Fluid of fused spheres as a model for protein solution
In this work we examine thermodynamics of fluid with "molecules" represented by two fused hard spheres, decorated by the attractive square-well sites. Interactions between these sites are of short-range and cause association between the fused-sphere particles. The model can be used to stud...
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irk-123456789-1558112019-06-18T01:29:29Z Fluid of fused spheres as a model for protein solution Kastelic, M. Kalyuzhnyi, Yu.V. Vlachy, V. In this work we examine thermodynamics of fluid with "molecules" represented by two fused hard spheres, decorated by the attractive square-well sites. Interactions between these sites are of short-range and cause association between the fused-sphere particles. The model can be used to study the non-spherical (or dimerized) proteins in solution. Thermodynamic quantities of the system are calculated using a modification of Wertheim's thermodynamic perturbation theory and the results compared with new Monte Carlo simulations under isobaric-isothermal conditions. In particular, we are interested in the liquid-liquid phase separation in such systems. The model fluid serves to evaluate the effect of the shape of the molecules, changing from spherical to more elongated (two fused spheres) ones. The results indicate that the effect of the non-spherical shape is to reduce the critical density and temperature. This finding is consistent with experimental observations for the antibodies of non-spherical shape. В цiй роботi ми дослiджуємо термодинамiку плину з “молекулами”, представленими двома спаяними твердими сферами, якi декорованi вузлами з притягувальними потенцiалами типу квадратної ями. Взаємодiя мiж цими вузлами є короткодiюча i спричиняє асоцiацiю мiж частинками спаяних сфер. Модель може бути використана для дослiдження несферичних (чи димеризованих) протеїнiв у розчинi. Термодинамiчнi величини системи розраховуються за допомогою модифiкацiї термодинамiчної теорiї збурень Вертгайма, i результати порiвнюються з новими симуляцiями методом Монте Карло при iзобаричноiзотермiчних умовах. Зокрема, нас цiкавить фазове розшарування рiдина-рiдина в таких системах. Модельний плин використовується для оцiнки ефекту форми молекул, що змiнюється вiд сферичної до бiльш видовженої (двi спаянi сфери). Результати вказують, що ефект несферичної форми має зменшувати критичну густину i температуру. Це узгоджується з експериментальними спостереженнями для антитiл iз несферичною формою. 2016 Article Fluid of fused spheres as a model for protein solution / M. Kastelic, Yu.V. Kalyuzhnyi, V. Vlachy // Condensed Matter Physics. — 2016. — Т. 19, № 2. — С. 23801: 1–12. — Бібліогр.: 41 назв. — англ. 1607-324X DOI:10.5488/CMP.19.23801 arXiv:1603.07149 PACS: 87.15.A-, 87.15.ad, 87.15.km, 87.15.nr, 82.60.Lf, 64.70.Ja http://dspace.nbuv.gov.ua/handle/123456789/155811 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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In this work we examine thermodynamics of fluid with "molecules" represented by two fused hard spheres, decorated by the attractive square-well sites. Interactions between these sites are of short-range and cause association between the fused-sphere particles. The model can be used to study the non-spherical (or dimerized) proteins in solution. Thermodynamic quantities of the system are calculated using a modification of Wertheim's thermodynamic perturbation theory and the results compared with new Monte Carlo simulations under isobaric-isothermal conditions. In particular, we are interested in the liquid-liquid phase separation in such systems. The model fluid serves to evaluate the effect of the shape of the molecules, changing from spherical to more elongated (two fused spheres) ones. The results indicate that the effect of the non-spherical shape is to reduce the critical density and temperature. This finding is consistent with experimental observations for the antibodies of non-spherical shape. |
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Kastelic, M. Kalyuzhnyi, Yu.V. Vlachy, V. |
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Kastelic, M. Kalyuzhnyi, Yu.V. Vlachy, V. Fluid of fused spheres as a model for protein solution Condensed Matter Physics |
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Kastelic, M. Kalyuzhnyi, Yu.V. Vlachy, V. |
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Kastelic, M. |
| title |
Fluid of fused spheres as a model for protein solution |
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Fluid of fused spheres as a model for protein solution |
