are rapid, occurring on the femtosecond
to picosecond timescale, whereas conformational changes in proteins happen on
the nanosecond to millisecond timescale. Still, solvent mobility affects
protein motions [10]. The coupling of water motions, the presence of ions, and
the protein dynamics are protein specific due to differing charge
distribution, hydrophobicity, and surface roughness [11].

Thus, protein unfolding in solution occurs via a complex pathway and the
properties of the hydration shell are of critical importance [12]. These
interactions are restrained in lyophilised samples with strongly reduced water
content. Conversely, complete unfolding in the solid state is usually not
observed as thermal decomposition occurs before sufficiently high temperatures
for unfolding are reached.

Instead of studying solvated biomolecules, here we removed all possible water
molecules from protein samples. This results in the molecules coming into
close contact but without aggregating, as was the subject of a previous study
by Stephens et al. [9]. We can thus study protein dynamics decoupled from
solvent effects. This is of fundamental interest, as well as of practical
importance, for lyophilisates of protein drugs [13].

Lyophilisates typically comprise cryo- and lyoprotectants, surfactants, and
the active biomolecule. The design of the products and the lyophilisation
process require good understanding of the underlying stabilisation mechanisms
of the formulation components [14, 13].

In this context, the residual water content in dried protein samples plays an
essential role at low temperatures. Lyophilised lysozyme samples with residual
hydration larger than $27\text{\,}\mathrm{\char
37\relax}\mathrm{(}\mathrm{m}\mathrm{/}\mathrm{m}\mathrm{)}$ water content
[15], well above the typical water content of lyophilisates of approximately
$1\text{\,}\mathrm{\char
37\relax}\mathrm{(}\mathrm{w}\mathrm{/}\mathrm{w}\mathrm{)}$ exhibit the so-
called protein dynamical transition (PDT), an increase in the mean square
displacement of molecules at a temperature of
$180\text{\,}\mathrm{K}220\text{\,}\mathrm{K}$ upon heating. The PDT that is
usually measured with neutron scattering is not to be confused with the glass
transition $T_{\text{g}}$ that occurs in amorphous samples, which is
characterised by a sudden change in relaxation times and heat capacity at
$T_{\text{g}}$ and is usually measured with DSC. [16] The PDT has been
observed at temperatures at around $200\text{\,}\mathrm{K}$ for different
lightly hydrated proteins and is thought to be due to the onset of motions
involving interactions of charged side chains with surrounding water
molecules. [17]

The PDT and the glass transition can both be described in terms of Goldstein’s
potential energy landscape (or surface, PES) containing many small minima
within larger basins [18, 19]. Moving between minima requires overcoming the
local, shallow activation energy barriers. With lower temperature, the
configurational entropy decreases, i.e. the number of available minima
decreases. Moving from one basin to the next requires a large amount of
activation energy and cooperative rearrangement of molecules. The protein
dynamical and the glass transitions can be understood as corresponding to
lowering energy barriers on that landscape. In the absence of hydration, a
temperature increase activates a different set of motions of activation
energies similar to the thermal energies supplied. This effect can be very
subtle.

Given the propensity of hydrogen and van der Waals bonding interactions in
solvated and lyophilised biopharmaceuticals, an ideal experimental method to
study such systems is terahertz time-domain spectroscopy, THz-TDS, due to the
match in photon and bonding energies [20]. In the terahertz range, the boson
peak (BP) and the coupling of dipoles to the vibrational density of states
dominate the absorption mechanisms [21]. The motions at terahertz frequencies
play an essential role in understanding solid-state protein dynamics. At
storage temperature (room temperature), a considerable number of vibrational
modes in the terahertz frequency range are active and contribute to the
formulation stability.

