# Influence of Hydration and Dehydration on the Viscoelastic Properties of
Snail Mucus by Brillouin Spectroscopy

D. F. Hanlon<EMAIL_ADDRESS>M. J. Clouter  G. Todd Andrews  Department of
Physics and Physical Oceanography, Memorial University of Newfoundland and
Labrador, St. John’s, NL, Canada, A1B 3X7

###### Abstract

Brillouin spectroscopy was used to probe the viscoelastic properties of
diluted snail mucus at GHz frequencies over the range -11 ∘C $\leq T\leq$ 52
∘C and of dehydrated mucus as a function of time. Two peaks were observed in
the spectra for diluted mucus: the longitudinal acoustic mode of the liquid
mucus peak varies with dilution but fluctuates around the typical value of 8.0
GHz. A second peak due to ice remained unchanged with varying dilution and was
seen at 18.0 GHz and appeared below the dilutions “freezing” point depression.
Only a single peak was found in all the dehydrated mucus spectra and was also
attributed to the longitudinal acoustic mode of liquid mucus. Anomalous
changes in the protein concentration dependence of the frequency shift,
linewidth, and “freezing” point depression and consequently, hypersound
velocity, compressibility, and apparent viscosity suggest that the
viscoelastic properties of this system is influenced by the presence of water.
Furthermore, this research uncovered three unique transitions within the
molecular structure. These transitions included the first stage of
glycoprotein cross-linking, followed by the steady depletion of free water in
the system, and eventually resulted in the creation of a gel-like state when
all remaining free water was evaporated.

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##  I Introduction

In recent years, there has been a growing interest in exploring the mechanical
properties of polymer-water solutions, driven by their significance in
understanding the functional characteristics of proteins Bailey  _et al._
(2019); Comez  _et al._ (2012a, 2016); Lupi  _et al._ (2011). Previous
research however has primarily focused on rheological studies of natural snail
mucus Denny (1980, 1984). Recent studies on natural snail mucus used Brillouin
light scattering spectroscopy to investigate the effect of temperature on its
viscoelastic properties Hanlon  _et al._ (2023). The findings in that work
revealed that snail mucus exhibits increased viscoelastic properties, such as
the sound velocity, storage and loss moduli, as well as apparent viscosity, in
comparison to water. The study also discovered a phase transition in which the
viscous liquid state of snail mucus shifts to a condition in which both liquid
mucus and solid ice phases coexist. This work sheds light on the peculiar
behaviour of snail mucus under different temperature conditions. The results
of this study imply that the glycoprotein-water interaction is responsible for
the increased viscoelastic behaviour and phase change observed in snail mucus.
The results of this Brillouin scattering investigation, as well as prior
rheological experiments, clearly indicate that the cross-linked network of
glycoproteins in snail mucus is the major reason of the greater mechanical
properties reported in snail mucus Denny (1980, 1984); Hanlon  _et al._
(2023). The previous Brillouin light scattering study, while it does
investigate the temperature dependence of the system, it does not explore how
the glycoprotein concentration influences the mechanical properties of snail
mucus. While a number of studies in the past have explored the viscoelastic
properties of polymer-water systems, the impact of glycoprotein concentration
on the viscoelastic properties such as complex modulus, sound velocity, and
apparent viscosity still remains poorly understood Bailey  _et al._ (2019);
Comez  _et al._ (2012a, 2016); Lupi  _et al._ (2011).

Snail mucus is a polymer-water solution composed of long chains of high
molecular weight glycoproteins in water. These glycoproteins are known to to
cross-link in the mucus. Protein concentrations in these solutions were
previously reported in the literature to range from 3 - 7% proteins by weight,
with the remaining composition consisting of water Denny (1980, 1984); Verdugo
_et al._ (1987). The interaction between proteins and water plays a
significant role in the functionality of these polymer-water solutions and
holds important implications for various technological advancements. Active
research is being conducted in the field of hydrogels, which are being
explored for their potential as non-toxic medical adhesives Li  _et al._
(2017). A comprehensive understanding of the limitations associated with these
hydrogels is essential for their effective fabrication in medical
applications. Furthermore, the cryoprotectants field is actively exploring
polymer-water solutions, particularly antifreeze glycoproteins, as potential
substitutes for traditional cryoprotectants. These solutions exhibit the
capability to significantly reduce the freezing temperature, even at low
concentrations, thereby presenting exciting prospects for cryopreservation
applications Bouvet and Ben (2003); Kwan  _et al._ (2020). In recent years,
there has been a increased research focused on protein-water solutions to gain
a deeper understanding of the dynamics of water in the vicinity of proteins
with emphasis on understanding the interesting behavior of water molecules
surrounding proteins and their interactions Lupi  _et al._ (2011); Comez  _et
al._ (2012a, 2016); Monaco  _et al._ (2001); Ebbinghaus  _et al._ (2010).

In this paper, Brillouin light scattering spectroscopy was used to
characterize the temperature dependence of diluted snail mucus as function of
glycoprotein concentration over the range -11 ∘C to 52 ∘C. We report values
for sound velocity, storage (bulk) moduli, apparent viscosity, and hypersound
absorption for all dilutions as a function of temperature. Additionally, we
show the influence of hydration on the previously discovered freezing point
depression attributed by the presence of glycoproteins Hanlon  _et al._
(2023). Furthermore, we investigate the influence dehydration has on the
viscoelastic properties of snail mucus by allowing the mucus to naturally
dehydrate over the course of $\sim$ 400 hours. The results of these two
experiments build upon previously published data on the viscoelastic
properties of natural snail mucus, offering new insights into the physics of
aqueous glycoprotein solutions and shedding light on the role of glycoproteins
in the phase behavior of these systems Hanlon  _et al._ (2023). In a wider
context, the present study enhances our understanding of phonon dynamics in
complex fluids, unraveling the complex interplay between water and
biomacromolecules, and illuminating the significance of glycoproteins in
biological systems.

##  II Methodology

###  II.1 Brillouin Light Scattering Spectroscopy

####  II.1.1 Brillouin Scattering in Liquids

Brillouin light scattering spectroscopy is an inelastic light scattering
technique which measures random thermal density fluctuations in liquids. The
intensity of light inelastically scattered by aqueous solutions given by
Iq($\omega$), with exchanged wave vector $q$, is dominated by density
fluctuations in the medium, which can be derived from the theoretical
framework of hydrodynamics Comez  _et al._ (2012b); Montrose  _et al._ (1968);
Tao  _et al._ (1992). The intensity can be represented with the following
expression,

$I_{q}(\omega)\propto
S_{q}=\frac{S_{q}M_{0}}{\pi\omega_{B}}\frac{M^{\prime\prime}}{[M^{\prime}-\rho\omega_{B}^{2}/q^{2}]^{2}+[M^{\prime\prime}]^{2}},$
(1)

where $S_{q}$ is the static structure factor, $\rho$ is the mass density,
$M_{0}$ is the relaxed (zero-frequency) longitudinal acoustic modulus, $q$ =
(4$\pi/\lambda$)$n\sin(\theta/2)$, $n$ is the refractive index of the medium,
(here assumed constant at $n=1.34$ Gugliando  _et al._ (2021)) and $\theta$ is
the scattering angle. Additionally, $\omega$ is the angular frequency of the
density fluctuations, and $\omega_{B}$ is the angular frequency of the density
fluctuations obtained through Brillouin scattering. The complex longitudinal
modulus, $M(\omega)=M^{\prime}(\omega)+iM^{\prime\prime}(\omega)$, often
referred to as the complex modulus, can be obtained directly via Brillouin
spectroscopy using the following relations Bailey  _et al._ (2019),

$M=M^{\prime}+iM^{\prime\prime}=\frac{\lambda_{i}^{2}\rho}{4n^{2}}f_{B}^{2}+i\frac{\lambda_{i}^{2}\rho}{8\pi
n^{2}}f_{B}\Gamma_{B},$ (2)

where $\Gamma_{B}$ is the full width at half maximum (FWHM) and $M^{\prime}$
is referred as storage modulus and is a measure of how much energy is stored
elastically in the system. The loss modulus, $M^{\prime\prime}$, is a measure
of how much energy is lost through heat in the system. $M^{\prime}$ and
$M^{\prime\prime}$ can be expressed in terms of sound velocity $v$ and
apparent viscosity $\eta=4\eta_{s}/3+\eta_{b}$, where $\eta_{s}$ and
$\eta_{b}$ are the shear and bulk viscosity, respectively, can be represented
by the following,

$M^{\prime}=\rho v^{2}$ (3)

and

$M^{\prime\prime}=2\pi\eta f_{B}$ (4)

For a $180^{\circ}$ backscattering geometry used in the present work,
application of energy and momentum conservation to the scattering process
reveals that the phonon velocity, $v$, and frequency shift of the incident
light, $f$, are related by

$v=\frac{f\lambda_{i}}{2n},$ (5)

where $n$ is the refractive index of the target material at the incident light
wavelength $\lambda_{i}$.

The spectral width of the Brillouin peaks is determined by the time that each
fluctuation interacts with the incident light, thus measuring its lifetime
(thermal relaxation) or the attenuation of these fluctuations Dil (1982).
Using both the frequency shift and FWHM, the frequency-independent sound
absorption coefficient can be calculated using the following Rouch  _et al._
(1976),

$\frac{\alpha}{f^{2}}=\frac{\Gamma_{B}}{2vf_{B}^{2}}.$ (6)

where, $f_{B}$, and $\Gamma_{B}$ are directly obtained from the Brillouin
spectra, and $v$ is is calculated from Eq. 5.

####  II.1.2 Apparatus

Brillouin spectra were obtained using a $180^{\circ}$ backscattering geometry
using the experimental setup described previously in Ref. Andrews (2018). A
Nd:YVO4 single mode laser provided light of wavelength $\lambda_{i}=532$ nm
and power of 100 mW, which was directed onto the samples inside a temperature-
controlled sample chamber described in great detail previously Hanlon  _et
al._ (2023). A high-quality anti-reflection-coated camera lens of focal length
$f=5$ cm and $f/\\#=2.8$ served to both focus incident light onto the sample
and to collect light scattered by it. After exiting this lens, the scattered
light was focused by a 40 cm lens onto the 450 $\mu$m-diameter input pinhole
of an actively-stabilized 3+3 pass tandem Fabry-Perot interferometer (JRS
Scientific Instruments). The interferometer had a free spectral range of 30
GHz, and possessed a finesse of approximately 100. The light transmitted by
the interferometer was directed onto a pinhole with a diameter of 700 $\mu$m
and subsequently detected by a low-dark count ($\lesssim 1$ s-1)
photomultiplier tube where it was converted to an electrical signal and sent
to a computer for storage and display.

####  II.1.3 Spectral Peak Analysis

Peak parameters including frequency shifts, and linewidths (FWHM), were
obtained by fitting Lorentzian functions to the Stokes and anti-Stokes peaks
and averaging the resulting parameters of best-fit. To obtain spectral
linewidths, the instrumental linewidth of 0.3 GHz was subtracted from FWHM
values obtained from the fits. Estimated uncertainties in peak parameters were
also obtained from the uncertainty in the fits.