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Fluid of fused spheres as a model for protein solution |
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Fluid of fused spheres as a model for protein solution |
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Fluid of fused spheres as a model for protein solution |
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fluid of fused spheres as a model for protein solution |
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Інститут фізики конденсованих систем НАН України |
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| citation_txt |
Fluid of fused spheres as a model for protein solution / M. Kastelic, Yu.V. Kalyuzhnyi, V. Vlachy // Condensed Matter Physics. — 2016. — Т. 19, № 2. — С. 23801: 1–12. — Бібліогр.: 41 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kastelicm fluidoffusedspheresasamodelforproteinsolution AT kalyuzhnyiyuv fluidoffusedspheresasamodelforproteinsolution AT vlachyv fluidoffusedspheresasamodelforproteinsolution |
| first_indexed |
2025-07-14T08:02:41Z |
| last_indexed |
2025-07-14T08:02:41Z |
| _version_ |
1837608637849141248 |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 2, 23801: 1–12
DOI: 10.5488/CMP.19.23801
http://www.icmp.lviv.ua/journal
Fluid of fused spheres as a model for protein solution
∗
M. Kastelic1, Yu.V. Kalyuzhnyi2, V. Vlachy1
1 Faculty of Chemistry and Chemical Technology, University of Ljubljana, Večna pot 113, 1000 Ljubljana, Slovenia
2 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
Received November 17, 2015, in final form December 1, 2015
In this work we examine thermodynamics of fluid with “molecules” represented by two fused hard spheres,
decorated by the attractive square-well sites. Interactions between these sites are of short-range and cause
association between the fused-sphere particles. The model can be used to study the non-spherical (or dimer-
ized) proteins in solution. Thermodynamic quantities of the system are calculated using a modification of
Wertheim’s thermodynamic perturbation theory and the results compared with new Monte Carlo simulations
under isobaric-isothermal conditions. In particular, we are interested in the liquid-liquid phase separation in
such systems. The model fluid serves to evaluate the effect of the shape of the molecules, changing from spher-
ical to more elongated (two fused spheres) ones. The results indicate that the effect of the non-spherical shape
is to reduce the critical density and temperature. This finding is consistent with experimental observations for
the antibodies of non-spherical shape.
Key words: non-spherical proteins, liquid-liquid transition, directional forces, aggregation, thermodynamic
perturbation theory
PACS: 87.15.A-, 87.15.ad, 87.15.km, 87.15.nr, 82.60.Lf, 64.70.Ja
1. Introduction
Aggregation of proteins in solution is both desired and undesired. It represents the first step in the
downstream processing, i.e., salting out of the proteins for the purpose of cleaning and application. It
is also one of the intermediate steps in the process of protein crystallization. The unwanted, pathologi-
cal, protein aggregation is known to cause several diseases. Very importantly, bio-pharmaceutical drugs
should be free of aggregates, otherwise they may be harmful. To increase the stability of protein in aque-
ous solutions is, therefore, an important technical problem. For an excellent review of the theoretical and
experimental studies of protein solutions see reference [1].
The class of proteins we are interested in here are the so-called globular proteins. A typical represen-
tative of this class is lysozyme, which was extensively studied both experimentally and theoretically (see
for example [1], Chapter 9). Despite their complicated 3D structure, many properties of protein solutions
can be successfully described using relatively simple models [2–8]. Globular proteins are in their native
form (we assume that during the experimental treatment they do not denature) most often pictured as
perfectly spherical objects. This naïve representation is in reality never satisfied, it is used merely to sim-
plify the calculations. There is a large list of non-spherical proteins, for example the shape of lysozyme
mentioned above is ellipsoidal, including antibodies, lactoferrin, and others, where more complex ge-
ometry of particles would need to be taken into account to generate realistic results. This is important
because the interactions leading to protein aggregation are directional and of short-range.