In the 1970s, Fröhlich suggested the existence of coherent vibrations at
around ${10}^{11}\text{\,}\mathrm{H}\mathrm{z}$ and postulated that such
motions enable biological functions, so-called biological control through
selective long-range interactions. Motions active at terahertz frequencies are
long-range motions, and the possible existence of a so-called Fröhlich
condensate in lyophilised protein formulations can hence be investigated. Thus
far no experimental evidence for such coherent states was found for
biomolecules in solution.

Vibrational confinement can dramatically reduce molecular mobility in
lyophilisates at temperatures close to room temperature and depends heavily on
the interaction between protein and excipients in the formulation [22]. Upon
increasing the temperature, the molecular mobility in the sample increases due
to the ability to access more of the local minimum in the PES until the free
volume is taken up entirely and the molecule becomes “jammed”. Any further
increase in mobility, and hence terahertz absorption, is no longer possible
until a higher energy barrier to another basin in the PES is overcome that is
associated with additional degrees of freedom for molecular motions.

In the spectral region accessible with most terahertz spectrometers, between
$0.3\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ and
$3.0\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$, the frequency-dependent infrared
absorption coefficient $\alpha(\omega)$ is theoretically related to the
reduced density of states $g(\omega)$ via
$\frac{g(\omega)}{\omega^{2}}\propto\frac{\alpha(\omega)}{\omega^{3}}$ [21].
The excess density of states, apparent in the reduced density of states, is
referred to as the boson peak (BP). The BP is a harmonic phenomenon due to
inherent disorder that anharmonic effects can obscure [23]. Utilising THz-TDS,
the onset temperature of molecular mobility was found to correlate with
anharmonic effects in the model glass-former glycerol. These anharmonic
effects resulted in an apparent shift of the BP centre frequency and could
thus be separated from purely harmonic contributions [21].

Markelz et al. [24] observed that in an amorphous sample, an increase in
absorption with temperature at a single frequency as measured with THz-TDS is
due to anharmonic effects, even at very low temperatures. While no actual PES
is perfectly harmonic, these effects are comparatively small at low
temperatures, e.g. the BP is not yet obscured or its apparent centre frequency
affected. We will hence refer to the temperature region below $T^{*}$, i.e. at
temperatures at which the BP is unaffected by anharmonic effects, as the
harmonic regime, and to the temperature region at which the BP is affected as
the anharmonic regime. This is expected to conceptually also apply to protein
samples.

In the present work, we investigated terahertz protein dynamics decoupled from
solvent effects in four different one-component lyophilised products, namely
sucrose (a widely used bulking agent, molecular weight
$0.34\text{\,}\mathrm{k}\mathrm{D}\mathrm{a}$), bacitracin (a polypeptide
antibiotic, molecular weight $1.4\text{\,}\mathrm{k}\mathrm{D}\mathrm{a}$),
lysozyme (a globular protein, molecular weight
$14.5\text{\,}\mathrm{k}\mathrm{D}\mathrm{a}$), and human serum albumin (HSA,
a globular protein and bulking agent, molecular weight
$66.5\text{\,}\mathrm{k}\mathrm{D}\mathrm{a}$), which are shown in Figure 1. A
possible Fröhlich condensate in lyophilised protein formulations would become
apparent by distinct spectral features emerging in the terahertz spectrum.

Figure 1: Visualisation of the four different key molecules of this study.

##  II Materials and Methods

### Materials

Sucrose (a commonly used cryo- and lyoprotectant, molecular weight
$0.34\text{\,}\mathrm{k}\mathrm{D}\mathrm{a}$) and HSA (globular model
protein, molecular weight $66.5\text{\,}\mathrm{k}\mathrm{D}\mathrm{a}$) were
purchased from Merck GmbH (Steinheim, Germany). Bacitracin (a polypeptide
antibiotic, molecular weight $1.4\text{\,}\mathrm{k}\mathrm{D}\mathrm{a}$) and
lysozyme (a globular protein, molecular weight
$14.5\text{\,}\mathrm{k}\mathrm{D}\mathrm{a}$) were purchased from Carl Roth
GmbH (Karlsruhe, Germany). The samples were prepared with highly purified
water (HPW; Sartorius Arium Pro, Sartorius, Göttingen, Germany) to reach a
total solid content of $10\text{\,}\mathrm{\char
37\relax}\mathrm{m}\mathrm{/}\mathrm{m}$ prior to lyophilisation.