###  II.2 Attenuated Total Reflectance IR Spectroscopy

####  II.2.1 Theory

Fourier Transform Infrared (FTIR) spectroscopy, a type of vibrational
spectroscopy, is a technique used to obtain an infrared spectrum of absorbance
or emission of a solid, liquid or gas Griffiths (1983). This technique allows
for the identification and characterization of various chemical compounds
which makes it useful in understanding the structure of complex liquid systems
and is often used in conjunction with other spectroscopy techniques in order
to characterize different samples Arunkumar  _et al._ (2019); Lewis  _et al._
(2013); Spiridon  _et al._ (2011); Griffiths (1983). The technique employs a
Michelson interferometer to acquire IR spectra of the material. FTIR data is
commonly gathered through an attenuated total reflectance (ATR) method. The
ATR technique utilizes a crystal prism, such as diamond or zinc selenide, in
direct contact with the sample Grdadolnik (2002). In this method, the infrared
radiation is directed onto the prism at a precise angle, subsequently
undergoing multiple internal reflections by the prism before being ultimately
absorbed by the sample Grdadolnik (2002).

####  II.2.2 Apparatus

All data was collected using a Vertex 70v vacuum Fourier transform infrared
(FTIR) spectrometer (Bruker, Billerica, MA, USA) with Platinum Diamond
attenuated total reflectance (ATR) attachment and a globar (blackbody) light
source. Data was collected using a KBr beam splitter (4000–400 cm-1) with 2
cm-1 resolution.

Before each measurement, a background spectra was first collected to calibrate
the system. Each sample was collected by using a pipette to extract the mucus,
which was then directly placed onto the ATR crystal. Approximately 0.5 ml of
mucus was used for each sample to ensure complete coverage of the crystal
surface.

####  II.2.3 Spectral Analysis

The data collection procedure involved initially collecting FTIR spectra on
ultra-pure water, as depicted in Fig. 1. This spectra served as a “baseline”
for all subsequent spectra obtained for diluted and dehydrated mucus. ATR
spectra initially collected on pure snail mucus and showed very good agreement
with spectra previously obtained for gastropod mucus. Furthermore, ATR spectra
were collected on all diluted mucus samples. As for the dehydrated mucus,
spectra were only collected after a noticeable change in Brillouin data was
observed. Additionally, the presence of proteins in the snail mucus is
indicated by the arrow in Fig. 1.

FTIR data in this work are displayed according to the common convention of ATR
intensity versus wave number, where the wave numbers are displayed with
decreasing values from left to right. The concentration of pure mucus, diluted
and dehydrated mucus was estimated by comparing the integrated intensity of
water absorbance bands to the same bands visible in spectra of diluted and
dehydrated mucus. Figure 1 shows FTIR spectra of ultra-pure water and pure
mucus. Both spectra show three prominent absorbance bands $\nu_{1}$, $\nu_{2}$
and $\nu_{3}$ (see Fig. 1 due to the symmetric O-H stretching, O-H bending
(scissors) vibration and a combination band of $\nu_{2}$ plus an additional
motion of molecule. These absorbance bands act as a fingerprint for water and
can help in identifying the presence of water in various systems.
Additionally, we see the presence of absorbance bands likely due to the
proteins, as indicated by the arrow in Fig. 1.

####  II.2.4 Concentration Determination

One of the primary objectives of utilizing FTIR in this study was to estimate
the concentration of water and proteins within the snail mucus. Prior research
suggests that an accurate concentration can be determined by comparing the
integrated intensity of the spectral water peak with that of an unidentified
peak of some system containing water Venyaminov and Prendergast (1997). For
example, Fig. 1 shows very similar ATR spectra, and by comparing the
absorbance bands of water, with those peaks at the same position in the mucus
we can determine the concentration of protein and water present. The intensity
of these absorbance peaks gives a sense of the abundance of molecules
absorbing the IR radiation Larkin (2017). To obtain the integrated intensity,
a Gaussian profile was used to fit to the absorbance bands shown in Fig. 2 due
to its ability to properly fit the bands of cross-linked networks Bradley
(2007) (see Table S1 in Supplementary file for values extracted from ATR
spectra).

In general, the shifts of all vibrational modes remained relatively constant.
The FWHM for the $\nu_{1}$ generally decreases with decreasing concentration,
$\nu_{2}$ remains constant for most of the concentrations, and $\nu_{3}$
increases with increasing concentration for those spectra where the peak is
observed. The most obvious changes in Fig. 2 is with the spectral intensity.
The vibrational mode $\nu_{1}$ is the most intense throughout all spectra, and
shows a general decrease from ultra-pure water to the spectra containing 45%
water. The intensity of the $\nu_{2}$ vibrational mode is relatively constant,
showing only slight variations between 15-40 counts over the whole range
studied. As well, the intensity of the vibrational mode at 2130 cm-1 remains
constant throughout.

####  II.2.5 Density Determination

The density of each diluted and dehydrated mucus solution was determined using
water and protein concentrations determined from ATR analysis and the
following expression for the density of a two-component solution Bailey  _et
al._ (2019):

$\rho=\frac{m_{W}+m_{M}}{\frac{m_{w}}{\rho_{w}}+\frac{m_{M}}{\rho_{M}}},$ (7)

where $m_{W}$ and $m_{M}$ denote the mass of water and dry mucus,
respectively, and $\rho_{W}=0.997$ g/cm3 O’Connor and Schlupf (1967) and
$\rho_{M}=1.571$ g/cm3 are the corresponding mass densities. The dry mucus
density $\rho_{M}$ was obtained from Eq. 7 using the known density of natural
snail mucus ($\rho=1.040$ g/cm3) determined in a previous study Hanlon  _et
al._ (2023) and compositional information obtained through ATR analysis.

Figure 1: ATR spectra collected on ultra-pure water and pure snail mucus at
room temperature. Figure 2: ATR-FTIR spectra of natural snail mucus. ATR
spectra collected on diluted snail mucus are indicated by the superscripts
$a-d$. The remaining spectra were collected on dehydrating mucus as a function
of time.

##  III Dilution Experiments

###  III.1 Sample Preparation

The diluted snail mucus samples utilized in this study were prepared by
combining natural snail mucus with ultra-pure water. To prevent dehydration,
the mucus was initially stored in a sealed container. Using a syringe, 1 ml of
mucus was extracted from the parent container and transferred to a glass
sample cell that was sealed at one end. Ultra-pure water was added to the cell
in 0.5 ml increments for each dilution. After water was added, the mucus-water
solution was stirred with a stirring stick for 10 minutes to ensure proper
mixing. This standardized procedure was followed for all dilutions to maintain
consistency.

Table 1: Sample descriptions for Diluted Snail Mucus | Dilution #1 | Dilution #2 | Dilution #2 | Dilution #4
---|---|---|---|---
Total Volume (mL) | 1.5 $\pm$ 0.1 | 2.0 $\pm$ 0.1 | 2.5 $\pm$ 0.1 | 1.5 $\pm$ 0.1
Protein Concentration (wt %) | 5 $\pm$ 0.7 | 4 $\pm$ 0.6 | 3 $\pm$ 0.5 | 2 $\pm$ 0.2
Water Concentration (wt %) | 95 $\pm$ 0.7 | 96 $\pm$ 0.6 | 97 $\pm$ 0.5 | 98 $\pm$ 0.2
Density (g/cm3) | 1.027 $\pm$ 0.007 | 1.020 $\pm$ 0.005 | 1.017 $\pm$ 0.005 | 1.013 $\pm$ 0.003

Following the mixing of solutions, the samples were left undisturbed to
equilibrate for 6 hours. Brillouin spectra were collected as a function of
temperature after each dilution. It should be noted that the glass sample cell
had physical limitations, and the maximum amount of water that could be added
was 2.5 ml. Additionally, an alternative sample was prepared by initially
adding 0.5 ml of mucus first and then adding 1.5 ml of water. Once each sample
was prepared, the glass sample cell was placed into the previously discussed
temperature-controlled sample chamber Hanlon  _et al._ (2023).

###  III.2 Results: Raw Spectral Signatures

Figure 3: Temperature dependence of Brillouin spectra for diluted gastropod
mucus with 5 wt% protein and 95 wt% water (Dilution #1 in Table 1). Direction
of decreasing temperature is indicated by the large arrow second from left. M
and Ih identify peaks due to mucus and ice, respectively.  Figure 4: Room
temperature Brillouin spectra (Anti-Stokes) of all diluted gastropod mucus
samples. Arrow represents direction of increasing protein concentration.

Figure 3 shows representative series of Brillouin spectra collected on diluted
snail mucus for temperature ranging from -11.0∘C - 50.0∘C. Figure 4 shows the
anti-stokes Brillouin peak for liquid mucus for all dilutions at room
temperature displaying the general difference in spectra for each dilution. As
evident from Fig. 4, as the dilution increases, the Brillouin shift decreases.

Figure 5: Plot of frequency shift, and FWHM of diluted snail mucus samples as
a function of temperature.

In all dilutions there were two sets of Brillouin peaks present in the
spectra, one of which occured at a shift of $\sim$ 8.0 GHz which decreased to
a value similar to that of water as the dilution increased. This peak was
present over the entire temperature range for all dilutions, however the
spectral intensity of this peak decreased with increasing dilution for
temperatures below the corresponding “freezing” point. A second peak occured
at approximately 18.0 GHz and has previously been deemed due to the
longitudinal acoustic modes of ice Ih Hanlon  _et al._ (2023). Throughout this
manuscript, the term “freezing” point is to indicate the point at which this
“ice peak” appears in the spectrum.

Figure 5 shows the frequency shift as a function of temperature for all
dilutions. For all dilutions there is a slight increase in the frequency shift
for increasing temperatures greater than its freezing point. Below their
respective freezing points however, dilutions 1-3 (see Table 1) all show a
rapid increase in frequency shift with decreasing temperature. Dilution 4 on
the other hand shows that below its ”freezing” point, the frequency shift
follows a similar trend to that of pure water, and decreases with decreasing
temperature. This result suggests that there is a transition occurring in
dilution#4 that is disrupting the network of glycoproteins and water molecules
present in snail mucus. This is further supported by the difference in
frequency shift between the most dilute (Dilution #4) and least dilute
(Dilution #1). This shows that for higher protein concentration (Dilution #1),
the frequency shift resembles that of pure mucus Ref. Hanlon  _et al._ (2023),
while the lowest protein concentration (Dilution #4) resembles that of ultra
pure water Ref. Hanlon  _et al._ (2023). Previous concentration dependent
Brillouin studies on polymer gels shows a similar trend, that is at very low
polymer concentrations the frequency shift is comparable to pure water, and at
high concentrations the frequency shift is drastically different Zhao and
Vanderwal (1997); Scheyer  _et al._ (1997); Pochylski  _et al._ (2006).

Figure 5 shows the FWHM as a function of temperature for all dilutions which
for T $\geq$ 0 ∘C decreases with increasing temperature. Below, this
temperature the FWHM in general increases. The inset figure shown in Fig. 5
shows the FWHM for temperature region T $\geq$ 0 ∘C. In this temperature
region, the FWHM for all dilutions shows slight variations between one another
which become more apparent towards T = 0 ∘C, and practically identical near T
= 50.0 ∘C. In general, the FWHM decreases with increasing dilutions, this is
best illustrated by the series of values at T = 5.0∘C. This is consistent with
the observation in the frequency shift data, that is as the dilution
increases, the FWHM approaches values close to ultra pure water. Previous
Brillouin scattering studies on aqueous polymer solutions had shown that
polymer concentration has a significant influence on the FWHM Bailey  _et al._
(2019); Van Den Bulcke  _et al._ (2000); Pochylski  _et al._ (2006). In fact,
Ref. Bailey  _et al._ (2019) had stated that the polymer concentration has the
largest influence on the Brillouin linewidth. Although the concentrations for
this dilutions experiments are similar, there is a noticeable change in the
FWHM between each dilution.