The shape of protein molecules influence their mutual interaction directly and indirectly. For exam-
ple, (i) depletion interaction is largely dependent on the shape of particles [9]. (ii) Experimentally, it is
observed that many of proteins with roughly spherical shape exhibit upper critical solution temperature
∗
We dedicate this contribution to our friend and coworker Professor A.D.J. Haymet on occasion of his 60
th
birthday.
©M. Kastelic, Yu.V. Kalyuzhnyi, V. Vlachy, 2016 23801-1
http://dx.doi.org/10.5488/CMP.19.23801
http://www.icmp.lviv.ua/journal
M. Kastelic, Yu.V. Kalyuzhnyi, V. Vlachy
at protein concentration equal to 240 g/L [1, 10, 11]. In contrast to that, Y-shaped antibodies exhibit the
shift toward much lower values, way down to 100 g/L [12, 13]. (iii) The hydrodynamic radii of the non-
spherical objects are different, therefore, their hydrodynamic and transport coefficients [14], as well as,
kinetic parameters [15] are modified. It is also known that classical nucleation theory has difficulties in
describing the crystallization of other than spherical (for example ellipsoidal) particles [1, 16].
Recently, we used a simple spherical model [8] to analyze experimental results for the cloud-point
temperatures in aqueous protein solutions with added salts [10, 11]. We modelled the solution as a one-
component system; the protein molecules were represented as perfect spheres, embedded in the contin-
uum solvent composed of water, buffer, and various simple salts. The attractive short-range interactions
between the proteins were due to the square-well sites located on the surface of protein molecules. The
model was numerically evaluated using Wertheim’s perturbation theory [17–19]. The short-range and di-
rectional nature of the interactions among proteins led to good agreement with the experimental data for
the liquid-liquid phase diagram in case of lysozyme and γ-crystallin solutions [10, 11]. With knowledge
of the experimental cloud-point temperature as a function of composition of electrolyte present in the
system, the model gave predictions for the liquid-liquid coexistence curves, the second virial coefficients,
and other similar properties under such experimental conditions.
One weakness of the model presented above was its simplified geometry. Neither lysozyme nor other
proteins are spherical, and some of them for example, lactoferrin [20] look more like two fused spheres.
The other weakness was that we assume for protein molecules to exist in form of monomers, which is
not true. Even in very dilute solutions, proteins can be at least partially dimerized. The purpose of the
present work is to investigate how the relaxing of these two basic assumptions influence the liquid-liquid
coexistence curve.
The models for the association of spherically symmetric particles into dimer molecules are of consid-
erable interest to science and technology and have been actively studied earlier. Of particular interest for
us are the models where there is an inter-penetration (“fusing” of cores) of the spherical particles upon
association so that the bonding length L is less than the core diameter σ. The “shielded sticky shell” and
the “shielded sticky point” models of Stell and co-workers [21–23] and their extensions [24–26], belong
to this group of models and are the starting point for our work. These types of the models were studied
using regular [21, 22, 24] and multi-density [23, 25–29] integral equation theories.
In the present study, we use spherical particles as building blocks, which are fused together to form
a new species. In this way, we compose the molecule with arbitrary spacing L between the centers of
spheres. Next, we decorate the surfaces of fused-spheremolecules with the attractive short-range binding
sites, which may cause the association of the newly formed molecules. Such an extension of the protein
model follows from our previous work [8]. Here, wewish to explore the effects of the non-spherical shape
on various thermodynamic properties.
