Proteins in solution: Fractal surfaces in solutions
The concept of the surface of a protein in solution, as well of the interface between protein and 'bulk solution', is introduced. The experimental technique of small angle X-ray and neutron scattering is introduced and described briefly. Molecular dynamics simulation, as an appropriate com...
Збережено в:
| Дата: | 2016 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2016
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/155787 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Proteins in solution: Fractal surfaces in solutions / R. Tscheliessnig, L. Pusztai // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13803: 1–8. — Бібліогр.: 35 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-155787 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1557872019-06-18T01:28:48Z Proteins in solution: Fractal surfaces in solutions Tscheliessnig, R. Pusztai, L. The concept of the surface of a protein in solution, as well of the interface between protein and 'bulk solution', is introduced. The experimental technique of small angle X-ray and neutron scattering is introduced and described briefly. Molecular dynamics simulation, as an appropriate computational tool for studying the hydration shell of proteins, is also discussed. The concept of protein surfaces with fractal dimensions is elaborated. We finish by exposing an experimental (using small angle X-ray scattering) and a computer simulation case study, which are meant as demonstrations of the possibilities we have at hand for investigating the delicate interfaces that connect (and divide) protein molecules and the neighboring electrolyte solution. Впроваджено концепцiю поверхнi протеїну у розчинi, а також границi роздiлу мiж протеїном та “об’ємним розчином”. Коротко описано експериментальний метод розсiяння X-променiв при малих кутах i нейтронного розсiяння. Обговорено моделювання молекулярної динамiки як обчислювального iнструменту для вивчення гiдратацiйної оболонки протеїнiв. Розроблено концепцiю протеїнових поверхонь з фрактальним вимiром. Статтю завершено описом конкретного випадку експериментального (з використанням розсiяння X-променiв при малих кутах) та комп’ютерного дослiдження, щоб продемонструвати наявнi можливостi дослiдження делiкатних границь роздiлу молекул протеїну i розчину електролiту. 2016 Article Proteins in solution: Fractal surfaces in solutions / R. Tscheliessnig, L. Pusztai // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13803: 1–8. — Бібліогр.: 35 назв. — англ. 1607-324X DOI:10.5488/CMP.19.13803 arXiv:1603.02441 PACS: 87.10.Tf, 87.14.E-, 87.15.A-, 87.15.N- http://dspace.nbuv.gov.ua/handle/123456789/155787 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The concept of the surface of a protein in solution, as well of the interface between protein and 'bulk solution', is introduced. The experimental technique of small angle X-ray and neutron scattering is introduced and described briefly. Molecular dynamics simulation, as an appropriate computational tool for studying the hydration shell of proteins, is also discussed. The concept of protein surfaces with fractal dimensions is elaborated. We finish by exposing an experimental (using small angle X-ray scattering) and a computer simulation case study, which are meant as demonstrations of the possibilities we have at hand for investigating the delicate interfaces that connect (and divide) protein molecules and the neighboring electrolyte solution. |
| format |
Article |
| author |
Tscheliessnig, R. Pusztai, L. |
| spellingShingle |
Tscheliessnig, R. Pusztai, L. Proteins in solution: Fractal surfaces in solutions Condensed Matter Physics |
| author_facet |
Tscheliessnig, R. Pusztai, L. |
| author_sort |
Tscheliessnig, R. |
| title |
Proteins in solution: Fractal surfaces in solutions |
| title_short |
Proteins in solution: Fractal surfaces in solutions |
| title_full |
Proteins in solution: Fractal surfaces in solutions |
| title_fullStr |
Proteins in solution: Fractal surfaces in solutions |
| title_full_unstemmed |
Proteins in solution: Fractal surfaces in solutions |
| title_sort |
proteins in solution: fractal surfaces in solutions |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/155787 |
| citation_txt |
Proteins in solution: Fractal surfaces in solutions / R. Tscheliessnig, L. Pusztai // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13803: 1–8. — Бібліогр.: 35 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT tscheliessnigr proteinsinsolutionfractalsurfacesinsolutions AT pusztail proteinsinsolutionfractalsurfacesinsolutions |
| first_indexed |
2025-07-14T08:01:24Z |
| last_indexed |
2025-07-14T08:01:24Z |
| _version_ |
1837608557672923136 |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 1, 13803: 1–8
DOI: 10.5488/CMP.19.13803
http://www.icmp.lviv.ua/journal
Proteins in solution: Fractal surfaces in solutions
R. Tscheliessnig1, L. Pusztai2
1 Austrian Centre for Industrial Biotechnology (ACIB), Muthgasse 11, A-1190 Wien, Austria
2 Wigner Research Centre for Physics, Hungarian Academy of Sciences,
Konkoly Thege út. 29-33, H-1121, Budapest, Hungary
Received November 23, 2015, in final form December 24, 2015
The concept of the surface of a protein in solution, as well of the interface between protein and ’bulk solution’, is
introduced. The experimental technique of small angle X-ray and neutron scattering is introduced and described
briefly. Molecular dynamics simulation, as an appropriate computational tool for studying the hydration shell
of proteins, is also discussed. The concept of protein surfaces with fractal dimensions is elaborated. We finish
by exposing an experimental (using small angle X-ray scattering) and a computer simulation case study, which
are meant as demonstrations of the possibilities we have at hand for investigating the delicate interfaces that
connect (and divide) protein molecules and the neighboring electrolyte solution.