Following from the Brillouin frequency shift and linewidth, values of
hypersound velocity, viscosity and sound absorption can be directly calculated
using the raw spectral signatures obtained from Brillouin spectra.

###  III.3 Results: Viscoelastic Behaviour

####  III.3.1 Sound Velocity

Figure 6 shows the sound velocity for diluted mucus as a function of
temperature and was calculated using Eq.. 5. From examining the hypersound
velocity obtained by Brillouin scattering can provide valuable insights into
the viscoelastic properties within the diluted snail mucus samples. In
general, the sound velocity exhibits interesting variations with temperature
across the dilutions. At temperatures above the freezing point of each
dilution, there is a gradual increase in hypersound velocity as the
temperature increases. This behavior suggests a typical thermal expansion,
where the molecules within the mucus exhibit increased vibrational energy and
contribute to the propagation of sound waves at higher velocities. However,
below their respective freezing points, dilutions 1-3 illustrate a pronounced
increase in sound velocity with decreasing temperature. This notable change
implies a structural transformation in the snail mucus network, likely induced
by addition of ice in mucus. The transition occurring in these dilutions
affects the arrangement and interaction of glycoproteins and water molecules,
leading to an alteration in the sound velocity. In comparison to the other
dilutions, dilution 4 demonstrates a distinct variation in sound velocity at
temperatures below its freezing point. The sound velocity in dilution 4
decreases with decreasing temperature, similar to how it does in pure water.
This finding implies that a phase transition is observed in dilution 4 which
indicate a break in the network of glycoproteins and water molecules in snail
mucus.

Figure 6: Hypersound velocity, apparent viscosity, and sound absorption for
diluted mucus as a function of temperature.

####  III.3.2 Apparent Viscosity

Figure 7: Natural logarithm of apparent viscosity of diluted snail mucus
samples versus temperature. Solid lines: Best fits of Eq. 8 to high
temperature data. Inset: Activation energy $E_{a}$ obtained from the linear
fits as a function of protein concentration.

Figure 6 shows the apparent viscosity for the diluted mucus sample as a
function of temperature. In general, for T $\geq$ 0∘C the apparent viscosity
decreases with increasing temperature for all dilutions. Although it is hard
to tell from Fig. 6, there is a systematic decrease in viscosity with
increasing dilution. Below T = 0∘C, the apparent viscosity does not show any
consistent behaviour, likely due to the presence of ice crystallites in this
region.

Fig. 7 shows the natural logarithm of the apparent viscosity of diluted snail
mucus as a function of temperature calculated directly from Eq.. 2 using the
density (see Table 1) and FWHM. The viscosity decreases with increasing
temperature in a similar manner to what has been previously reported for snail
mucus along with other polymer-water systems So  _et al._ (1994, 1995);
Esquivel-Sirvent  _et al._ (1993, 1994).

In the high temperature region (25∘C $\leq$ T $\leq$ 50∘C), $\ln(\eta)$
depends linearly on $1/T$ so we fit an Arrhenius relationship of the form

$\eta=\eta_{0}e^{E_{a}/k_{B}T}$ (8)

to this data and extracted the activation (enthalpy) energy $E_{a}$. Here,
$\eta_{0}$ is a prefactor that contains the entropic contribution to the
viscosity and $k_{B}$ is the Boltzmann constant Lupi  _et al._ (2011). The
resulting values for the activation energy $E_{a}$ of all diluted mucus have
values of $\sim$ 13.3 $\pm$ 0.3 kJ/mol (see Table 2). Considering the data
shown in Fig 7, dilutions 1 - 3 all exhibit similar values of the natural
logarithm of apparent viscosity as the temperature decreases. However, the
value for dilution 4 is approximately 2% smaller than the other dilutions and
is very similar to the value of water previously reported Hanlon  _et al._
(2023). This observed deviation we see in the natural logarithm of apparent
viscosity between dilutions 3 and 4 is an indication of a structural change
occurring in the mucus at the low protein concentration.

Because the activation energy Ea is consistent across all dilutions, as well
as prior findings for snail mucus, the change in apparent viscosity was
attributed to the pre-entropic component $\eta_{0}$ Hanlon  _et al._ (2023).
However, that study did not investigate the influence of protein concentration
on the apparent viscosity or the activation energy. The concentration
dependence of activation energy for polymer-water solutions previously studied
displayed the same findings presented here Lupi  _et al._ (2011); Comez  _et
al._ (2016); Monaco  _et al._ (2001); Comez  _et al._ (2012a). That is, as
concentration increases, the apparent viscosity increases, but $E_{a}$ stays
the same for the high temperature region (T $\geq$ 20∘C). The results shown
here further suggest that there is a slowing down of water dynamics near the
protein as opposed to a strengthening in the H bonds present, and this is
attributed a decrease in configurational entropy. Otherwise, we should see an
increase in the slope of Fig. 7. This entropy, as mentioned previously, is
included in the pre-factor $\eta_{0}$. This result is consistent with previous
studies that conduct similar Brillouin studies where the activation energy was
determined for a number of different concentrations of diluted water–tert-
butyl alcohol solutions Lupi  _et al._ (2011). Like the work presented in this
manuscript, Ref. Lupi  _et al._ (2011) also showed that the activation energy
Ea for all dilutions had the same value.

The intermediate temperature region (-2.5∘C $\leq$ T $\leq$ 25∘C) displays a
non-Arrhenius behaviour as is apparent in Fig. 7. Previously it was stated
that the reasoning behind this could be due to the onset of cooperative
motions at molecular level associated with a power law, or a Vogel–Fulcher-
Taumann global dependence of viscosity Lupi  _et al._ (2011). Previous results
obtained for snail mucus suggested that a factor that could be influencing the
behaviour seen here could be attributed to the difference in density
fluctuation relaxation times and structural relaxation times Lupi  _et al._
(2011); Comez  _et al._ (2012a).

Additionally, in the region below the freezing point, we can only say that the
likely rapid increase in $E_{a}$ is due the coexistence of ice and mucus,
which has previously been reported Hanlon  _et al._ (2023).

Table 2: Best-fit parameters for fit of function $\displaystyle\ln(\eta)=\ln\eta_{0}+E_{a}/k_{B}T$ to experimentally determined apparent viscosity. Temperature | Dilution | $\ln\eta_{0}$ | $E_{a}$ | R2
---|---|---|---|---
Range | # | (cP) | kJ/mol |
| 1 | -5.2 | 13.1 $\pm$ 0.2 | 0.992
25.0∘C $\leq T\leq$ 52.0∘C | 2 | -5.4 | 13.3 $\pm$ 0.3 | 0.993
| 3 | -5.1 | 13.3 $\pm$ 0.3 | 0.994
| 4 | -5.2 | 13.3 $\pm$ 0.2 | 0.989

####  III.3.3 Hypersound Attenuation

Figure 6 presents the frequency-independent hypersound absorption coefficient,
obtained through Eq. 6, as a function of temperature for diluted mucus. At
temperatures above $T=0$ ∘C, the absorption coefficient generally decreasing
with increasing temperature. Conversely, as the temperature decreases, the
absorption coefficient tends to increase. However, it is important to note
that the situation becomes more complex below $T=0$ ∘C due to the presence of
ice crystallites in the system. The presence of these crystallites introduces
scattering effects that are not accounted for in the sound absorption formula.
The observed decrease in sound absorption with increasing temperature provides
valuable insights into the dynamic behavior of the glycoprotein-water
structure. It suggests that temperature has an influence on the acoustic
properties of the system, indicating there are structural changes occurring in
the system, although these changes are likely to be minimal over the dilution
range studied. This implies that the dilution of the system, within the
investigated range, is not heavily influenced by the addition of water to the
system. Other factors, such as the composition and concentration of
glycoproteins, or the presence of other impurities may have a stronger
influence of the sound absorption.

####  III.3.4 Storage & Loss Modulus

Figure 8: Room temperature Brillouin spectra collected on natural snail mucus
dehydrating as a function of time. Arrow indicates direction of increasing
time.

Figure 8 shows the storage modulus calculated using the first term in Eq. 2
for all dilutions as a function of temperature. In general, for $T\geq 0$ ∘C
the storage modulus shows a gradual increase in value for all dilutions with
increasing temperature. Below $T=0$ ∘C, the storage modulus for dilutions 1-3
behave similarly, all showing a rapid increase with decreasing creasing
temperature. However, the storage modulus for dilution 4 begins to decrease
with decreasing temperature, similar to the behaviour previously observed for
water Hanlon  _et al._ (2023).

Figure 8 also shows the loss modulus calculated used the second term in in Eq.
2 for all dilutions as a function of temperature. The loss modulus exhibits a
general decrease in value for $T\geq 0$ ∘C for all dilutions. Furthermore, the
loss modulus for dilutions 1-3 all increase with decreasing temperatures for
$T\leq 0$ ∘C. Dilution 4 also increases with decreasing temperature, but we
have limited data for this dilution below 0 ∘C liquid mucus Brillouin peaks
became very weak.

####  III.3.5 Freezing Point Depression

Figure 9: Plot of freezing point depression as a function of protein
concentration. Dashed curve: Best fit of Eq. 9 to the experimental data.

Figure 9 illustrates the relationship between protein concentration (wt %) and
freezing point depression. The observed behavior aligns with the Langmuirian
adsorption model, previously used to explain the adsorption of antifreeze
glycoproteins onto ice crystal surfaces Burcham  _et al._ (1986). This model
can be described by the following expression,

$\Delta T=\frac{\Delta T_{m}Kc}{1+Kc}$ (9)

where, $\Delta T$ represents the freezing point depression, $\Delta T_{m}$ is
the maximum freezing point depression, $K$ denotes the intensity of
interaction between glycoprotein and ice, and $c$ represents the
concentration. Although the Langmuir model is typically used to describe the
adsorption of molecules onto solid surfaces, we find it suitable for our
system due to its ability to characterize and consider solute-solvent
interactions. Consequently, we fit Equation 9 to our freezing point depression
data, and the resulting fit is displayed in Figure 9. The fitting yields
$\Delta T_{m}=-5.85^{\circ}$C and $K=0.123$ (wt%-1).

The trend that we see here in the freezing point depression is similar to
previously studied sugar and water solutions and also similar to previous
antifreeze glycoprotein studies URAJI  _et al._ (1996); Ebbinghaus  _et al._
(2010). Although the relationship between the freezing point shown here for
snail mucus and the sugar solutions are similar, the concentrations required
for sugars to deplete ice growth required much higher sugar concentrations to
depress the freezing point by a similar amount to that of snail mucus. In
fact, it took nearly five times as much sugar as it did glycoproteins to
depress the freezing point to a value equal to that in snail mucus URAJI  _et
al._ (1996). The freezing point depression, previously reported for antifreeze
glycoproteins showed an overall increase as the concentration increased. As
well, the magnitude of the increase was much higher for lower concentrated
solutions similar to the snail mucus. This is an indication that these
glycoprotein systems influence the system in a similar way.

##  IV Dehydration Experiments

###  IV.1 Sample Preparation

The dehydrated mucus was prepared by initially adding $\sim 2$ ml of natural
snail mucus to a clear cuvette. The snail mucus was left unsealed at one end
and was allowed to dehydrate while room temperature Brillouin spectra was
collected. Temperature dependent Brillouin spectroscopy could not be completed
due to the current limitations with the temperature-controlled sample chamber.