### Lyophilisation

Lyophilisation stoppers (B2-TR coating, West) and DIN 10R vials (Fiolax,
Schott, Germany) were cleaned with highly purified water and dried at
$333\text{\,}\mathrm{K}$ for $8\text{\,}\mathrm{h}$. The vials were filled
with $3\text{\,}\mathrm{m}\mathrm{L}$ solution and subsequently semi-
stoppered. The product temperature in vials at different positions on the
shelf was recorded with a thermocouple. Formulations were freeze-dried
according to the protocol in Table 1 using an FTS LyoStar 3 freeze dryer (SP
Scientific, Warminster, Pennsylvania, USA). The end of primary drying was
controlled by comparative pressure measurement between a Pirani and MKS
sensor. The vials were stoppered after secondary drying under nitrogen
atmosphere at $800\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$ and
crimped with flip-off seals.

Table 1: Lyophilisation protocol.

Step | Ramp ($\text{\,}\mathrm{K}\mathrm{/}\min$) | Shelf temperature ($\text{\,}\mathrm{K}$) | Pressure ($\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{b}\mathrm{a}\mathrm{r}$) | Hold time ($\text{\,}\mathrm{h}$)
---|---|---|---|---
Freezing | 1.0 | 223 | 1 atm | 3
Primary drying | 0.5 | 253 | 60 | 50
Secondary drying | 0.4 | 323 | 60 | 5

### Differential Scanning Calorimetry (DSC)

The $T_{\text{g}}$ of the lyophilisates was determined with a DSC
821${}^{\text{e}}$ (Mettler Toledo, Giessen, Germany). 5 to
$10\text{\,}\mathrm{m}\mathrm{g}$ of crushed lyo cake were filled into
$40\text{\,}\mathrm{\SIUnitSymbolMicro L}$ aluminium crucibles (Mettler
Toledo, Giessen, Germany) under controlled humidity conditions
($\leq$$10\text{\,}\mathrm{\char 37\relax}$ relative humidity) and sealed
hermetically. The samples were heated from $280\text{\,}\mathrm{K}$ to
$415\text{\,}\mathrm{K}$ at a ramp rate of
$2\text{\,}\mathrm{K}\mathrm{m}\mathrm{i}\mathrm{n}^{-1}$. The $T_{\text{g}}$
was determined as the midpoint of the phase transition.

### Terahertz Time-Domain Spectroscopy (THz-TDS)

The lyophilised cake was broken up under a dry nitrogen atmosphere contained
within a glove bag (AtmosBag, Merck UK, Gillingham, UK), the powder was gently
mixed using an agate mortar and pestle and then pressed into a thin pellet
(thickness less than $800\text{\,}\mathrm{\SIUnitSymbolMicro m}$, diameter
$13\text{\,}\mathrm{mm}$) using a manual press (load $3\text{\,}\mathrm{t}$,
Specac Ltd, Orpington, UK). The pellet was sealed between two z-cut quartz
windows of $2\text{\,}\mathrm{m}\mathrm{m}$ thickness each and fixed to the
cold finger of a cryostat (ST-100, Janis, Wilmington, MA, USA).