Different versions of the model of dimerizing particles, represented by the tangentially bonded chain
molecules, have been studied earlier [30, 31]. In this type of themodel, dimerization occurs due to square-
well bonding site, placed on the surface of one of the hard-sphere terminal monomer of each chain. The-
oretical description of the model was carried out using first order thermodynamic perturbation theory
(TPT1) of Wertheim [17, 18]. There are two major features of our model that set it apart from the models
studied earlier, i.e., (i) in our model the molecules are represented by the two hard-sphere monomers
fused at a distance L, which is less than the contact distance σ and (ii) the molecules upon association can
form a three-dimensional network. Due to the former feature of the model, a straightforward application
of Wertheim’s multi-density approach fails to produce accurate results [29, 32, 33]. In the present work,
we use a modified version of the TPT1, which takes into account the change of the overall packing frac-
tion of the system due to the association forces [29, 34, 35]. The accuracy of our modified TPT1 approach
is checked by the newly generated Monte Carlo simulation data.
2. Model, theory, and simulations
2.1. Model
We introduce a one component model of spherical particles, decorated with additional binding sites
of two different types, A and B. The binding site A is placed within the sphere, with the displacement
23801-2
Fluid of fused spheres
Figure 1. (Color online). Spherical particles with diameter σ are capable of, due to the attraction among
sites of type A, penetrating to form fused sphere molecules. The cross interactions A–B are prohibited. In
this figure, KB = 3.
dA Éσ/2, while an arbitrary number KB of binding sites of type B is located on the surface of the sphere
(the displacement dB =σ/2). The model is visualized in figure 1. We consider a special case, where we
exclude the cross interactions among sites A and B. The total pair potential is written as follows:
u(r) = uR(r )+
B∑
M=A
uM M (xM M ), (2.1)
where uR is the pair potential for hard spheres, and uAA and uBB are inter-particle site-site potentials.
The vector r (r = |r|) connects the centers of hard spheres, and xM M denotes the inter-particle vector
connecting two sites of the type M . As mentioned above, uM M is the orientation dependent square-well
potential between the sitesM ∈ {A,B}, defined as follows:
uM M (xM M ) =
{
ε′M M =−εM M −ξM M , for |xM M | < aM M ,
0, for |xM M | Ê aM M .
(2.2)
The site A causes inter-penetration of particles (see figure 3). Note that we need the term ξAA→∞ to com-
pensate for the hard sphere repulsion. For the periphery sites B, we do not need such a term, therefore
ξBB→ 0. To fuse hard cores at separation L, we choose dA = L/2 and take the limit εAA→∞.
2.2. Theory
An appropriate theoretical approach to be used is the first-order Wertheim’s thermodynamic pertur-
bation theory (TPT1) [17, 18]. According to this theory, the Helmholtz free energy of the system can be
written as a sum of several terms:
A = Aid+ Ahs+ AA–A+ AB–B, (2.3)
where Aid+ Ahs = AR is the free energy of the reference system represented by the hard-sphere sys-
tem [36] and AA–A+ AB–B = Aass is the contribution due to A–A and B–B interactions. Following Chapman
et al. [19], we have:
β(A− AR)
N
= βAA–A
N
+ βAB–B
N
, (2.4)
βAA–A
N
= ln XA − 1