###  IV.2 Results: Raw Spectral Signatures

Figure 10 shows room temperature Brillouin spectra collected on natural snail
mucus as a function of time. This series of spectra while although it is a
function of time, it is inherently a function of protein concentration as
determined by the FTIR data (see Fig 2). Only a single peak was found in all
spectra collected for the dehydrating mucus. However, preliminary dehydration
experiments had shown that for a completely dried snail mucus sample, there
was the presence of additional peaks. For the purpose of this manuscript this
will not be examined here.

Figure 10: Room temperature Brillouin spectra collected on natural snail mucus
dehydrating as a function of time. Arrow indicates direction of increasing
time. Figure 11:  Plot of frequency shift and storage (top plot) modulus, and
FWHM and apparent viscosity (bottom plot) as a function of time for the
dehydrated mucus sample. Shaded region indicates the transition from a high to
low hydrated state.

Figure 11 shows the frequency shift for the dehydrated mucus as a function of
time (concentration). As can be seen from this plot, there are two points
where there is a dramatic change in the frequency shift. The first occurring
at $\sim$ 75 hours. Before this point, there is an increase in frequency shift
from 8.1 to 8.5 GHz, or at a rate of $\sim$ 0.004 GHz/h. After this first
point, there is a sudden increase in the frequency shift at a rate of $\sim$
0.02 GHz/h up to a shift of $\sim$ 12 GHz at 300 hours, as indicated by the
dashed line in Fig. 11. The second point occurs at a shift of 12 GHz where the
frequency then remains constant. The general increase in frequency shift with
increasing polymer concentration (time) is expected and consistent with
previous Brillouin scattering studies on aqueous polymer solutions Bailey  _et
al._ (2019); Bedborough and Jackson (1976); Bot  _et al._ (1995); Palombo and
Fioretto (2019).

Figure 11 shows the FWHM as a function of time for dehydrated mucus. Like the
frequency shift data, there exists two points where there is a sudden change
in the FWHM of the dehydrated mucus. The first point also occurs at $\sim$ 75
hours, the same as the frequency shift. Above this point, like the frequency
shift, there is a linear increase up to the the first transition point. Below
the transition, the FWHM increases rapidly. However unlike the frequency shift
result, the second transition point occurs at about 200 hours. The FWHM has
been previously shown to be heavily influenced by polymer concentration Bailey
_et al._ (2019); Van Den Bulcke  _et al._ (2000). This polymer influence has
been proposed previously by other Brillouin studies to the fact that waters
mobility is restricted in the presence of polymers hydration shell Lupi  _et
al._ (2011); Comez  _et al._ (2012a).

It has been shown previously in similar Brillouin scattering studies on
polymer-water systems that as the concentration in these systems increases,
there is a transition from a high-hydration (liquid-like) to a gel like state
that occurs Bailey  _et al._ (2019); Palombo and Fioretto (2019). The shaded
region indicated in Fig. 11 illustrates the point and region at which we
believe this system transitions from a high hydration to a low hydration
state. This transition will be further investigated in Section V.

###  IV.3 Results: Viscoelastic Behaviour

####  IV.3.1 Sound Velocity

Figure 12 shows the sound velocity as a function of time for the dehydrated
mucus. Just like the frequency shift and FWHM results displayed in Fig. 11,
there exists two transition points where the behaviour of sound velocity
changes. The first is at 75 hours, where before this transition point, there
is a slight linear increase from $\sim$ 1600m/s to 1700 m/s. After the
transition the sound velocity displays a sharp linear increase in value from
1700 m/s to 2400 m/s where the second transition occurs. After the second
transition at 300 hours the sound velocity plateaus and maintains a constant
velocity of $\sim$ 2400 m/s. This result is an indication that as the mucus
dehydrates the structure of this system is becoming more rigid and “gel-like”.
If this were not the case, then one would expect the sound velocity to
increase only slightly since for liquids, especially those with high water
content have sound velocities comparable to that of water Hanlon  _et al._
(2023); Lupi  _et al._ (2011); Comez  _et al._ (2012a).

Figure 12: Sound velocity, apparent viscosity and sound absorption as a
function of time for the dehydrated mucus sample. Shaded region shows the
transition from a high to low hydrated state.

####  IV.3.2 Apparent Viscosity

Figure 12 shows the natural logarithm of the apparent viscosity for dehydrated
snail mucus calculated using Eq. 2. The apparent viscosity data exhibits two
distinct transition points at approximately 75 and 200 hours. Above the first
transition, there is a gradual linear increase in apparent viscosity.
Subsequently, the apparent viscosity continues to rise linearly until reaching
a value of approximately 3.5 cP at the second transition point. Beyond this
second transition, the apparent viscosity remains relatively constant.

This data indicates that as the elapsed time increases, the viscosity
generally increases. Furthermore, the behavior of the apparent viscosity
differs above and beyond the first transition point, indicating the occurrence
of a structural transition. These changes in apparent viscosity provide clear
evidence of modifications in the structure of the glycoprotein-water network.
The increase in viscosity might be associated with factors such as enhanced
intermolecular interactions or strengthening of the cross-linked glycoprotein-
water network.

Additionally, the fact that the viscosity remains constant after the second
transition point suggests that the structure has reached a stable state and is
not easily able to change.

####  IV.3.3 Hypersound Attenuation

Figure 12 shows the hypersound absorption coefficient for dehydrated mucus
calculated using Eq. 6. The observed behavior of the hypersound absorption
coefficient for dehydrated mucus is unlike the previous data shown in this
section, the hypersound absorption shows an unusual behaviour. Initially,
there is a gradual increase in the damping of sound waves, reaching a maximum
absorption point at approximately 100 hours. Following this, there is a
gradual decrease leading to a minimum hypersound absorption at around 250
hours, which then remains constant. This unusual behavior of the hypersound
absorption highlights the complex nature of mucus dehydration and its impact
on sound wave propagation properties.

This result is rather intriguing and suggests that as the system transitions
to a gel-like state, the sound absorption by the glycoprotein-water network
decreases. One possible reason that may account for the sound absorption in
this is the increased structure associated with the gel state. In a liquid
state, the molecules are relatively more mobile and can easily vibrate in
response to sound waves. This molecular motion leads to effective sound energy
dissipation and absorption. However, as the system transitions to a gel-like
state, the increased molecular interactions and structural rigidity can
restrict the movement of molecules. This restricted mobility reduces the
ability of the gel state to effectively absorb sound energy.

####  IV.3.4 Storage & Loss Modulus

Figure 13: Storage and Loss modulus of dehydrating snail mucus as a function
of temperature.

Figure 13 shows the storage modulus as a function of temperature calculated
using the first term in Eq. 2. The storage modulus shows two transition points
just like the other quantities detailed in this work occurring at 75 and 300
hours. Before the first transition, the storage modulus increases linearly
from $\sim$ 2.9 GPa at 0 hours to about 3.0 GPa at 75 hours. After the first
transition, the storage modulus increases rapidly, up to about 6 GPa where it
approaches the second transition. After the second transition, the storage
modulus remains constant for the entirety of this experiment. The storage
modulus as we know, is a measure of how much energy is stored elastically in
the system. From this, Fig. 13 tells us that as the protein concentration
increases, there is a general increase in the energy stored elastically in the
mucus, where it eventually reaches a maximum value of 6 GPa at 300 hours.

Similarly, Fig. 13 also shows the loss modulus calculated using the second
term in Eqn. 2 as a function of temperature. Like the storage modulus, the
loss moduli shows a very similar behaviour. That is, there are two transition
present, both also at 75 hours and 300 hours. Before the first transition,
there is a linear increase from 0.005 to 0.01 GPa in the loss modulus. After
the first transition, there is non-linear increase up to a value of 0.03 GPa
at 300 hours. Below the second transition (at 300 hours) the loss modulus
appears to slightly increase. The loss modulus is a measure of how much energy
is lost through heat in a system, what the data presented here shows is that
as the protein concentration increases, the amount of energy lost through heat
also increases.

The result from a previous Brillouin scattering study on the protein
concentration of a collagen gel shows a similar plot to Fig. 13. However, as
suggested by this study, the loss modulus should exhibit an increase in value
once it transitions to a gel like state which is present in the data for loss
modulus, albeit slight.

##  V General Discussion

###  V.1 Structural Transition: Influence of Glycoproteins

Figure 14: Plots of room-temperature Brillouin peak frequency shift and FWHM
(left panel) and derived viscoelastic properties (right panel) as a function
of protein concentration in snail mucus solutions. Additionally, a simple
schematic is provided to illustrate the observed structural transitions. These
transitions, along with the concentration at which they occur is depicted in
the right panel. The symbols in this figure correspond to $\circ$ \- free
water, $\bullet$ \- bound water, $\medwhitestar$ \- cross-links and lines
represent glycoproteins.

This study on natural snail mucus has revealed the presence of at least three
structural transitions, each characterized by a discontinuous change observed
in both raw Brillouin scattering data and calculated viscoelastic properties.
These transitions mark significant shifts in the physical properties of the
mucus, indicating changes in the molecular arrangement, intermolecular forces
or conformational dynamics in the glycoprotein-water network.

Figure 14 shows the frequency shift and FWHM of both diluted obtained from
room temperature Brillouin spectroscopy of both diluted and dehydrated snail
mucus, plotted as a function of protein concentration. As mentioned above, we
observe three distinct transitions in the raw spectral data, along with
calculated quantities throughout this manuscript. The first transition occurs
at approximately 3% protein concentration. Both the frequency shift and FWHM
show a sharp increase in value at this point. Despite only having one data
point for diluted mucus (dilution 4) above this point, the frequency shift and
FWHM is significantly different than dilutions 1 - 3. Additionally, if we
consider the temperature dependence of frequency shift and FWHM shown
previously (see Fig 11) we can see that data for dilution 4 resembles that of
pure water obtained previously by Brillouin spectroscopy Hanlon  _et al._
(2023). The dilution data presented throughout this work suggests that this
first transition is likely attributed to the proteins becoming first cross-
linked in the system. Further support for this is through the calculated
viscoelastic properties, speed of sound, apparent viscosity and sound
absorption as shown in the right panel of Fig. 14. Furthermore, the
temperature dependence of the viscoelastic properties for each dilution
previously shown (see Figures 6 \- 8) all increase in value as the protein
concentration increases, a particularly strong increase is observed between 2
and 3% protein concentration. Perhaps, the most obvious increase in the
viscoelasticity is observed in Fig. 7, where there is a clear difference
between the natural logarithm of apparent viscosity for dilution 3 and 4.
Recall, that this change was associated with the entropic pre-factor
$\eta_{0}$, which is subject to conformational changes Lupi  _et al._ (2011);
Comez  _et al._ (2012a). It has been stated before that cross-linked protein-
water systems increase the elastic and acoustic properties of the system
Hanlon  _et al._ (2023); Bailey  _et al._ (2019); Denny (1980, 1984) which
would also support our idea proposed here.