Key words: protein solution, protein hydration, protein surface, small angle scattering
PACS: 87.10.Tf, 87.14.E-, 87.15.A-, 87.15.N-
1. Introduction
The appearance of aqueous solutions of even large proteins is, in many cases, similar to that of
dilute solutions of simple salts: the liquid may be completely transparent, even though the size of so-
lute molecules may be two orders of magnitude larger than that of the solvent particles (i.e., dozens of
nanometers). This is made possible by strong interactions between the charged ‘surface’ of a protein and
the dipolar solvent molecules that surround a large particle; sometimes even tiny changes of the condi-
tions (of e.g., composition, temperature) can alter the situation completely and make protein molecules
aggregate and precipitate (see, e.g., reference [1, 2]).
The ‘surface’ of protein molecules in aqueous solutions may be considered as being defined by the
hydration sphere of a macromolecule. The natural tool for studying the hydration structure, within dis-
tances of a few Å, would be wide angle X-ray (and/or neutron) scattering — just as it is routinely done
for solutions of simple salts (see, e.g., reference [3]). However, due to a large number of components in
a solution (water, protein, stabilizers), as well as due to the complicated internal structure and relatively
lowmolar concentration of the protein, this route has not been very frequently chosen; examples of such
studies are references [4, 5].
Perhaps surprisingly, it is themicroscopic dynamics of the hydration sphere that has beenmore exten-
sively studied than the static structure: this can be readily understood by considering that most of the dy-
namical studies are based on examining the dynamics of water molecules only. NMR spectroscopy [6, 7],
dielectric relaxation spectroscopy [8, 9], as well inelastic neutron scattering [10, 11] have all been applied
for the purpose. More recently, terahertz (THz) spectroscopy has been used for tracking changes of the
broadly defined hydration layer, up to a thickness of about 1 nm [12, 13].
In the pursuit of revealing the surface of a protein molecule in solutions, small angle scattering
(SAS) [14–16] is our chosen experimental method for the present report. SAS provides a (or arguably,
the only) viable experimental possibility for studying the shape of a biomolecule in solution, as it has
been exemplified in references [16–18]. Unfortunately, the interpretation of SAS data is far from being
straightforward: this issue is considered in detail later in this work (see below).
© R. Tscheliessnig, L. Pusztai, 2016 13803-1
http://dx.doi.org/10.5488/CMP.19.13803
http://www.icmp.lviv.ua/journal
R. Tscheliessnig, L. Pusztai
In any case, to make the surface of a biomolecule ‘visible’, one needs to possess a high (most prefer-
ably, atomic) resolution picture of the molecule in solution. No experimental technique is capable of pro-
viding such pictures so far: for this reason, we must turn to computer simulations, such as the molecular
dynamics (MD) method [19]. Proteins and their solutions have been targeted by MD for quite some time,
due to the pioneering works of Karplus and co-workers (see, e.g., reference [20]). The MD methodology
will also be made use of extensively in the present work; more details will be provided in due course.
One way of defining the surface of a protein is to evaluate the ‘solvent inaccessible’ volume of the
biomolecule. In the cube method [21], the biomolecule is placed in a parallelepiped-shape box which is
subdivided into small cubes with edges of 0.5–1.5 Å. The boundary of the biomolecule is determined by
examining whether each cube belongs to the biomolecule or to the solvent (see, e.g., reference [22]). A
more complicated method is to calculate the ‘electron envelope’ of the macromolecule: an algorithm for
this is implemented in the program CRYSOL17 [23].