Samples were analysed with a Terapulse 4000 spectrometer (Teraview Ltd,
Cambridge, UK) in transmission under vacuum (pressure
$<$20\text{\,}\mathrm{mbar}$$). Each sample and reference spectrum was
calculated from the co-average of 1000 waveforms which were acquired with a
resolution of $0.94\text{\,}{\mathrm{cm}}^{-1}$ and transformed to the
frequency domain via fast Fourier transform. The absorption coefficient was
calculated following the method by Duvillaret et al. [25]

At the beginning of each measurement, the sample was cooled down from room
temperature to $80\text{\,}\mathrm{K}$ and left to equilibrate for at least
$30\text{\,}\mathrm{min}$. The temperature was subsequently increased in steps
of $10\text{\,}\mathrm{K}$ up to a maximum temperature of
$440\text{\,}\mathrm{K}$. The system was allowed to equilibrate for
$8\text{\,}\mathrm{min}$ at each temperature increment before reference (two
z-cut quartz windows with no sample in between) and sample measurement.

##  III Results and discussion

###  III.1 Thermal analysis of $T_{\text{g}}$ by DSC

In DSC data a clear $T_{\text{g}}$ was only found for sucrose (at
$340\text{\,}\mathrm{K}$, shown in Figure 2). The peptide and proteins did not
show a clear step in heat capacity corresponding to $T_{\text{g}}$; instead, a
gradual decrease in heat flow was observed, potentially linked to structural
changes at elevated temperatures. The time scales of protein unfolding are
strongly temperature-dependent and it is possible that the mobility was
insufficient to maintain equilibrium between folded and unfolded states during
the DSC measurements [26]. In all three pure macromolecule samples, an
inflection point occurred, namely at $384\text{\,}\mathrm{K}$ (bacitracin),
$330\text{\,}\mathrm{K}$ (lysozyme), and $337\text{\,}\mathrm{K}$ (HSA).

Figure 2: DSC of sucrose, bacitracin, lysozyme, and HSA lyophilisates. A clear
$T_{\text{g}}$ was only found for sucrose. Inflection points for the other
samples were found at $348\text{\,}\mathrm{K}$ (bacitracin),
$330\text{\,}\mathrm{K}$ (lysozyme), and $337\text{\,}\mathrm{K}$ (HSA).

###  III.2 THz-TDS provides insights into molecular mobility

The THz-TDS spectra of the four materials show a similar profile (Figure 3).
The change in absorption coefficient with temperature was most pronounced for
sucrose. Figure 4 shows the absorption coefficient extracted at a frequency of
$1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ for different samples. The
absorption coefficient measures the change in dipole moments caused by inter-
and intramolecular motions in the sample of interest. As the temperature
increases, larger-scale motions become available, resulting in changes of the
dipole moments. Each sample is characterised by a distribution of states, each
with its own onset temperature. In smaller systems, like sucrose, the
distribution of states is narrower in temperature, and the total number of
states is lower than in more complex systems.

The rate of change in absorption with temperature increased at both transition
temperatures (see Figure 4). $T^{*}_{\text{g}}$ was found at
$340\text{\,}\mathrm{K}$, which agreed very well with the $T_{\text{g}}$
measured by DSC and literature values [27]. $T^{*}$, which cannot be measured
by DSC, was found at $230\text{\,}\mathrm{K}$.

In contrast, the changes in the absorption coefficient of the protein samples
can be very gradual with temperature. The PDT commonly occurs around
$200\text{\,}\mathrm{K}$. We therefore analysed the average rate of absorption
change in three temperature intervals: $T<$200\text{\,}\mathrm{K}$$,
$$200\text{\,}\mathrm{K}$<T<$300\text{\,}\mathrm{K}$$, and
$T>$300\text{\,}\mathrm{K}$$ (Table 2).

Table 2: Rate of absorption change with temperature of sucrose, bacitracin,
lysozyme, and HSA lyophilisates.

|  $\text{d}\alpha/\text{d}T$
(${10}^{-2}\text{\,}{\mathrm{cm}}^{-1}\text{\,}{\mathrm{K}}^{-1}$)
---|---
| Sucrose | Bacitracin | Lysozyme | HSA
$T<$200\text{\,}\mathrm{K}$$ | $2.9\pm 0.4$ | $2.7\pm 0.2$ | $3.5\pm 0.8$ | $2.9\pm 0.2$
$$200\text{\,}\mathrm{K}$<T<$300\text{\,}\mathrm{K}$$ | $7.1\pm 0.6$ | $4.6\pm 0.2$ | $9.0\pm 0.8$ | $5.5\pm 0.8$
$T>$300\text{\,}\mathrm{K}$$ | $15.2\pm 3.3$ | $3.0\pm 0.3$ | $5.3\pm 0.8$ | $1.9\pm 0.6$