The second transition we observe occurs at a protein concentration of
approximately 7% and has been attributed to the depletion of any free water in
the snail mucus. Bound and free water exist in polymer gel networks naturally
and together make up the entire water content in such systems Peppas and
Merrill (1977); Peppas and Khare (1993); Hoffman (2012). Bound water refers to
the water molecules that are tightly associated or “bound” to the protein-
water network, while free water, on the other hand, represents the water that
is not bound to the network and is “free” to move in the system. The greater
amount of bound water in a system can increase the cross-linking density which
in turn causes the molecular structure more rigid Hoffman (2012). The observed
behavior in both the raw Brillouin data and calculated viscoelastic quantities
supports the notion that this transition is caused by the initial depletion of
free water in the mucus. First considering, Fig. 14, we can see a general
increase in both the frequency shift and FWHM as the protein concentration
increases after a protein concentration of 7% up to $\sim$ 50%. The increase
in both frequency shift and FWHM with protein concentration in this region is
a clear indication of increased rigidity. Additionally, the viscoelastic
properties presented in Fig. 14 show an increase in the sound velocity and
viscosity with increasing concentration, while $\alpha/f^{2}$ decreases with
increasing concentration. The increase in both sound velocity and viscosity
observed in this region signifies a significant shift in the molecular
structure, indicating a transition towards increased rigidity. The rise in
sound velocity, is a measure of how quickly sound propagates through a medium,
highlighting the increased stiffness of the glycoprotein-water network.
Simultaneously, the increased apparent viscosity, a measure of a fluids
resistance to flow, further supports this transition by indicating a greater
resistance to molecular rearrangements. Moreover, the decrease in the sound
absorption in this region provides more justification for this claim. In pure
liquids, molecules have higher mobility, allowing them to vibrate and
dissipate sound energy effectively. The random arrangement of molecules in a
liquid state provides more opportunities for sound waves to interact with the
medium and be absorbed. Thus, as the protein concentration in the system
increases, and the internal molecular structure becomes more rigid, then the
molecular mobility will be restricted leading to a lower sound absorption.

Lastly, the third transition observed in this work occurs at a protein
concentration of 54% and has been deemed due to the transition to a gel state
and it is suggested that some of the remaining bound water be evaporated at
this state. As illustrated by the raw spectral data in Fig. 14, there is a
clear and distinct transition occurring at a protein concentration of 54%,
where both the frequency shift and FWHM go from increasing to a constant
value. Due to the relatively high frequency shift and FWHM in this region,
along with the fact that there is still a considerable amount of water left in
the system, as determined by the ATR data, this transition was therefore
attributed to a gel-like state. Further support for this is based on the
calculated viscoelastic properties shown in Fig. 14. Both, the sound velocity,
viscosity and sound absorption all show their respective values remain
constant for this high protein concentration region. One would expect that for
a gel state, the structure be more rigid mainly for two reasons. Firstly, a
decrease in water content typically results in increased rigidity of the gel-
like structure. Secondly, the reduction in free water, with only bound water
being incorporated into the glycoprotein network, can contribute to an
increase in molecular arrangement structural and therefore greater rigidity.
In the gel state, as the free water content decreases, the molecular
interactions among glycoproteins and bound water become more pronounced. This
increased interaction contributes to a more structured and rigid gel network.
Moreover, the presence of bound water molecules in the gel network restricts
the movement of proteins and other components. Previous Brillouin scattering
studies on gelatin gels as a function of protein concentration show behaviour
nearly identical to what is shown here Bailey  _et al._ (2019). In that
Brillouin study, there is a plateau at high concentration values, like that
shown here. Likewise, the FWHM shows a brief plateau at concentrations near
50% protein, similar to our results.

##  VI Conclusion

Brillouin light scattering spectroscopy was employed to investigate the
viscoelastic properties and phase behavior of diluted and dehydrated snail
mucus. The temperature range of -11∘C $\leq T\leq$ 52 ∘C was explored for all
dilutions of the snail mucus, while the dehydrated mucus was examined with
respect to time and corresponding protein concentration. Intriguingly,
deviations in both temperature and concentration dependencies of spectral peak
parameters and derived viscoelastic properties validate the previously
identified liquid-to-solid phase transition at -2.5∘C, while additionally
revealing a correlation between concentration and this transition temperature
Tpt. Collectively, both the dilution and dehydration data suggest three
distinct structural transitions occurring at protein concentrations of 3%, 7%
and 54%. These transitions were attributed to the initiation of cross-linking
in the glycoprotein water network, depletion of free water, and the transition
to a gel state. These results support previous Brillouin studied on gels and
provide further evidence for the intricate role of glycoproteins within these
systems Bailey  _et al._ (2019). Notably, this research offers fresh insights
into the significance of hydration water in protein-water systems of
biological relevance. Furthermore, it underscores the effectiveness of
Brillouin spectroscopy in studying this unique class of natural materials,
expanding the range of systems to which this technique has proven to be highly
valuable.

## References

  * Bailey  _et al._ (2019) M. Bailey, M. Alunni-Cardinali, N. Correa, S. Caponi, T. Holsgrove, H. Barr, N. Stone, C. P. Winlove, D. Fioretto, and F. Palombo, arXiv preprint arXiv:1912.08292 (2019).
  * Comez  _et al._ (2012a) L. Comez, L. Lupi, M. Paolantoni, F. Picchiò, and D. Fioretto, The Journal of Chemical Physics 137, 114509 (2012a).
  * Comez  _et al._ (2016) L. Comez, M. Paolantoni, P. Sassi, S. Corezzi, A. Morresi, and D. Fioretto, Soft Matter 12, 5501 (2016).
  * Lupi  _et al._ (2011) L. Lupi, L. Comez, C. Masciovecchio, A. Morresi, M. Paolantoni, P. Sassi, F. Scarponi, and D. Fioretto, The Journal of chemical physics 134, 02B603 (2011).
  * Denny (1980) M. Denny, Nature 285, 160 (1980).
  * Denny (1984) M. Denny, Am. Zool. 24, 23 (1984).
  * Hanlon  _et al._ (2023) D. F. Hanlon, M. J. Clouter, and G. T. Andrews, arXiv preprint arXiv:2306.06807 (2023).
  * Verdugo  _et al._ (1987) P. Verdugo, I. Deyrup-Olsen, M. Aitken, M. Villalon, and D. Johnson, J. Dent. Res. 66, 506 (1987).
  * Li  _et al._ (2017) J. Li, A. D. Celiz, J. Yang, Q. Yang, I. Wamala, W. Whyte, B. R. Seo, N. V. Vasilyev, J. J. Vlassak, and Z. Suo, Science 357, 378 (2017).
  * Bouvet and Ben (2003) V. Bouvet and R. N. Ben, Cell Biochem. Biophys. 39, 133 (2003).
  * Kwan  _et al._ (2020) C. S. Kwan, A. R. Cerullo, and A. B. Braunschweig, ChemPlusChem 85, 2704 (2020).
  * Monaco  _et al._ (2001) G. Monaco, D. Fioretto, L. Comez, and G. Ruocco, Physical Review E 63, 061502 (2001).
  * Ebbinghaus  _et al._ (2010) S. Ebbinghaus, K. Meister, B. Born, A. L. DeVries, M. Gruebele, and M. Havenith, Journal of the American Chemical Society 132, 12210 (2010).
  * Comez  _et al._ (2012b) L. Comez, C. Masciovecchio, G. Monaco, and D. Fioretto, in  _Solid state physics_ , Vol. 63 (Elsevier, 2012) pp. 1–77.
  * Montrose  _et al._ (1968) C. Montrose, V. Solovyev, and T. Litovitz, The Journal of the Acoustical Society of America 43, 117 (1968).
  * Tao  _et al._ (1992) N. Tao, G. Li, and H. Cummins, Physical Review B 45, 686 (1992).
  * Gugliando  _et al._ (2021) E. Gugliando, M. Cordaro, R. Fusco, A. F. Peritore, R. Siracusa, T. Genovese, R. D’Amico, D. Impellizzeri, R. DiPaola, S. Cuzzocrea, and R. Crupi, Sci. Rep. 11, 3638 (2021).
  * Dil (1982) J. Dil, Rep. Prog. Phys. 45, 285 (1982).
  * Rouch  _et al._ (1976) J. Rouch, C. C. Lai, and S. H. Chen, J. Chem. Phys. 65, 4016 (1976).
  * Andrews (2018) G. T. Andrews, in  _Handbook of Porous Silicon_ , edited by L. Canham (Springer International, 2018) Chap. 53, pp. 691–703.
  * Griffiths (1983) P. R. Griffiths, Science 222, 297 (1983).
  * Arunkumar  _et al._ (2019) R. Arunkumar, C. J. Drummond, and T. L. Greaves, Frontiers in Chemistry 7, 74 (2019).
  * Lewis  _et al._ (2013) S. Lewis, A. Lewis, and P. Lewis, Vibrational Spectroscopy 69, 21 (2013).
  * Spiridon  _et al._ (2011) I. Spiridon, C. A. Teaca, and R. Bodirlau, BioResources 6, 400 (2011).
  * Grdadolnik (2002) J. Grdadolnik, Acta Chimica Slovenica 49, 631 (2002).
  * Venyaminov and Prendergast (1997) S. Y. Venyaminov and F. G. Prendergast, Analytical biochemistry 248, 234 (1997).
  * Larkin (2017) P. Larkin,  _Infrared and Raman spectroscopy: principles and spectral interpretation_ (Elsevier, 2017).
  * Bradley (2007) M. Bradley, Thermo Fisher Scientific, Madison, USA, Application Note 50733 (2007).
  * O’Connor and Schlupf (1967) C. L. O’Connor and J. P. Schlupf, J. Chem. Phys. 47, 31 (1967).
  * Zhao and Vanderwal (1997) P. Zhao and J. Vanderwal, Polymer Gels and Networks 5, 23 (1997).
  * Scheyer  _et al._ (1997) Y. Scheyer, C. Levelut, J. Pelous, J. Cook, M. Johnson, F. Prochazka, and D. Durand, Physica B: Condensed Matter 234, 445 (1997).
  * Pochylski  _et al._ (2006) M. Pochylski, F. Aliotta, Z. Blaszczak, and J. Gapiński, The Journal of Physical Chemistry B 110, 20533 (2006).
  * Van Den Bulcke  _et al._ (2000) A. I. Van Den Bulcke, B. Bogdanov, N. De Rooze, E. H. Schacht, M. Cornelissen, and H. Berghmans, Biomacromolecules 1, 31 (2000).
  * So  _et al._ (1994) J. H. So, R. Esquivel-Sirvent, S. S. Yun, and F. B. Stumpf, J. Acoust. Soc. Am. 96, 3807 (1994).
  * So  _et al._ (1995) J. H. So, R. Esquivel-Sirvent, S. S. Yun, and F. B. Stumpf, J. Acoust. Soc. Am. 98, 659 (1995).
  * Esquivel-Sirvent  _et al._ (1993) R. Esquivel-Sirvent, B. Tan, I. Abdelraziq, S. S. Yun, and F. B. Stumpf, J. Acoust. Soc. Am. 93, 819 (1993).
  * Esquivel-Sirvent  _et al._ (1994) R. Esquivel-Sirvent, S. S. Yun, and F. B. Stumpf, J. Acoust. Soc. Am. 95, 557 (1994).
  * Burcham  _et al._ (1986) T. Burcham, D. Osuga, Y. Yeh, and R. Feeney, Journal of Biological Chemistry 261, 6390 (1986).
  * URAJI  _et al._ (1996) T. URAJI, H. KOHNO, H. YOSHIMURA, M. SHIMOYAMADA, and K. WATANABE, Food Science and Technology International, Tokyo 2, 38 (1996).
  * Bedborough and Jackson (1976) D. S. Bedborough and D. Jackson, Polymer 17, 573 (1976).
  * Bot  _et al._ (1995) A. Bot, R. Schram, and G. Wegdam, Colloid and polymer science 273, 252 (1995).
  * Palombo and Fioretto (2019) F. Palombo and D. Fioretto, J. Chem. Rev. 119, 7833 (2019).
  * Peppas and Merrill (1977) N. A. Peppas and E. W. Merrill, Journal of biomedical materials research 11, 423 (1977).
  * Peppas and Khare (1993) N. A. Peppas and A. R. Khare, Advanced drug delivery reviews 11, 1 (1993).
  * Hoffman (2012) A. S. Hoffman, Advanced Drug Delivery Reviews 64, 18 (2012).