2
XA + 1
2
, (2.5)
βAB–B
N
= KB
(
ln XB − 1
2
XB + 1
2
)
. (2.6)
23801-3
M. Kastelic, Yu.V. Kalyuzhnyi, V. Vlachy
Here, β = (kBT )−1
and kB is Boltzmann’s constant as usual, T is the absolute temperature, and N is the
number of spheres. Further, XM defines the average number fraction of particles, which are not bonded
through the binding siteM . Parameters XA and XB are determined by the mass-action law [19]
XA = 1
1+ρ(
XA∆AA +KBXB∆AB
) , (2.7)
XB = 1
1+ρ(
XA∆BA +KBXB∆BB
) , (2.8)
where ρ = N /V is the number density of spheres and ∆M N connects the pair distribution function of
hard spheres g hs(r ) (reference system) and the binding potential for sites M and N . The corresponding
∆M N parameters are:
∆M N = 4π
dM+dN+aM N∫
dM+dN
g hs(r ) f̄M N (r )r 2dr ∀ M , N ∈ {A,B}. (2.9)
Expression for the solid-angle averaged Mayer function
f̄M N (r ) =
∫ ∫
fM N
(
xM N (r)
)
dΩM dΩN (2.10)
was initially derived by Wertheim [37] and further generalized here to be
f̄M N (r ) = exp
(−βε′M M
)−1
24dM dN r
(aM N +dM +dN − r )2(2aM N −dM −dN + r ). (2.11)
To suppress the cross interactions A–B, we set ∆AB =∆BA = 0, which finally yields two independent equa-
tions, written in a quadratic form
ρ∆AAX 2
A
+XA−1 = 0, (2.12)
ρKB∆BBX 2
B
+XB−1 = 0. (2.13)
2.2.1. Association parameters ∆AA and ∆BB
The association parameter∆AA is related to XA via equation (2.12) and to the free energy contribution
due to A–A binding, by equation (2.5). For the complete association limit, i.e., fusing of hard cores at
separation L, no monomer spheres are present, so XA = 0. We re-write the association parameter ∆AA
and introduce the cavity correlation function yhs(r ) to obtain
∆AA = 4π
2dA+aAA∫
2dA
yhs(r )ehs(r ) f̄AA(r )r 2dr, (2.14)
where ehs(r ) = exp[−βuR(r )]. Note that, as already mentioned before, 2dA = L. By applying the sticky
limit approximation [37], that is by assuming the constant value of yhs
within the integration domain, we
obtain
∆AA = yhs(r = 2dA)IAA, (2.15)
IAA = 4π
2dA+aAA∫
2dA
ehs(r ) f̄AA(r )r 2dr. (2.16)
The integral given by equation (2.16) is not used in further calculations and, accordingly, it will not be
considered in more detail here.
Parameter∆BB determines the degree of association of fused spheres and the free energy contribution
due to the B–B binding, see equations (2.6) and (2.13). Notice that due to the association between A-type
23801-4
Fluid of fused spheres
of the sites, the packing fraction of fused spheres ηeff is different from the packing fraction originally
present (un-fused) hard spheres η. These fractions are related as follows:
ηeff = D(l∗)η, (2.17)
D(l∗) = 1
2
(
1+ 3
2
l∗− 1
2
l∗3
)
, (2.18)
where η=πρσ3/6 is the packing fraction of hard spheres and l∗ = L/σ is reduced A–A bonding distance.
Using the sticky limit approximation [37] for ∆BB [equation (2.9)], we have:
∆BB = g hs(r =σ,η= ηeff)IBB, (2.19)
IBB = 4π
σ+aBB∫
σ
f̄BB(r )r 2dr . (2.20)
The integral in IBB can be evaluated analytically. We have used the Carnahan-Starling approximation for
the contact value of g hs
at the effective packing fraction of fused spheres ηeff
g hs(r =σ,η= ηeff) = 2−ηeff
2(1−ηeff)3 . (2.21)
2.2.2. Cavity correlation function y hs
The last unknown quantity in equation (2.15) is the cavity correlation function of hard sphere fluid,
yhs
. It is calculated by using the Tildesley-Streett expression for pressure of the hard dumbbell fluid [38].