Figure 3: Terahertz spectra for sucrose, bacitracin, lysozyme, and HSA
lyophilisates. Absorption mostly increases with temperature. Blue:
$80\text{\,}\mathrm{K}$, red: $420\text{\,}\mathrm{K}$.

It is known that the $T_{\text{g}}$ of sucrose decreases with increasing water
content [27]. The agreement between the $T^{*}_{\text{g}}$ we measured with
THz-TDS as well as DSC with literature values for dry sucrose matrices
indicates that the experimental setup and procedures ensure a very low water
content. Additionally in THz-TDS, any water molecules that could have
potentially adsorbed to the sample surface during preparation may be removed
by the vacuum (of $<$$20\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$) in
the measurement chamber.

Figure 4: Absorption at $1\text{\,}\mathrm{THz}$ for sucrose, bacitracin,
lysozyme, and HSA lyophilisates. The vertical line in the plot of sucrose
marks the $T_{\text{g}}$ as determined with DSC. Error bars are standard error
for $n$ measurements ($n=5$ for sucrose, $n=4$ for bacitracin, $n=3$ for
lysozyme and HSA).

Generally, the rate of change in absorption with temperature decreased at
temperatures above $300\text{\,}\mathrm{K}$ for the larger molecular weight
systems. This phenomenon was previously observed in other (more complex)
lyophilised formulations and attributed to high-temperature macromolecular
confinement [22]. We observed that the confinement effect depends on the shape
and size of the molecules. This effect cannot be observed in small organic
molecular systems like sucrose where only few degrees of freedom of dihedral
motion are available and hence it is not possible to reach a “jammed
conformation”. Bacitracin and lysozyme show a similar restriction of motions
to that observed in HSA, i.e. a change of the PES. The effect is slightly more
pronounced for lysozyme due to its increased size and hence higher number of
internal degrees of freedom. Between
$310\text{\,}\mathrm{K}330\text{\,}\mathrm{K}$, the absorption coefficient
does not increase and the overall absorption change becomes less above
$300\text{\,}\mathrm{K}$. At temperatures above $330\text{\,}\mathrm{K}$, the
absorption increases again with temperature, as is also the case for BSA
formulations measured previously [22]. This temperature corresponds to an
energy barrier of $2.7\text{\,}\mathrm{k}\mathrm{J}\,{mol^{-1}}$ (equal to
$0.65\text{\,}\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}\,{mol^{-1}}$), which is
significantly lower than the energy barrier of unfolding of BSA in solution,
which is on the order of
$64\text{\,}\mathrm{k}\mathrm{J}\mathrm{/}\mathrm{m}\mathrm{o}\mathrm{l}267\text{\,}\mathrm{k}\mathrm{J}\mathrm{/}\mathrm{m}\mathrm{o}\mathrm{l}$
[28].

The confinement is most pronounced in HSA, the largest macromolecule studied.
Here, the absorption coefficient even decreases slightly at temperatures
between $310\text{\,}\mathrm{K}360\text{\,}\mathrm{K}$. This decrease in
absorption could be because the molecules may lose some of the degrees of
freedom that they already gained at lower temperatures as the conformational
jamming increases once they become trapped in a steep minimum on the potential
energy landscape, as shown schematically in Figure 5. Even after the jammed
conformation is overcome, $\text{d}\alpha/\text{d}T$ in HSA is less than half
compared to lysozyme.

Figure 5: Schematic of a possible potential energy landscape topology; with
sufficient thermal energy, a sample can explore different hypersurface
configurations and can become trapped in shallow minima.