# Restricted Airspace Protection using Multi-UAV Spatio-Temporal Multi-Task
Allocation

Shridhar Velhal1 and Suresh Sundaram2 1Shridhar Velhal is a PhD student at
Department of Aerospace Engineering, Indian Institute of Science, Bengaluru,
India<EMAIL_ADDRESS>Sundaram is an Associate Professor at
Department of Aerospace Engineering, Indian Institute of Science, Bengaluru,
India<EMAIL_ADDRESS>
###### Abstract

This paper addresses the problem of restricted airspace protection from
invaders using the cooperative multi-UAV system. The objective is to detect
and capture the invaders cooperatively by a team of homogeneous UAVs (called
evaders) before invaders enter the restricted airspace. The problem of
restricted airspace protection problem is formulated as a Multi-UAV Spatio-
Temporal Multi-Task Allocation problem and is referred as MUST-MTA. The MUST-
MTA problem is solved using a modified consensus-based bundled auction method.
Here, the spatial and time constraints are handled by combining both spatial
and temporal loss component. The solution identifies the sequence of spatial
locations to be reached by the evader at specific time instants to neutralize
the invaders. The performance of MUST-MTA with consensus approach is evaluated
in a simulated environment. The Monte-Carlo simulation results clearly
indicate the efficacy of the proposed approach in restricted airspace
protection against intruders.

###### Index Terms:

Restricted Airspace protection, Multi-Task Allocation, Consensus Based
Auction, Spatio-Temporal Task.

##  I INTRODUCTION

Rapidly evolving technologies in autonomous unmanned aerial vehicles (UAV’s)
and associated developments in low-cost sensor have created a significant
interest among researchers, in using them for various civil and military
applications. Particularly, the autonomous aerial vehicles are often used for
logistics [1], medical [2], agriculture [3], security and surveillance [4],
[5]. The increase in the use of UAV in lower altitude introduces many
challenges in privacy, safety and security [6]. These UAV’s may be flying over
critical infrastructure such as nuclear facility, airport, chemical
industries, ports and so on. Protecting restricted airspace from the UAV’s
physical attack is really a challenging problem. Detecting and responding to
the UAV’s invaders over a restricted airspace plays an important role.

First time in the literature, this paper address a cooperative multi-UAV
system for restricted airspace protection from UAV invaders. In a typical
Restricted Airspace Protection (RAP) problem as shown in Fig. 1, the multiple
evaders revolve around the airspace and cooperatively neutralize the multiple
invaders moving towards the airspace. At any given time, evaders will be able
to detect the invaders and estimate the time and location of arrival in the
region of engagement (spatio-temporal tasks). Note that these task are
available only at a specific time and if these task are not handled by the
evaders than invader will be able to attack the airspace successfully. Under a
full communication scenario, the problem of RAP is formulated as a multiple-
UAV spatio-temporal multi-task allocation problem. It is referred as MUST-MTA.
The dynamically varying number of tasks, spatial and temporal dimensions adds
to the complexity in cooperative task allocation between multiple evaders
which minimize the overall effort.

Figure 1: Restricted airspace protection using Multi-UAV system

A composite loss function which computes the effort made by individual evaders
to move in a specific sequence to reach the location of the task at a specific
time instant. Note that the tasks assigned to the evader are unique. The
problem of MUST-MTA is formulated as a linear integer programming and solved
using modified consensus-based bundled algorithm (CBBA) proposed in [7]. The
algorithm utilizes market-driven decision strategy for decentralized multi-
task allocation with time constraints and consensus routine to resolve
conflict between the evaders. The algorithm allocates the spatio-temporal task
which forms the path for the evaders in the region of engagement. The
performance of the proposed MUST-MTA for RAP problem has been evaluated in a
simulated environment. Further, Monte-Carlo simulation studies are carried-out
to understand the effect of time-separation between the intruders on the
point-of-failure.

The main contribution of the paper is: Formalization of the RAP problem into a
multi-UAV spatio-temporal multi-task allocation problem. The presence of
spatial and temporal dimensions and dynamic environments makes the solution
for MUST-MTA challenging. A composite loss function is defined to handle the
spatio-temporal nature of the task. The linear integer programming problem is
solved using modified consensus based bundle allocation.

This paper is organized as follows. The related work is discussed in II.
Section III defines the RAP problem, spatio-temporal tasks, formulate the
MUST-MTA problem, and present the modified CBBA method. Section IV provides
the simulation results of the restricted airspace protection problem. Finally,
a conclusion is given in Section V

##  II Related Work

One of the important challenges in the use of multi-UAV system for real-world
applications is a complex task allocation problem between agents under
unknown/uncertain environment. The objective in the complex task allocation is
to find optimal strategy that will assign a set of tasks to the UAV such that
multi-UAV system achieves its goal. More detailed review of task allocation
and taxonomy of task allocation can be found in [8], [9]. Recent task
allocation literature focus on dynamic allocation of spatially located tasks
using market driven strategies [10], game theoretic strategies [11], Hungarian
method [12, 13] and consensus based task allocation [7, 14, 15, 16]. Recently,
in [17, 18] fisher market clearing based task allocation approach is presented
to handle dynamically allocated spatial task which requires certain time to
complete the task as a time-window constraints. The spatial task with time
window constraint is solved using heuristic methods, where a penalty is
imposed on delayed execution of tasks. More detail on existing algorithms on
multi-task allocation with time-window constraints can be found in [19]. The
above-mentioned works are not suitable for RAP because the tasks are dynamic
and are available only at a specific time instant.

Issac et al [20] introduced the concept of territory guarding differential
game played by invaders and evaders. The goal of evader is to capture the
invader as far as from the territory, and goal of invader is to avoid
capturing and reach as close as possible towards the territory. Several
research works are available in the literature to address territory guarding
problem [21, 22, 23, 24, 25]. These works either uses single invader and
single evader or two evader to handle single invader. Recently, deep
reinforcement learning based defensive escort team is proposed to avoid active
collision [26]. Here, escort team tries to safely navigate the payload by
positioning escort around the payload. (Payload co-relates to the restricted
airspace in RAP.) As obstacles are repelled by escorting agents, agents will
need to position them in a way such that they repel the obstacles on route
cooperatively. Although these approaches handle dynamic tasks, the intruders
are not directed as in RAP problem. Further, in RAP, evaders have to reach the
spatial location at specific instant to neutralize the invaders.

##  III  Restricted Airspace Protection Problem

###  III-A Problem Definition

The restricted airspace depends on the critical infrastructure; Any shaped
infrastructure is approximated by circle. The scenario of RAP for a time
instant t is shown in Fig. 2. A cooperative multi-UAV team called as
‘evaders’, will protect the airspace from intruders. The evaders will operate
only inside the territory and neutralize the intruder in a ring around the
restricted airspace referred to as ‘region of engagement’. One evader can
neutralize an intruder when they come close within a neutralizing distance
(r). Evaders will detect the positions of intruders. Depending upon the
position information. Each evader needs to cooperatively decide for a sequence
of tasks, following which it can neutralize invaders.

Figure 2: Restricted Airspace (Territory) protection problem

The terms used in the RAP problem are as follows,

  * •

$Territory$: It is restricted airspace to be protected.

  * •

$Evader$: A UAV which protects the restricted airspace.

  * •

$Invader$: A UAV which tries to enter the restricted airspace.

  * •

$Region\ of\ Engagement$: A region (ring) around the restricted airspace where
invader can be neutralized.

  * •

$Task$: $\bf{\alpha}(\alpha^{s},\alpha^{t})$ : A task is defined by every
intruder; intruder will penetrate from location $\alpha^{s}$ at time
$\alpha^{t}$.

  * •

$Neutralizing\ point$ ($\alpha^{s}$): A location in RoE from where an intruder
tries to enter.

  * •

$Time\ of\ intrusion$ ($\alpha^{t}$): A time at which intruder will enter
restricted airspace if not neutralized.

  * •

$Path$ ${\bf p}_{i}$: It is a sequence in which evader $i$ will execute the
tasks.

Evaders ($E_{1},E_{2},..E_{N}$) are less in number than intruders
($I_{1},I_{2},..,I_{K}$). Evaders will fly with maximum speed $v_{E}^{max}$.
Only one evader is sufficient to neutralise one intruder; more over once an
intruder is neutralised, evader is free to do another task. The path ${\bf
p}_{i}=\\{\alpha_{x},\alpha_{x},\alpha_{z}\\}$ means the evader $i$ will
execute task $\alpha_{x}$, $\alpha_{y}$, and $\alpha_{z}$ sequentially.

The following assumptions are made for RAP problem,

  1. A1)

Each intruder $I_{j}$ is moving with a constant speed $v_{j}$, directed
towards the centre of the restricted airspace.

  2. A2)

All evaders are homogeneous. Using same sensors they will identify the
intruders position and velocity. All evaders can communicate with each other.

  3. A3)

The evader has higher velocity than that of intruder.

  4. A4)

Intruders are not attacking at the same time.

  5. A5)

All intruders and evaders are operating at the same height.

The assumption A4 is very critical because evader will be helpless when a
large number of the intruders approach at different locations at the same time
instant.

Intruder $I_{j}$ will try to enter the airspace with a velocity of $v_{j}$,
directed towards the centre. Evader has to neutralize the intruder $I_{j}$ at
location $\alpha_{j}^{s}$ from where intruder tries to enter. A task
${\bf{\alpha}}_{j}(\alpha_{j}^{s},\alpha_{j}^{t})$ is a spatio-temporal task
defined such that, evader should reach a location $\alpha_{j}^{s}$ at a
specific time $\alpha_{j}^{t}$.

###  III-B Spatio-temporal task

A task ${\bf{\alpha}}_{j}$ is generated by each intruder $I_{j}$. The tasks
${\bf{\alpha}}_{j}$ consists of two dimensions namely, spatial and temporal.
The spatial dimension is to reach a location $\alpha_{j}^{s}$ and temporal
dimension is to complete the task at time $\alpha_{j}^{t}$. Each task has to
be executed at a specific location at a specific time. Hence task is at a
distinct point on the graph with two dimensions: space and time. For clarity,
spatial location is converted to angular position on region of engagement.
Figure 3 shows a typical spatio-temporal tasks identified at a time t. In
general, spatio-temporal tasks have n-dimension of spatial location and one
temporal dimension.

Figure 3: Typical spatio-temporal tasks identified at a time instant

###### Lemma III.1

If the number of the intruder tries to enter territory at the same time with
no radial separation, is more than the number of the evaders then solution is
infeasible.

Consider $N_{i}$ number of intruders are trying to intrude at time $t$. In
spatio-temporal task, it is required to execute $N_{i}$ tasks at a time $t$.
As the intruder are not radially separated, they will reach RoE at different
location at same time instant. The number of locations to be reached at time
$t$ are more than the number of evaders available; Hence, all intruders can
not be neutralized. The solution for these cases is infeasible.