In general, due to a large variety of the ways the beta sheets and alpha helices are put into sequences
in protein molecules, the surface of such molecules is rather complicated. In the present contribution, we
consider that in general (or at least, in a large number of cases) the boundary of a protein molecule may
have a fractal dimension. We pursue this idea by presenting theoretical and experimental arguments; we
finish with providing computer simulation results based on simple concepts.
2. Scattering from fractal surfaces
What is a surface in terms of scattering theories? Small angle scattering may provide information
on surface areas that are larger and more uniform than that of a biomolecule (see references [14–16]);
we must, therefore, take a more indirect way. In fact, connections between the measured intensity and
fractal (or ‘rough’) surfaces have already been sought for [24–26]; note that these investigations have not
considered protein surfaces directly.
While the ‘reaction coordinate’, i.e., the location of a site of importance (e.g., of a scattering site)
within the investigated volume in a slit pore seems obvious, as it follows from the symmetry of the pore,
it is a complex task to determine if we deal with soft matter, e.g., proteins. Let us take a rather simple
protein: it will be formed by alpha helical domains (a typical one is indicated orange in figure 1) and
joined by random coils. For the present considerations, we have chosen a well-known globular protein,
selected out of thousands of possibilities: Bovine serum albumin (BSA) (for its crystalline structure see
reference [27]). We determine its point of reference (in other words, the ‘origin’ of the system). This is a
crucial step because it will mathematically determine what we term a fractal surface.
We assume scattering sites in the vicinity of, or indeed, within amino acids.We compute their centroid
and from their relative distance we compute the pair densities. The chosen system lacks any symmetry
Figure 1. (Color online) The point of reference. Left-hand panel: Any protein is a complex structure of
scattering sites. The issue to decide upon: which is the one we refer to? What are then reaction coor-
dinates? Red spheres mark the sites for which the Fourier transform of pair and radial distributions
provides comparable results. Their centroid is colored magenta, while blue spheres mark the centroid of
all of the protein sites. (From left to right: the important part considered is gradually enlarged.)
13803-2
Proteins: Fractal surfaces in solutions
and that is why we use equation (2.2) to determine the point of reference and compute, with respect to it,
the radial density function.
First, we sketch a mathematical methodology to access structural information from small angle and
neutron scattering data; this information will be related to the issue of the surface of a protein. We link
the distribution of scattering sites to the definition of α stable distributions [28, 29].
We assume that {Xi } and {X j } are random variables. Here, they are distances of scattering sites, with
respect to sites i or site j of those variables, and they are distributed according to a particular probability
density φ(ζ). The distribution is called stable if the probability density p(ζ) of any linear combination
Y =λ1X1 +λ2X2 then,
γ(ζ) =λ1λ2
∞∫
0
dζ′φ
(
(ζ−ζ′)λ1
)
φ(ζ′λ2). (2.1)
The distributions coincide subject to rescaling, i.e.,
〈F (γ(ζ))[Q]〉 = 〈|F (φ(λζ))[Q]|2〉. (2.2)
It is a significant extension to the formulation of Kotlarchyk’s work [30], as it includes scaling to relate
the pair density of scattering sites of a protein with their radial distribution.
Kotlarchyk relates the scattering intensity, I (Q), to the protein form factor P (Q) and the protein-
protein structure factor S(Q) by
I (Q) = P (Q)
{
1+β(Q)[S(Q)−1]
}
, (2.3)
wherein the term β(Q) = P?(Q)/P (Q) takes into account the possible anisotropic form of a protein. In
the previous paper [31] we introduced the fractal pendant to Debye’s formula:
JD [Q ζb] = JD/2−1(Q ζb)
(Qζb)D/2−1
. (2.4)
We did give clear evidence [31] that the fractal dimension, D , may be related to the Debye screening
length, and that it is not necessarily D = 3.