####  III.2.1 The low-frequency (boson peak) region

The BP can be visualised by plotting the absorption coefficient $\alpha$
divided by $\nu^{3}$ over the frequency $\nu$ (Figure 6), [21] and the maximum
value of $\alpha/\nu^{3}$ was found from smoothed data. If the BP occurs below
$0.3\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$, the data are not extrapolated,
and no maximum is reported to avoid extrapolation errors.

In small molecular systems, for example, in glycerol, the dynamics of glasses
upon heating usually fall into two regimes [29]. At very low temperatures, the
terahertz spectra are dominated by harmonic excitations. The BP itself is
harmonic and, therefore, a temperature-independent phenomenon. Its centre
frequency is constant. Above $T^{*}_{\text{g}}$, the system leaves the
harmonic minimum on the potential energy landscape. Shallower minima increase
the amount of anharmonicity and absorption. Once anharmonic effects dominate
at $T^{*}$, they lead to an apparent frequency shift of the BP maximum and
obscure it completely at temperatures close to $T^{*}_{\text{g}}$[21].

The maximum frequency of the BP of the protein lyophilisates is preserved in
the harmonic regime at temperatures below approximately
$150\text{\,}\mathrm{K}200\text{\,}\mathrm{K}$. However, it decreases in the
anharmonic regime with increasing temperature before being obscured by
anharmonic effects that appear to shift it outside the experimentally
accessible region. The BP in sucrose is the least pronounced of the samples
measured and appears to be masked by anharmonic effects already at very low
temperatures.

The BP occurs close to the Ioffe-Regel crossover frequency at which the mean-
free path of transverse waves becomes equal to their wavelength, meaning that
there is a crossover from wave-like to random-matrix-like physics. Once global
mobility sets in above the glass transition temperature, anharmonicity and
mobility increase with temperature until a “critical” mobility is reached,
completely obscuring the BP.

At very similar temperatures, the proteins reconfigure and get trapped in a
different conformation, decreasing mobility overall. Interestingly, the
inflection point observed in the DSC data coincides with the temperature
regime just above the initial trapping. Therefore, it is hypothesised that the
trapping and/or the conformational change inducing the trapping result in a
subtle heat capacity change with increasing temperature and that the maximum
in the DSC data corresponds to partial unfolding.

Figure 6: BP visualisation for sucrose, bacitracin, lysozyme, and HSA
lyophilisates at different temperatures. Boson peaks have been highlighted
with crosses.

####  III.2.2 Anharmonicity parameter $a$

The derivative parameter $a$ can be utilisied to characterise terahertz
spectra at frequencies above and below the BP more reliably. It reflects the
slope of the absorption spectra averaged over a specific frequency range [21].

We chose three different frequency ranges:
$0.35\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}0.55\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$
(below the BP),
$0.90\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}1.10\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$
(above the BP at the spectrometer’s highest SNR), and
$1.45\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}1.65\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$
(at even higher frequencies above the BP).

The lyophilisates show a markedly different behaviour at frequencies below and
above the BP (Figure 7). In the protein samples, the plateau just above room
temperature, seen in the absorption coefficient at
$1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ (Figure 4), can only be observed at
frequencies above the BP. At lower frequencies, the increase of $a$ with
temperature is monotonous. For bacitracin, that increase is approximately
linear with temperature, while for lysozyme and HSA, a subtle transition can
be observed at around $200\text{\,}\mathrm{K}$ and $300\text{\,}\mathrm{K}$,
respectively.

While the protein dynamical transition has thus far only been observed in
hydrated samples, these increases in $a$ point to the existence of thermally
activated modes that involve only the protein molecules themselves. Even
without solvent molecules, parts of the proteins retain some flexibility.

Given that the increase in $a$ appears in a similar temperature range, for
example, in lysozyme, it could be caused by side chain motions where the
activation energy is not affected by the presence of the solvent [24].