By choosing KB = 0, dA É σ/2 and applying sticky limit conditions, i.e., εAA →∞, aAA → 0 while keeping
the integral in equation (2.14) finite, our model reduces to the hard dumbbell fluid. We modify the mass
action law [equation (2.12)], by inserting equation (2.15) with XA = ρ0/ρ, where ρ0 stands for the number
density of spheres, not bonded through binding site A (monomers). The result represents a different form
of equation (110) of Wertheim’s paper [37]
ρ = ρ0 +ρ2
0IAAyhs(r = L). (2.22)
Following Wertheim [37], we get the expression for the excess pressure in the form:
β(P −PR) =−1
2
(ρ−ρ0)
{
1+ρ ∂ ln[yhs(r = L)]
∂ρ
}
. (2.23)
We are now in position to obtain the cavity correlation function yhs
of hard sphere system. We use the
Carnahan-Starling equation of state [36] for the reference system (PR) and the Tildesley-Streett equation
of state [38] for the perturbed hard dumbbell system (P ).
• Carnahan-Starling EOS:
βPR
ρ
= 1+η+η2 −η3
(1−η)3 . (2.24)
• Tildesley-Streett EOS:
βP
ρd
= 1+(1+Ul∗+V l∗3)ηeff+(1+W l∗+X l∗3)η2
eff−(1+Y l∗+Z l∗3)η3
eff
(1−ηeff)3 , (2.25)
where ρd = ρ/2 is the number density of hard dumbbells. The set of numerical parametersU , V ,W , X ,
Y , Z is given in table 1.
23801-5
M. Kastelic, Yu.V. Kalyuzhnyi, V. Vlachy
Table 1. Parameters in the Tildesley-Streett EOS [38].
U V W X Y Z
0.37836 1.07860 1.30376 1.80010 2.39803 0.35700
Within the framework of Wertheim’s theory, we must set ρ0 = 0 in equation (2.23) to recover the
fluid of hard dumbbell particles (no monomers present). Next, we use equations (2.17), (2.18), (2.23) and
equations of state [(2.24) and (2.25)] to obtain the derivative
ρ
∂ ln[yhs(r = L)]
∂ρ
= −
∑6
i=1 aiη
i
1+∑6
i=1 biηi
. (2.26)
The set of equations which determine ai and bi [D ≡ D(l∗)] are as follows:
A = (1+Ul∗+V l∗3)D, (2.27)
B = (1+W l∗+X l∗3)D2, (2.28)
C = (1+Y l∗+Z l∗3)D3, (2.29)
with the arrays
a1 = A+3D −8, b1 =−3(1+D),
a2 =−3A+B +15D −3D2 +4, b2 = 3(1+3D +D2),
a3 = 3A−3B −C −3D −15D2 +D3, b3 =−(1+9D +9D2 +D3),
a4 =−A+3B +3C −3D +3D2 +5D3, b4 = a2D,
a5 =−B −3C +3D2 −D3, b5 = a1D2,
a6 =C −D3, b6 = D3.
It is obvious ρ∂ ln[yhs(r = L)]/∂ρ = η∂ ln[yhs(r = L)]/∂η, therefore equation (2.26) can be easily inte-
grated to yield:
ln[yhs(r = L)] = −
η∫
0
∑6
i=1 ai t i−1
1+∑6
i=1 bi t i
dt . (2.30)
The integral was checked to be non-singular for all investigated η and l∗ values. Numerical results for
ln[yhs(r )] are for a few fluid densities shown in figure 2.
Figure 2. Logarithm of the cavity distribution function yhs
of hard spheres for ρσ3
is equal to: 0.4 (dashed
line), 0.6 (dashed-dotted line) and 0.8 (solid line). The limit of yhs
, limr→σ yhs(r ), coincides with equa-
tion (2.21) for the “non-effective” packing fractions η.
23801-6
Fluid of fused spheres
2.2.3. Other thermodynamic properties
Next, we calculate the excess pressure Pass = P −PR and the excess chemical potential µass = µ−µR
needed in phase diagram calculations as also excess internal energy Eass = E −ER, due to association.