We rewrite the scattering intensity:
I (Q) =FD (γ(ζR ))[Q] =
∞∫
0
dζRζ
D−1
R γ(ζR )JD [Q ζb], (2.5)
as a function of the pair density of the protein scattering sites γ(ζR ). The pair density is a function of the
relative distances, ζR , between individual protein scattering sites. The protein form factor
P (Q) = 〈|F (Q)|2〉 =F 2
D (φ(ζb))[Q] =
∞∫
0
dζbζ
D−1
b φ(ζb)JD [Q ζb]2, (2.6)
however, is the Fourier transform of the radial probability density. It is a function of ζb , the distance with
respect to an arbitrary site, within the protein. Commonly, the center of mass of the protein centroid is
chosen.
In order to compute the protein anisotropy, β(Q), we need
P?(Q) = |〈F (Q)〉|2 =FD (φ(ζb))[Q]2 =
∞∫
0
dζbζ
D−1
b φ(ζb)JD [Q ζb]
2
. (2.7)
We shall not explore a detailed deduction but draft the essence, and provide motivation from the
observation of scalability by wet lab and computer experiments
I (Q) =FD (γ(ζR ))[Q] =
∞∫
0
dζRζ
D/λ−1
R γ(ζR )JD/λ[Q ζb]. (2.8)
13803-3
R. Tscheliessnig, L. Pusztai
Due to the scaling capability and alpha stability of the pair density and radial probability density, we
are allowed to introduce ζb =λζR and rewrite the scattering intensity in terms of the protein form factor:
P (Q) ∝
∞∫
0
dζbζ
D−1
b φ(ζb/λ)JD [Q ζb]2 =λD
∞∫
0
dζRζR
D−1φ(ζR )JD [QλζR ]2
∝
∞∫
0
dζRζR
D/λ−1φ(ζR )JD/λ[QζR ]2. (2.9)
The above is a set of equations that we term as ‘fractal scattering theory’.
3. Small angle neutron and X-ray scattering from biological soft matter
In this section we briefly discuss the origin of the fractal dimension D .
Small angle neutron and small angle X-ray scattering data were collected to obtain structural infor-
mation for BSA (concentration: 5 mg/ml) in three different aqueous salt environments (i.e., in different
electrolyte solutions). The data are displayed in figure 2. These three different environments contained
zero ammonium sulfate (state 1), 0.7 mol/kg ammonium sulfate (state 2), and 1.2 mol/kg (state 3). The
pH of the solutions is very close to neutral (just below 7) and these electrolyte concentrations are far
below the salting-out limit of the protein. Detailed description of the experiments can be found in refer-
ences [31, 32].
We argue that the parameter D is considered to be of electrostatic origin and proportional to the
salt concentrations in bulk solutions [32]. In figure 2 (right-hand panel) the pair density function of the
initial crystallographic model is shown as dashed blue line with grey marker. The fits (solid lines, left-
hand panel) were obtained by calculating the pair density function (solid blue line, right-hand panel) and
scattering intensity using equation (2.4), withD = 3. Clearly, irrespective of the ion concentrations, we do
not see significant changes in terms of the electronic contrast.
The SANS measurements were complemented by SAXS measurements for identical solutions. SAXS
data are presented in figure 3. Note the discrepancy between the results of the two experimental tech-
niques. Though the systems are identical, their scattering intensities I (Q) differ.
Typically, for small angle X-ray scattering, one is tempted to interpret the changes in I (Q) by the
changes in their individual pair density distributions, and then, consequently, argue the changes in the
protein conformation. However, this line of arguments is not supported by small angle neutron scattering
data.
Inwhat follows, we interpret the data differently: we leave the pair correlation untouched and change
the parameterD , whichmay be interpreted as a fractal Dimension. Note that the fits of experimental data
Figure 2. (Color online) SANS data of BSA at different salt concentrations. Left-hand panel: SANS data of
BSA dissolved in three different environments which contained zero ammonium sulfate (blue squares),
0.7 mol/kg (i.e., per kg of solution) ammonium sulfate (red circles), and 1.2 mol/kg (NH4)2SO4 (brown
triangles). Solid lines give fits of experimental data. Right-hand panel: Dashed blue line with greymarkers
indicates the pair density computed from a crystallographic model of BSA, whereas solid blue line marks
the pair density that corresponds to fits shown in the left-hand panel.