Sucrose shows a more pronounced increase of $a$ over temperature, and a
distinct increase is observed at $T^{*}_{\text{g}}$. This discontinuity is
expected as the glass transition temperature marks the onset of global
mobility.

Figure 7: Anharmonicity parameter evaluated in different frequency ranges for
sucrose, bacitracin, lysozyme, and HSA lyophilisates. Error bars are standard
error for $n$ measurements ($n=5$ for sucrose, $n=4$ for bacitracin, $n=3$ for
lysozyme and HSA).

####  III.2.3 Molecular confinement is observed in the spectrum at frequencies
beyond the BP

While the absorption coefficient at $1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$
is located relatively close to the BP maximum and is strongly influenced by
anharmonic effects, a different behaviour may be observed at higher
frequencies.

In the following we use a model developed by Chumakov et al. [29] to
investigate the effect of subtle spectral changes on the extrapolated
vibrational density of states (VDOS). This model was previously utilised to
investigate the model glass former glycerol and is now applied to more complex
biomolecules.

The exponential function $\alpha=A\nu^{3}\exp{(-\nu/\nu_{c})}+C$ is fitted to
the higher frequency part of the experimentally accessible spectrum. If that
function is plotted at even higher frequencies, a peak with a centre frequency
of $3\nu_{c}$ is observed. It has to be noted that this is a feature based on
the fitting function and not an accurate prediction of the centre frequency of
the VDOS. The experimental data available spans a frequency range from the
Ioffe-Regel crossover up to approximately
$2.3\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$, while the actual VDOS may
exhibit an underlying multi-peak structure [30] and show more features beyond
the peak itself [31].

Chumakov et al. found excellent agreement in the frequency range of
$1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}1.7\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$
in glycerol, where an exponential decay in the reduced density of states was
observed [29]. The reduced density of states and the absorption coefficient
measured with THz-TDS are related by a factor of $\nu^{3}$.

Figure 8: Extrapolated spectra at higher frequencies, for sucrose, bacitracin,
lysozyme, and HSA lyophilisates, fit to frequencies above the BP maximum.

The peak described by this model is narrowest and least intense in sucrose and
gets broader but more intense in the protein samples (Figure 8). Some
temperature dependence of the centre frequency is also apparent (Figure 9). In
all samples, the centre frequency of the VDOS decreases with increasing
temperature, thereby shifting the VDOS to lower frequencies and increasing the
absorption coefficient measured at the shoulder (e.g. at
$1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$). It is possible that the frequency
shift may follow a Bose-Einstein distribution as previously observed for
crystalline modes where thermal excitation was mediated by phonons populating
an anharmonic potential. A redshift of a mode is observed when phonons are
excited by sufficient thermal energy [32]. However, because the data are only
extrapolated, we refrain from fitting a model and will simply discuss it in
broader, qualitative terms. In future, it might be beneficial to measure
similar samples on a spectrometer with higher spectral bandwidth to be able to
extract more accurate data.

Figure 9: Extrapolated centre frequency for sucrose, bacitracin, lysozyme, and
HSA lyophilisates. Error bars are standard error for $n$ measurements ($n=5$
for sucrose, $n=4$ for bacitracin, $n=3$ for HSA). For the lysozyme
lyophilisate, two separate measurements are shown with error bars the
respective $95\text{\,}\mathrm{\char 37\relax}$ confidence intervals of the
fit.

For sucrose, the change in the centre frequency is pronounced above
$T^{*}_{\text{g}}$, whereas in bacitracin, the decrease is gradual over the
entire temperature range but slightly increased between
$180\text{\,}\mathrm{K}$ and $350\text{\,}\mathrm{K}$. The centre frequency of
lysozyme generally decreases with temperature below $300\text{\,}\mathrm{K}$
and is constant above. The centre frequency of HSA lyophilisates is constant
upon heating to a temperature of $210\text{\,}\mathrm{K}$, followed by a
decrease in $\nu_{\text{BP}}$ and reaching a minimum at
$340\text{\,}\mathrm{K}$, followed by a slight increase. In all cases, the
change in the centre frequency is less at temperatures at which confinement is
observed, indicating that the VDOS does not shift. If the molecules are
trapped in a conformation, one may assume that the vibrations are temperature
independent.