###  III-C Multi-UAV Spatio-Temporal Multi-Task Allocation

Each intruder $j$ generates a spatio-temporal task $\alpha_{j}$ and evaders
have to execute all tasks cooperatively. As intruders are more than evaders,
evader has to do multiple tasks. The tasks are assigned to evaders by solving
a MUST-MTA problem. Here, intruders are modelled as tasks and evaders are the
agents who execute the task. In the rest of the paper, agents are denoted by
$i,k$ and task is denoted by $j$.

Multi-task allocation is a decision making whether a task is assigned to an
agent or not. Also, if tasks are sequential, then MTA solves for the sequence
in which tasks to be done. The Loss function is used to quantify the task,
depending upon that tasks are assigned to agents. The spatio-temporal task has
both spatial and temporal loss function. Composite loss function has been
designed by uniting spatial and temporal loss functions.

####  III-C1 Spatial component $L_{ij}^{s}({\bf p}_{i})$

It is the component defined for agent $i$, to execute spatial task
$\alpha_{j}^{s}$ on path ${\bf p}_{i}$ .

$L_{ij}^{s}({\bf p}_{i})={\|(\alpha_{j}^{s}-E_{i}^{p}({\bf
p}_{i}))\|}_{2}+\eta{\|(\alpha_{j}^{s}-I_{j})\|}_{2}$ (1)

where, $\alpha_{j}^{s}$ is spacial requirement of task $j$, $E_{i}^{p}({\bf
p}_{i})$ is the location of previous task on path ${\bf p}_{i}$. For first
task on path $E_{i}^{p}=E_{i}$, the location of agent $i$, $\eta\in(0,1)$ is
scaling factor. The first term in Eq.(1) is the effective distance travelled
for reaching the spatial location along path ${\bf p}_{i}$. The second term in
Eq.(1) is distance of intruder from neutralising point; this value is constant
for each task independent of agent.

The effective distance travelled by an agent on path $\\{A,B,C\\}$ is computed
sequentially. Effective distance for reaching $B$ is distance between $AB$.
The effective distance to reach $C$ along path $\\{A,B,C\\}$ is the distance
between $BC$ as the distance $AB$ is already considered for reaching $B$.

####  III-C2  Temporal component $L_{ij}^{t}$

Firstly we compute, time at which intruder $j$ enters RoE, reffered as time of
intrusion ($alpha_{j}^{t}$),

$\displaystyle\alpha_{j}^{t}$
$\displaystyle=\frac{{\|(I_{j}-\alpha_{j}^{s})\|}_{2}}{v_{j}}$ (2)

The temporal component $L_{ij}^{t}({\bf p}_{i})$ is defined for agent $i$, to
execute task $j$ at time $\alpha_{j}^{t}$ along path ${\bf p}_{i}$, $I_{j}$ is
locatopn of intruder $j$

$L_{ij}^{t}({\bf
p}_{i})=\begin{cases}(1+\alpha_{j}^{t})\left(\alpha_{j}^{t}-\alpha_{j^{p}}^{t}({\bf
p}_{i})\right)&\text{if $j$ is feasible on ${\bf p}_{i}$ }\\\ \infty&\text{if
$j$ is infeasible }\end{cases}$ (3)

where $\alpha_{j^{p}}^{t}({\bf p}_{i})$ is the time of arrival of previous
task on path ${\bf p}_{i}$; $\alpha_{j^{p}}^{t}=0$ for first task on path.

The task $j$ is feasible on path ${\bf p}_{i}$ if task is executed with
positive time step, and spatially feasible. The positive time step is
mathematically represented as $\left(\alpha_{j}^{t}-\alpha_{j^{p}}^{t}({\bf
p}_{i})\right)>0$. The spacial feasibility means distance between evader and
the spatial task, is reachable with in the temporal requirement of task;
mathematically written as,

$\frac{{\|(\alpha_{j}^{s}-E_{i}^{p}({\bf
p}_{i}))\|}_{2}}{v_{max}^{E}}<\left(\alpha_{j}^{t}-\alpha_{j^{p}}^{t}({\bf
p}_{i})\right)$

.

####  III-C3 Composite loss function $L_{ij}$

The composite loss function is defined for a spatio-temporal task as,

$L_{ij}({\bf p}_{i})=f\left(L_{ij}^{s}({\bf p}_{i}),L_{ij}^{t}({\bf
p}_{i})\right)$ (4)

The function $f$ can be any nonlinear function; here $f(x,y)=xy$ hence above
Eq.(4) reduce to

$L_{ij}({\bf p}_{i})=L_{ij}^{s}({\bf p}_{i})L_{ij}^{t}({\bf p}_{i})$ (5)

Now, MTA is defined as an linear integer programming.

####  III-C4 Multi-task allocation problem

The goal of the task allocation algorithm is to assign each of the $N_{t}$
tasks to the available $N$ agents such that single task is assigned to only
one agent. The cost associated with assigning a task $j$ to the agent $i$ is
$c_{ij}$. $\delta_{ij}$ is decision variable for assigning agent $i$ to task
$j$ The task assignment problem is defined as

$\displaystyle\min_{\delta_{ij}}\quad$
$\displaystyle\sum_{i=1}^{N_{a}}\left(\sum_{j=1}^{N_{t}}c_{ij}\delta_{ij}\right)$
(6) $\displaystyle{\rm such\ that}\qquad$
$\displaystyle\sum_{i=1}^{N_{u}}\delta_{ij}\leq 1\qquad\forall j\in{\cal J}$
(6a) $\displaystyle\delta_{ij}\in\\{0,1\\}\qquad\forall(i,j)\in{\cal
I}\times{\cal J}$ (6b)

the condition Eq.(6a) enforces that the task can be assigned to only one
agent. Eq. (6b) is decision variable either agent $i$ is assigned to task
$j$or not.

The objective is to find the sequence (path) assigned to individual evader
such that overall goal is achieved, The is cost of over all goal given below:

$\displaystyle c_{ij}[{\bf p}_{i}]$
$\displaystyle=\begin{cases}\min_{n\leq|{\bf p}_{i}|}L_{i}^{{\bf
p}_{i}\oplus_{n}\\{j\\}}-L_{i}^{{\bf p}_{i}}&\text{if $j\notin{\bf p}_{i}$
}\\\ \infty&\text{if $j\in{\bf p}_{i}$ }\end{cases}$ (7) $\displaystyle{\rm
where,}$ $\displaystyle\qquad L_{i}^{{\bf p}_{i}}=\sum_{j}L_{ij}^{{\bf
p}_{i}}$ (8)

$|.|$ is the cardinality of path, and $\oplus_{n}\\{j\\}$ adds the $j$ after
nth element. As the task $j$ is added at any location, the cost of the new
task is the difference between new path cost and original path cost.

Remark: The loss function computed in equations (1), (3), and (5) considers
the path, but their computation is based on only previous task listed in path,
and not on the complete path ${\bf p}_{i}$. Hence, for the cost of the path is
computed by summing loss function of all tasks in ${\bf p}_{i}$. The cost of
the newly added task is the difference in path cost due to the new task.

###  III-D Modified Consensus-based bundled auction algorithm

Algorithm 1 modified CBBA for agent $i$ at iteration $q$

1:procedure input ${\bf b}_{i}(q-1)$, ${\bf p}_{i}(q-1)$, ${\bf y}_{i}(q-1)$,
${\bf z}_{i}(q-1)$

2:${\bf b}_{i}(q)={\bf b}_{i}(q-1)$; ${\bf p}_{i}(q)={\bf p}_{i}(q-1)$

3:${\bf y}_{i}(q)={\bf y}_{i}(q-1)$; ${\bf z}_{i}(q)={\bf z}_{i}(q-1)$

4:conflict resolved $=0$

5:while conflict resolved $=0$ do % Auction Algorithm

6: $c_{ij}=\min_{n\leq|{\bf p}_{i}|}L_{i}^{{\bf
p}_{i}\oplus_{n}\\{j\\}}-L_{i}^{{\bf p}_{i}},\ \ \forall j\in{\cal
J}\backslash{\bf b}_{i}$

7: $h_{ij}={\mathbb{I}}(c_{ij}<y_{ij}),\qquad\forall j\in{\cal J}$

8: $J_{i}=argmin_{j}\ c_{ij}.h_{ij}$

9: $n_{i,J_{i}}=argmin_{j}\ L_{i}^{{\bf p}_{i}\oplus_{n}\\{j\\}}$

10: ${\bf b}_{i}={\bf b}_{i}\oplus_{end}{J_{i}}$

11: ${\bf p}_{i}={\bf p}_{i}\oplus_{n_{i,J_{i}}}{J_{i}}$

12: ${y}_{i,J_{i}}(q)=c_{i,J_{i}}$

13: ${z}_{i,J_{i}}=i$

14: Call Consensus Algorithm

15:end while

(Remark: minimization over all $\infty$ value is taken as $\infty$. All
$\infty$ means that the task $j$ is infeasible along path ${\bf p}_{i}$)

Algorithm 2 Consensus by agent $i$ at iteration $q$

1:procedure input ${\bf b}^{k}$, ${\bf p}^{k}$, ${\bf y}^{k}$, ${\bf z}^{k}$
(data received from agent $k$ via synchronized communication ) $i\neq k$ , $\
m\neq\\{i,k\\}$

2:if $z_{kj}^{k}=k\ \&\ z_{ij}^{i}=k$ then

3: Update

4:end if

5:if $z_{kj}^{k}=k\ \&\ y_{kj}<y_{ij}$ then

6: Update

7:end if

8:if $z_{kj}^{k}=i\ \&\ z_{ij}^{i}=k$ then

9: Reset

10:end if

11:if ${z}_{kj}^{k}=m\ \&\ {z}_{ij}^{i}\neq m$ then

12: if ${y}_{mj}(=y_{kj})<{y}_{ij}$ then

13: Reset

14: end if

15:end if

16:if ${\bf z}_{i}={\bf z}_{k}\qquad\forall i,k\in{\cal I}$ then

17: conflict resolved $=1$

18:end if

19:Update : $y_{ij}=y_{kj},z_{ij}=z_{kj}$

20:Reset : $y_{ij}=\infty,z_{ij}=\emptyset$

The CBBA [7] is a distributed task allocation algorithm in which agents bids
for a bundle of tasks which would be executed in a specific path. Path ${\bf
p}_{i}$ is the sequence in which agent $i$ is assigned the task. The composite
for spatial temporal task $c_{ij}({\bf p}_{i})$ is the cost associated with
agent $i$ doing task $j$ in path ${\bf p}_{i}$. The formulated cost function
is minimized in CBBA. In CBBA, bids which win the task (smallest bidding value
for task $j$) are denoted by ${{\bf y}_{i}}$, the wining agent’s list is
denoted by ${{\bf z}_{i}}$. ${\bf p}_{i}$ is the path of performing tasks and
${\bf b}_{i}$ bundle of tasks. The path ${\bf p}_{i}$ gives the sequence in
which tasks will be executed by agent $i$.

The Algorithm is initialized, as winning value of cost ${\bf y}_{i}=\infty$
and allocation ${\bf z}_{i}={\bf 0}$. The bundle and path are initialized as
null, ${\bf p}_{i}={\bf b}_{i}=\emptyset$. Agents bids independently in the
auction irrespective of other agents, hence computation of bidding is
distributed. The conflicts among the agents are resolved by consensus; which
requires communication for exchanging information of bidding value, winning
value, and path.