13803-4
Proteins: Fractal surfaces in solutions
Figure 3. (Color online) SAXS data of BSA at different salt concentrations. Left-hand panel: SAXS data of
BSA dissolved in three different environments which contained zero ammonium sulfate (blue squares),
0.7 mol/kg ammonium sulfate (red circles), and 1.2 mol/kg (brown triangles). Solid lines give fits of exper-
imental data, forD = 3.2,D = 2.9 andD = 2.5. Right-hand panel: Solid blue line indicates the pair density
computed from a crystallographic model of BSA. All data presented in the left-hand panel were computed
from it for different fractal dimensions.
have been achieved by changingD = 3.2,D = 2.9 andD = 2.5without changing the protein conformation.
We left the density distribution of the protein untouched. The linear relation of D to the ionic strength of
the particular solution is obvious. The higher the salt concentration the lower D should be put.
For a quick exploration of the effects of varyingD , we use a computational approach, usingmolecular
dynamics simulations (see next section).
4. The (fractal) surface of biomolecules: demonstration via computer
simulation
Having defined three different quantities, i.e., the form factor, the structure factor and the anisotropic
factor [β(Q)], it is time to explore these and put them in relation to computational approaches, such as
density functional theory [33]. Therefore, we set up three systems. We discuss two of them qualitatively,
whereas the third one we explore in detail. Since many theoretical systems, especially in the density
functional theory, deal with slit pores [33], we shall start with these.
From a mathematical point of view it is difficult to compute the pair distribution of an infinite planar
slit pore numerically, as one would need to compute the pairwise densities over all sites of a slit pore. The
sum, or moments of the sum, would not necessarily converge: one might think of particle interactions
that produce in plane pair densities that we may consider α stable. The common way out is to measure
and compute density distributions perpendicular to the surface.
Let us switch to spherical coordinates: we do so for different reasons. They seem mathematically
easier as well as they are very frequently applicable in soft matter as many a system investigated is of
spherical symmetry. In fact, it may be the experiment as well that imposes spherical symmetry to the
measured data, just as small angle X-ray and neutron scattering certainly do (see the previous section).
Let us rethink the planar slit pore to be an infinite spherical one. Then, we have to consider the point
of reference, in order to define a reaction coordinate, ζ. For planar slit pores, its particular symmetry
suggests to place the point of reference in the center of the slit pore. This may also be used for finite and
infinite spherical slit pores. Now, a spherical slit pore will consist of two concentric spheres. The inner
shell has a radius of r∞ while the outer one, a radius of r∞+∆. We define a radial density distribution by
exploiting the shift property of the Fourier transform:
〈|F (φ(λ (r∞+ζ)))[Q]|2〉 = 〈|F (φ(λζ))[Q]|2〉. (4.1)
We hereby reinterpret the planar slit pore to be an infinite spherical slit pore. We shift the point of the
origin next to one planar surface since it would be numerically cumbersome to compute the pair distri-
bution of an infinite spherical slit pore, but by the use of equation (2.2). Next, we drop the inner spherical
surface and replace it by a ‘protein’. We use a simple Lennard Jones (LJ) model. We used the molecular
13803-5
R. Tscheliessnig, L. Pusztai
dynamics package LAMMPS [34]. All parameters listed in the subsequent paragraphs are reduced to the
wall LJ parameters. To save computational time, we rescale the protein by a factor of five.
We compute the centroids of each amino acid and replace these by LJ sites. ‘Pair styles’, i.e., specific
parameters for the particular pairs of sites (for details, see the LAMMPS Manual [35]) between protein
and liquid were put to ε= 0.1 andσ= 2.5. The protein is positioned in its appropriate center, as computed
from (2.2). For simplicity, we fix the protein “amino acids” by springs to the centroids. The spring constant
that kept sites of the protein was put to k = 10. This value was chosen so that the liquid may slightly
penetrate the protein. Interactions within the protein were turned off. The protein is dissolved in a LJ
liquid. For liquid-liquid interactions, we constructed a hybrid potential by superposing two pair styles, a
lj/soft/cut and a gauss/cut. LJ parameters for liquid-liquid interactions were set to ε= 0.05 and σ= 1.5. A
repelling Gaussian potential was added to the liquid-liquid interactions, whose amplitude was set to 0.05.
A repelling distance of ζ= 1.0 and a variance of 1 were used. The liquid comprised 3553 sites.
The construction of wall and liquid is enclosed in a spherical wall of a diameter ζd = 32. It is a LJ wall
of type wall/lj93 (according to LAMMPS terminology) and parameterized as ε= 1.0 and σ= 1.0.