Starting with the pressure, we have
β(P −PR) = ρ2 ∂
[
β(A− AR)/N
]
∂ρ
= ρ
B∑
M=A
(∂[β(A− A R)/N
]
∂XM
)(
η
∂XM
∂η
)
. (2.31)
By inserting the appropriate derivatives from equations (2.4), (2.12), and (2.13), we get the final expres-
sion for the excess pressure. The second term B is evaluated at ηeff, see equation (2.21), therefore upon
differentiation we get an additional factor D(l∗)
β(P −PR) = βP AA +βP BB, (2.32)
βP AA = −ρ
2
(1−XA)
{
1+η∂ ln[yhs(r = L)]
∂η
}
, (2.33)
βP BB = −ρ
2
KB(1−XB)
{
1+D(l∗)η
∂ ln[g hs(r =σ)]
∂η
∣∣∣
η=ηeff
}
. (2.34)
The expression η∂ ln[yhs(r = L)]/∂η is obtained from equation (2.26), while the second derivative
∂ ln[g hs(r = L)]/∂η
∣∣
η=ηeff is obtained analytically at η= ηeff from equation (2.21)
∂ ln[g hs(r = L)]
∂η
∣∣∣
η=ηeff
= 5−2ηeff
(1−ηeff)(2−ηeff)
. (2.35)
The excess chemical potential µass =µ−µR is obtained through the relation
µass = Aass
N
+ Pass
ρ
. (2.36)
The logarithmic term ln XA in equation (2.5) is divergent for the complete association limit (∆AA À 1),
therefore we re-write this term by using equation (2.15) as follows:
ln XA = ln
(−1+ √
1+4ρ∆AA
2ρ∆AA
)
≈ ln
( √
4ρ∆AA
2ρ∆AA
)
=−1
2
ln[ρyhs(L)]− 1
2
ln[IAA(β)]. (2.37)
The second term in equation (2.37) is independent of density and, accordingly, does not contribute to the
pressure. The expression for P is the same as derived before [equations (2.32)–(2.34)]. The equilibrium
conditions require the equality of chemical potential at a constant temperature (see equations below), so
the second term in equation (2.37) cannot affect the coexistence curve. The equilibrium conditions read:
µ(ρI,T ) = µ(ρII,T ), (2.38)
P (ρI,T ) = P (ρII,T ), (2.39)
where ρI and ρII are the two coexisting densities. At this step, the phase diagram can be constructed by
applying equations (2.38)–(2.39) as it is in more detail explained in the previous work [8].
Another thermodynamic quantity is the excess internal energy Eass = E −ER, obtained as
E −ER
N
= ∂
[
β(A− AR)/N
]
∂β
= EA–A
N
+ EB–B
N
, (2.40)
EA–A
N
= −ρ
2
X 2
A
∂∆AA
∂β
, (2.41)
EB–B
N
= −ρ
2
(KBXB)2 ∂∆BB
∂β
. (2.42)
Since EA–A/N is divergent, the only relevant part is EB–B/N . Derivative ∂∆BB/∂β is obtained analytically
from equations (2.19)–(2.20), since g hs
is β independent. Thermodynamic functions for the reference
system of hard spheres, βAR/N , βPR, uR and ER/N , can be found elsewhere [36].
23801-7
M. Kastelic, Yu.V. Kalyuzhnyi, V. Vlachy
2.3. N , P, T Monte Carlo simulation
To validate the accuracy of the modified TPT1 approach, we performed Monte Carlo computer sim-
ulations in the N ,P,T ensemble [39]. We assumed fused spheres with one and two binding sites on
each sphere, where the prescribed arrangement of sites was preserved during the simulation. Simulated
molecules are schematically shown in figure 3. We adopted the sampling method suggested by Tildesley
and Streett [40], where a single displacement parameter was needed to describe the translation and ro-
tation of fused spheres. The simulation box contained 250 fused spheres (molecules), which is equivalent
to 500 penetrating (original, un-fused) spheres. We defined the cycle with 250 attempts to move the ob-
ject and by 1 attempt to change the volume box. Next, we defined the block to be equal to 5×104
cycles.