The algorithm 1 presents the steps for auction. In which each agent bids for
feasible targets independently. During this procedure, they also fix their
path of execution of the task. Each agent bids greedily for all feasible
tasks; this bidding information is shared with all agents via synchronous
communication. As bidding is performed independently, there may be conflicts
between agents for the task. Consensus obtained for resolving these conflicts
as given in algorithm 2. In consensus, a task is given to the lowest cost
valued agent for that task, and this will be updated in the path, bundle, and
winning agent vectors. If a new task is added to the path, the task is added
at a location, where it minimizes the path cost. When a task is removed from
the agent then the entire bundle needs to be removed. The removal of a task
changes the path of that agent, as other tasks are selected based on the path
which is no longer valid; the entire path needs to be cancelled.

##  IV Simulation Results and Discussion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 4: Snapshots of the intruder tasks allocated to the evaders, at
different time instants The evaders $E_{1},E_{2},E_{3}$ are represented by
colours blue, red, magenta respectively. The intruders assigned to an evader
are coloured with the colour of evader. The path selected by evader is shown
by symbols in sequence : *, $\diamond$, $\square$, $\lhd$, $\rhd$, $\otimes$

The simulations are performed for RAP problem. The restricted airspace is
selected as a circle of radius of $R=100$ m, The number of the evaders is
$K=3$ and the number of the intruder is set $N_{i}=6$. The evader will
neutralise an intruder within neutralising distance $(r)$ set to $20$ m. The
velocity of all intruder is constant and selected as $v_{I}=3$ m/s. The
maximum velocity of all evaders is constrained by ${v}_{E}^{max}=4.5$ m/s.
When intruder is neutralised, new intruder is added at random location, in the
neighbouring region of territory. The motion of evader and invader are
computed using kinematic equations. This simulation are conducted in MATLAB
R2019b in windows 10 environment.

###  IV-A Case study and discussion

Fig. 5 shows the spatio-temporal tasks for test scenario discussed in Fig. 4.
Total 15 spatio-temporal are avilable during time interbval of $0-150$ sec The
temporal tasks are well separated, and hence all tasks can be executed, and
this has been shown in results. The temporal separation plays a critical role
in the feasibility of the task. As tasks are temporally separated so all tasks
are executed successfully and shown in 4

Figure 5: Spatio-temporal tasks used for simulation case study

Fig 4 shows the screenshots the task assignment computed for a test scenario
of RAP problem. Total 15 intruders are neutralised in this case, in which
intruders are added at random locations. The task allocation algorithm is
solved at every sampling time instant. The path of agents are shown in Fig. 4,
at different time instants. Figure 4(a) shows the initial position of intruder
and the evaders. The intruders are allocated to all three agents, but one
intruder is unassigned. One intruder is unassigned because that task is
infeasible with the path chosen by agents. All agents move with the time that
task may become feasible as observed in 4(b).

The agents $E_{1}$ and $E_{3}$ neutralise their respective first intruder; new
intruders from the random location is approaching towards the terrain as shown
in Fig. 4(c). Furthermore, agent $E_{1}$ and $E_{3}$ are assigned to one task
each while, agent $E_{2}$ is assigned to four tasks. Since it is cooperative
task different number of task by agents. In Fig. 4(d), it is observed that
previous task allocation gets reallocated by the entry of a new intruder. The
tasks of agent $E_{2}$ are allocated to agent $E_{1}$. Reallocation is because
of two reasons; agent $E_{2}$ moves away from task due to newly added
intruder; $E_{2}$ changes path, and the task becomes infeasible in a new path.
Fig 4(h) shows that task on the top right is unassigned, this is due to that
fact that the task is feasible by agents $E_{1}$, but agent $E_{1}$ has
another task which is less costly than that task. However, after a few time
steps, task is taken by agent $E_{3}$ 4(i). This happens due to sub-optimality
in CBBA [7].

Monte-Carlo simulation is performed by varying the radial distance between
intruders. As velocity of the intruder is constant the radial distance
provides a temporal separation between tasks. A randomized simulations with
200 epoch are carried for a different radial distance for a newly added
intruder. For an epoch, maximum of 30 intruders are added. When an intruder
penetrates the restricted airspace, RAP fails. Figure. 6 shows the percentage
of success rate with different minimum radial distances; as the intruders gets
more radial separation the success percentage increases. For the low value of
radial separation, the tasks become infeasible as explained in Lemma III.1.
The number of the evaders are varied to study the minimum evaders required a
given restricted airspace. It is expected that more number of the evaders can
execute for less separation tasks The less number of evaders protecting
restricted airspace requires more temporal separation between tasks.

Figure 6: Monte-Carlo simulations by varying radial distance between intruders
and varying number of evader

##  V Conclusion

Restricted airspace protection (RAP) from UAV intruders using the cooperative
multi-UAV system is addressed. The movement of intruder towards the restricted
airspace leads to the spatio-temporal task (capturing the intruder) for the
multi-UAV (evaders) system. RAP problem has been formulated as a multi-UAV
spatio-temporal multi-task allocation (MUST-MTA) problem. This paper provides
a multi-task allocation solution to a spatio-temporal task using modified
consensus-based bundle algorithm. The cost function is modified via composite
loss function, which unites spatial and temporal components. The simulation
validates the efficacy of the proposed method. Simulated results show that the
tasks can be executed if they have sufficient temporal separation. As the
number of the evaders increases, the temporal separation requirement decreases
gradually.

## References

  * [1] K. Kuru, D. Ansell, W. Khan, and H. Yetgin, “Analysis and optimization of unmanned aerial vehicle swarms in logistics: An intelligent delivery platform,” _IEEE Access_ , vol. 7, pp. 15 804–15 831, 2019.
  * [2] J. C. Rosser Jr, V. Vignesh, B. A. Terwilliger, and B. C. Parker, “Surgical and medical applications of drones: a comprehensive review,” _JSLS: Journal of the Society of Laparoendoscopic Surgeons_ , vol. 22, no. 3, 2018.
  * [3] U. R. Mogili and B. Deepak, “Review on application of drone systems in precision agriculture,” _Procedia computer science_ , vol. 133, pp. 502–509, 2018.
  * [4] K. Harikumar, J. Senthilnath, and S. Sundaram, “Mission aware motion planning (map) framework with physical and geographical constraints for a swarm of mobile stations,” _IEEE transactions on cybernetics_ , 2019.
  * [5] ——, “Multi-uav oxyrrhis marina-inspired search and dynamic formation control for forest firefighting,” _IEEE Transactions on Automation Science and Engineering_ , vol. 16, no. 2, pp. 863–873, 2018.
  * [6] A. Solodov, A. Williams, S. Al Hanaei, and B. Goddard, “Analyzing the threat of unmanned aerial vehicles (uav) to nuclear facilities,” _Security Journal_ , vol. 31, no. 1, pp. 305–324, 2018.
  * [7] H. L. Choi, L. Brunet, and J. P. How, “Consensus-based decentralized auctions for robust task allocation,” _IEEE Transactions on Robotics_ , vol. 25, no. 4, pp. 912–926, 2009.
  * [8] G. A. Korsah, A. Stentz, and M. B. Dias, “A comprehensive taxonomy for multi-robot task allocation,” _The International Journal of Robotics Research_ , vol. 32, no. 12, pp. 1495–1512, 2013.
  * [9] A. Khamis, A. Hussein, and A. Elmogy, “Multi-robot task allocation: A review of the state-of-the-art,” in _Cooperative Robots and Sensor Networks 2015_. Springer, 2015, pp. 31–51.
  * [10] E. G. Jones, M. B. Dias, and A. Stentz, “Learning-enhanced market-based task allocation for oversubscribed domains,” in _2007 IEEE/RSJ International Conference on Intelligent Robots and Systems_. IEEE, 2007, pp. 2308–2313.
  * [11] R. Cui, J. Guo, and B. Gao, “Game theory-based negotiation for multiple robots task allocation,” _Robotica_ , vol. 31, no. 6, pp. 923–934, 2013\.
  * [12] S. Chopra and M. Egerstedt, “Heterogeneous multi-robot routing,” in _2014 American Control Conference_. IEEE, 2014, pp. 5390–5395.
  * [13] S. Chopra, G. Notarstefano, M. Rice, and M. Egerstedt, “A Distributed Version of the Hungarian Method for Multirobot Assignment,” _IEEE Transactions on Robotics_ , vol. 33, no. 4, pp. 932–947, 2017.
  * [14] L. Brunet, H. L. Choi, and J. P. How, “Consensus-based auction approaches for decentralized task assignment,” _AIAA Guidance, Navigation and Control Conference and Exhibit_ , no. August, 2008.
  * [15] R. Zlot and A. Stentz, “Market-based multirobot coordination for complex tasks,” _The International Journal of Robotics Research_ , vol. 25, no. 1, pp. 73–101, 2006.
  * [16] M. P. Fanti, A. M. Mangini, G. Pedroncelli, and W. Ukovich, “A decentralized control strategy for the coordination of agv systems,” _Control Engineering Practice_ , vol. 70, pp. 86–97, 2018.
  * [17] S. Amador, S. Okamoto, and R. Zivan, “Dynamic multi-agent task allocation with spatial and temporal constraints,” in _Twenty-Eighth AAAI Conference on Artificial Intelligence_ , 2014.
  * [18] S. A. Nelke, S. Okamoto, and R. Zivan, “Market clearing-based dynamic multi-agent task allocation,” _ACM Transactions on Intelligent Systems and Technology (TIST)_ , vol. 11, no. 1, pp. 1–25, 2020.
  * [19] E. Nunes, M. Manner, H. Mitiche, and M. Gini, “A taxonomy for task allocation problems with temporal and ordering constraints,” _Robotics and Autonomous Systems_ , vol. 90, pp. 55–70, 2017.
  * [20] D. Wishart, “Differential games. a mathematical theory with applications to warfare and pursuit, control and optimization,” _Physics Bulletin_ , vol. 17, no. 2, p. 60, 1966.
  * [21] K.-H. Hsia and J.-G. Hsieh, “A first approach to fuzzy differential game problem: guarding a territory,” _Fuzzy sets and systems_ , vol. 55, no. 2, pp. 157–167, 1993.
  * [22] Y.-S. Lee, K.-H. Hsia, and J.-G. Hsieh, “A strategy for a payoff-switching differential game based on fuzzy reasoning,” _Fuzzy sets and systems_ , vol. 130, no. 2, pp. 237–251, 2002.
  * [23] C. V. Analikwu and H. M. Schwartz, “Reinforcement learning in the guarding a territory game,” in _2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)_. IEEE, 2016, pp. 1007–1014.
  * [24] H. Raslan, H. Schwartz, and S. Givigi, “A learning invader for the “guarding a territory” game,” _Journal of Intelligent & Robotic Systems_, vol. 83, no. 1, pp. 55–70, 2016.
  * [25] C. V. Analikwu and H. M. Schwartz, “Multi-agent learning in the game of guarding a territory,” _Int. J. Innov. Comput. Inf. Control_ , vol. 13, no. 6, pp. 1855–1872, 2017.
  * [26] A. Garg, Y. A. Hasan, A. Yañez, and L. Tapia, “Defensive escort teams via multi-agent deep reinforcement learning,” _arXiv preprint arXiv:1910.04537_ , 2019.