We performed simple NVE simulations and initially gave all sites to a velocity of 3. After equilibrations
of 500 steps, we performed simulations of 5000 steps. The systemwas reduced to a configuration as shown
in figure 4. In the right-hand panel of figure 4 we find the radial distribution of the protein (solid blue
line) and the radial distribution of the LJ liquid (solid red line with grey markers). Both were normalized
to their maximum value. We rescaled these results to run them comparable to the experimental data.
Arrows in figure 4 right-hand panel mark three regions. The reaction coordinate up to the blue arrow is
termed protein. We attribute the linear regime (in-between blue and white arrow) to the protein surface,
whereas the planar regime (in-between white and red arrow) is attributed to the LJ liquid bulk. The
corrugation in the radial density of the LJ liquid around 8 nm proves the liquid-like state of it.
In figure 5, scattering profiles for protein plus protein surfaces of different thicknesses are displayed.
All these complexes are in the linear regime shown by the insert of figure 4. While the blue line refers to
the hypothetical scattering profile of a blank protein, the orange lines refer to scattering profiles and pair
densities of the protein embedded in LJ liquid of different thickness. The pair densities are self-similar.
The larger is the construct, then the corresponding scattering profile is found more to the left.
In the structure model in figure 5 we discriminate the protein (blue beads) from the protein surface
(i.e., the ‘hydration shell’ of the protein, orange beads). The protein surface was determined as follows.
For each amino acid we computed ten closest LJ sites. These form the protein surface. Clearly, we do see
areas of low density of LJ sites in the protein surface surrounded by areas of high density of LJ sites.
It is evident that within the hydration shell, the local density of LJ sites differ. Their distribution is
(though influenced by the parameters chosen) altogether a consequence of the protein morphology. It is
a key difference from planar surfaces, where we expect a homogeneous distribution perpendicular to the
Figure 4. (Color online) MD simulation of a protein dissolved in a LJ liquid. Left-hand panel: a protein
(blue beads) is dissolved in a LJ liquid (grey beads). The simulation box is not periodic: both types of
particles are enclosed in a spherical wall. Right-hand panel: the radial density, φ(ζ) is displayed for the
protein (blue) and the LJ liquid (red line with grey markers). We distinguish the protein, the protein
surface and the LJ liquid bulk.
13803-6
Proteins: Fractal surfaces in solutions
Figure 5. (Color online) Self similar SAXS signals. Left-hand panel: SAXS profiles computed from pair
density distributions. The blue solid line mimics the scattering profile computed from the protein crys-
tallographic model. There was no background added or subtracted. The orange (solid and dashed) lines
give scattering profiles computed from the protein and from particles from the LJ liquid in the proximity
of the protein. Right-hand panel: The blue solid line gives the pair density distribution for the LJ protein.
The full and dashed orange lines indicate pair density distributions computed from the LJ protein and
particles from the LJ liquid that are in the proximity of the protein. They scale invariantly.
surface.
Another difference is the linearity of the hydration shell, while the spherical surface already enforces
a layered structure. This seems to suggest that protein fractal morphology extends the Henry regime to
higher bulk densities— a conjecture that needs clarification in the future.
5. Summary and outlook
In this work we provide a (somewhat limited) collection of mathematical formulae that may be use-
ful to link theoretical findings of classical density functional theory to experimental results derived from
scattering techniques, such as small angle neutron and small angle X-ray scattering.We discuss the neces-
sity of these and their fractal flavour. Though we lack a detailed mathematical discussion of the possible
physical origin, we have experimental evidence that may be found in the electrostatics of the system in-
vestigated. We compare experimental data from small angle neutron scattering to the data of small angle
X-ray scattering. While neutron scattering data do not change upon different salt concentrations, small
angle X-ray data do. These changes in the scattering data can be explained by a fractal dimension, which
is of electrostatic origin. We performedmolecular dynamics simulation and presented a structure model.
We distinguish protein from protein surface and find scale invariance for both.
Acknowledgements
ACIB is supported by the Federal Ministry of Economy, Family and Youth (BMWFJ), the Federal Min-
istry of Traffic, Innovation and Technology (BMVIT), the Styrian Business Promotion Agency SFG, the
Standortagentur Tirol and ZIT-Technology Agency of the City of Vienna through the COMET-Funding Pro-
grammanaged by the Austrian Research Promotion Agency FFG. LP aknowledges financial support from
the National Research, Development and Innovation Office of Hungary (NKFIH), grant no. SNN 116198.