The higher frequencies seem to be more affected by the activation of modes at
around $200\text{\,}\mathrm{K}$. Modes at higher frequencies typically involve
a lower reduced mass than low-frequency vibrations. The effect on the higher
frequency modes, therefore, is an indication that the modes that become active
at around $200\text{\,}\mathrm{K}$ may involve side chains or functional
groups rather than the heavy protein backbone, which would influence the lower
frequencies instead. In this respect, using a frequency of
$1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ to analyse the systems is
beneficial as it provides insight into anharmonic effects, disappearance of
the BP at low temperatures, and activation of modes as well as jamming at
increased temperatures.

###  III.3 Fröhlich condensates

Fröhlich postulated the existence of coherent vibrations in the gigahertz to
terahertz range that help facilitate the biological function of biomolecules
[33, 34]. In terahertz spectra, such coherent modes appear as peaks in the
absorption spectrum that sharpen when the temperature is decreased.
Simultaneously, the absorption coefficient at neighbouring frequencies
decreases [35]. No such features were found in any measurement (Figure 3) Our
data hence show no evidence for Fröhlich coherence or the existence of a
Fröhlich condensate or other quantum effects. In protein solution, terahertz
vibrations are propagated by the solvent and coupled to dielectric
relaxations. This may lead to the fast dissipation of any postulated coherence
or localised states due to the similarity in their frequencies. In this case,
only extraordinarily rigid or well-ordered molecular structures would be
observable. However, we do not find such evidence for inherent coherent states
for the four molecules under investigation.

##  IV Conclusions

The importance of solvents and the solvation shell surrounding proteins for
their function is widely recognized. In the present work, we tried to obtain
insights into protein-protein interactions in the dry state analysing
lyophilisates of sucrose, bacitracin, lysozyme, and HSA by terahertz
spectroscopy.

The glass transition temperature, $T_{\text{g}}$, was identified in sucrose by
DSC, whereas $T_{\text{g}}$ could not be identified for bacitracin, lysozyme,
and HSA. However, THz-TDS demonstrated an increase in mobility with
temperature. An increase in temperature led to the activation of modes
involving only the protein molecules, resulting in an increase in the absolute
absorption coefficient and the anharmonicity parameter. The anharmonicity
parameter showed a markedly different behaviour below and above the BP centre
frequency. Utilising the theoretical model by Chumakov et al., the higher
frequencies were also evaluated, and a redshift of the VDOS was predicted.
Anharmonicity began to influence the spectra in sucrose at $T^{*}$ and
resulted in an apparent shift of the centre frequency of the BP. A further
increase in temperature and, thereby, mobility, led to the dissipation of the
BP. For the larger proteins lysozyme and HSA, jamming was observed at
increased temperatures after the dissipation of the BP and could only be
overcome by a further increase in temperature. Future experiments making use
of a higher- bandwidth spectrometer can investigate the impact of temperature
change on the VDOS. In the frequency range investigated, we could not find
evidence for the occurrence of Fröhlich coherence.

###### Acknowledgements.

All authors would like to thank Walter Schirmacher for insightful discussions
about the nature of the boson peak and theoretical aspects. JK thanks the
EPSRC Cambridge Centre for Doctoral Training in Sensor Technologies and
Applications (EP/L015889/1) and AstraZeneca for funding. MLA would like to
thank Erasmus+ for funding. All authors would like to thank the Cambridge-LMU
Strategic Partnership for funding. For the purpose of open access, the authors
have applied a Creative Commons Attribution (CC BY) licence to any Author
Accepted Manuscript version arising from this submission.

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