References
1. Arakawa T., Timasheff S.N., Biochemistry, 1982, 21, No. 25, 6545; doi:10.1021/bi00268a034.
2. Arakawa T., Timasheff S.N., Biochemistry, 1984, 23, No. 25, 5912; doi:10.1021/bi00320a004.
3. Mile V., Pusztai L., Dominguez H., Pizio O., J. Phys. Chem. B, 2009, 113, 10760; doi:10.1021/jp900092g.
4. Makowski L., J. Struct. Funct. Genomics, 2010, 11, 9; doi:10.1007/s10969-009-9075-x.
5. Makowski L., Bardhan J., Gore D., Lal J., Mandava S., Park S., Rodi D.J., Ho N.T., Ho C., Fischetti R.F., J. Mol. Biol.,
2011, 408, No. 5, 909; doi:10.1016/j.jmb.2011.02.062.
13803-7
http://dx.doi.org/10.1021/bi00268a034
http://dx.doi.org/10.1021/bi00320a004
http://dx.doi.org/10.1021/jp900092g
http://dx.doi.org/10.1007/s10969-009-9075-x
http://dx.doi.org/10.1016/j.jmb.2011.02.062
R. Tscheliessnig, L. Pusztai
6. Bax A., Protein Sci., 2003, 12, No. 1, 1; doi:10.1110/ps.0233303.
7. Halle B., Philos. Trans. R. Soc. London, Ser. B, 2004, 359, 1207; doi:10.1098/rstb.2004.1499.
8. Nandi N., Bhattacharyya K., Bagchi B., Chem. Rev., 2000, 100, No. 6, 2013; doi:10.1021/cr980127v.
9. Murakra R.K., Head-Gordon T., J. Phys. Chem. B, 2008, 112, 179; doi:10.1021/jp073440m.
10. Frölich A., Gabel F., Jasnin M., Lehnert U., Oesterhelt D., Stadler A.M., Tehei M., Weik M., Wood K., Zaccai G.,
Faraday Discuss., 2009, 141, 117; doi:10.1039/B805506H.
11. General discussion, Faraday Discuss., 2009, 141, 175; doi:10.1039/B818384H.
12. Dexheimer S., Terahertz Spectroscopy: Pprinciples and Applications, Taylor & Francis, London, 2007.
13. Leitner D.M., Havenith M., Gruebele M., Int. Rev. Phys. Chem., 2006, 25, No. 4, 553;
doi:10.1080/01442350600862117.
14. Glatter O., Kratky O., Small Angle X-ray Scattering, Academic Press, New York, 1982.
15. Feigin L.A., Svergun D.I., Structure Analysis by Small-angle X-ray and Neutron Scattering, Plenum Press, New
York, 1987.
16. Putnam C.D., Hammel M., Hura G.L., Tainer J.A., Q. Rev. Biophys., 2007, 40, 191; doi:10.1017/S0033583507004635.
17. Zhang F., Roth R., Wolf M., Roosen-Runge F., Skoda M.W.A., Jacobs R.M.J., Sztucki M., Schreiber F., Soft Matter,
2012, 8, 1313; doi:10.1039/C2SM07008A.
18. Zhang F., Roosen-Runge F., Sauter A., Roth R., Skoda M.W.A., Jacobs R.M.J., Sztucki M., Schreiber F., Faraday
Discuss., 2012, 159, 313; doi:10.1039/c2fd20021j.
19. Allen M.P., Tildesley D.J., Computer Simulation of Liquids, Clarendon Press, Oxfords, 1987.
20. Karplus M., McCammon J.A., Nat. Struct. Biol., 2002, 9, No. 9, 646; doi:10.1038/nsb0902-646.
21. Fedorov B.A., Denesyuk A.I., FEBS Lett., 1987, 88, 114; doi:10.1016/0014-5793(78)80620-6.
22. Hubbard S.R., Small-angle x-ray scattering studies of calcium-binding proteins in solution, Ph.D. Thesis, Stanford
University, 1987.
23. Svergun D., Barberato C., Koch M.H.J., J. Appl. Crystallogr., 1995, 28, 768; doi:10.1107/S0021889895007047.
24. Wong P-Z., Bray A.J., J. Appl. Crystallogr., 1988, 21, 786; doi:10.1107/S0021889888004686.
25. Schmidt P.W., J. Appl. Crystallogr., 1991, 24, 414; doi:10.1107/S0021889891003400.
26. Foster T., Safran S.A., Sottmann T., Strey R., J. Chem. Phys., 2007, 127, 204711; doi:10.1063/1.2748754.
27. Majorek K.A., Porebski P.J., Dayal A., Zimmerman M.D., Jablonska K., Stewart A.J., Chruszcz M., Minor W., Mol.
Immunol., 2012, 52, 174; doi:10.1016/j.molimm.2012.05.011.
28. Zolotarev V.M., One-dimensional Stable Distributions, Translations of Mathematical Monographs Series, Vol. 65,
American Mathematical Society, Providence, 1986.
29. Mandelbrot B.B., The Fractal Geometry of Nature, Henry Holt and Company, New York, 1983.
30. Kotlarchyk M., Chen S.-H., J. Chem. Phys., 1983, 79, 2461; doi:10.1063/1.446055.
31. Horejs C., Gollner H., Pum D., Sleytr U.B., Peterlik H., Jungbauer A., Tscheliessnig R., ACS Nano, 2011, 5, 2288;
doi:10.1021/nn1035729.
32. Tscheliessnig R., Sommer R., Überbacher R., Pusztai L., Székely N., Jungbauer A., Peterlik H., Soft Matter, 2016
(submitted).
33. Pizio O., Sokołowski S., Condens. Matter Phys., 2014, 17, 23603; doi:10.5488/CMP.17.23603.
34. Plimpton S.J., J. Comput. Phys., 1995, 117, 1; doi:10.1006/jcph.1995.1039.
35. http://lammps.sandia.gov/doc/Manual.html.
Протеїни в розчинi: фрактальнi поверхнi у розчинах
Р. Челєсснiг1, Л. Пустаї2
1 Австрiйський центр промислової бiотехнологiї (ACIB), A-1190 Вiдень, Австрiя
2 Вiгнерiвський дослiдницький центр з фiзики, Угорська академiя наук, Будапешт, H-1121, Угорщина
Впроваджено концепцiюповерхнi протеїну у розчинi, а також границi роздiлу мiж протеїном та “об’ємним
розчином”. Коротко описано експериментальний метод розсiяння X-променiв при малих кутах i нейтрон-
ного розсiяння. Обговорено моделювання молекулярної динамiки як обчислювального iнструменту для
вивчення гiдратацiйної оболонки протеїнiв. Розроблено концепцiю протеїнових поверхонь з фракталь-
ним вимiром. Статтю завершено описом конкретного випадку експериментального (з використанням
розсiяння X-променiв при малих кутах) та комп’ютерного дослiдження, щоб продемонструвати наявнi
можливостi дослiдження делiкатних границь роздiлу молекул протеїну i розчину електролiту.
Ключовi слова: розчин протеїну, гiдратацiя протеїнiв, поверхня протеїну, розсiяння при малих кутах
13803-8
http://dx.doi.org/10.1110/ps.0233303
http://dx.doi.org/10.1098/rstb.2004.1499
http://dx.doi.org/10.1021/cr980127v
http://dx.doi.org/10.1021/jp073440m
http://dx.doi.org/10.1039/B805506H
http://dx.doi.org/10.1039/B818384H
http://dx.doi.org/10.1080/01442350600862117
http://dx.doi.org/10.1017/S0033583507004635
http://dx.doi.org/10.1039/C2SM07008A
http://dx.doi.org/10.1039/c2fd20021j
http://dx.doi.org/10.1038/nsb0902-646
http://dx.doi.org/10.1016/0014-5793(78)80620-6
http://dx.doi.org/10.1107/S0021889895007047
http://dx.doi.org/10.1107/S0021889888004686
http://dx.doi.org/10.1107/S0021889891003400
http://dx.doi.org/10.1063/1.2748754
http://dx.doi.org/10.1016/j.molimm.2012.05.011
http://dx.doi.org/10.1063/1.446055
http://dx.doi.org/10.1021/nn1035729
http://dx.doi.org/10.5488/CMP.17.23603
http://dx.doi.org/10.1006/jcph.1995.1039
http://lammps.sandia.gov/doc/Manual.html
Introduction
Scattering from fractal surfaces
Small angle neutron and X-ray scattering from biological soft matter
The (fractal) surface of biomolecules: demonstration via computer simulation
Summary and outlook
|