Initially, we performed 1 block, to equilibrate the system, while 4 independent blocks were needed to
calculate thermodynamic properties via the block averaging. Simulations were performed for three l∗
values: 0.2, 0.6, 1.0, and four different pressures PkBT /σ3
: 0.5, 1.0, 2.0, and 4.0, for each model object
visualized in figure 3. The acceptance rate of trial configurations was between 0.2 and 0.6.
Figure 3. (Color online). Different molecules in N ,P,T Monte Carlo simulations: fixed binding sites on the
opposite poles, KB = 1 (a), and more complex geometry with two binding sites on each sphere, KB = 2 (b).
In the last example we set αBB = π/2 with perpendicular orientation of lines, connecting sites B on each
sphere. Center-to-center separation L (l∗ = L/σ) and displacement distance dB =σ/2 were fixed.
3. Results and discussion
3.1. Thermodynamic properties: Theory against Monte Carlo simulations
To test the accuracy of TPT1 we performed N ,P,T Monte Carlo simulations for values of KB equal to
1 and 2 (three l∗ values for each KB). We chose to compare the pressure P and the internal energy due
to B–B binding, EB–B. Since we used the complete A–A association limit within TPT1, the latter quantity,
EB–B, was the one that could be directly compared to computer simulations. We fixed the temperature
T ∗ = kBT /ε= 1, while the pair potential characteristics are given in table 2. The comparison between the
Table 2. Pair potential parameters used for testing TPT1 againt simulations.
aBB: 0.1σ
εBB: 5.0ε
dB: 0.5σ
theory and simulations is presented in figure 4. We found very good agreement for the pressure, while
the theoretical predictions for EB–B were less accurate. In case of l∗ = 1we obtained very good agreement
for KB = 1 and fair agreement for KB = 2 (black lines and corresponding symbols in EB–B/NFSkBT sub-
figures). If we reduced the l∗ values (blue and red lines, symbols), the deviations became larger, though
the qualitative picture remained correct. Deviations at low l∗ could be caused by the facts that: (i) fusing
of two spheres at small l∗ is not a small perturbation regarding the reference system of hard spheres,
23801-8
Fluid of fused spheres
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.0 0.1 0.2 0.3 0.4
E
B
−
B
/N
F
S
k
B
T
ηeff
(a)
0
1
2
3
4
5
0.0 0.1 0.2 0.3 0.4
P
k
B
T
/σ
3
ηeff
(b)
-1.8
-1.5
-1.2
-0.9
-0.6
-0.3
0.0
0.0 0.1 0.2 0.3 0.4
E
B
−
B
/N
F
S
k
B
T
ηeff
(c)
0
1
2
3
4
5
0.0 0.1 0.2 0.3 0.4
P
k
B
T
/σ
3
ηeff
(d)
Figure 4. (Color online). Pressure P and association energy EB–B per pair of fused spheres, thus NFS =
N /2. The calculations are presented by lines and the corresponding simulation results by symbols. We
studied three different l∗ values: 0.2 (red,N), 0.6 (blue, ■) and 1.0 (black, •). Panels, (a) and (b), belong
to KB = 1 and the panels, (c) and (d), to the case with KB = 2. Calculation apply to T∗ = kBT /ε = 1; pair
potential parameters are listed in table 2, note that L = 2dA. Uncertainties of simulation are within the
size of symbols.
and (ii) the arrangement of binding sites B is fixed during the simulation, which is not the case in TPT1,
where the orientation average over all geometries was assumed.
3.2. Effect of protein’s shape on the liquid-liquid phase diagram
To illustrate the influence of protein shape on the liquid-liquid phase behavior, we compared the
phase diagrams for two versions of the model: model (I) of two fused hard spheres and the model (II)
of equivalent hard sphere, which was defined as a limiting case of (I), when L → 0 and σ→ deqv. The
latter was chosen in such a way, that the volume of two fused spheres in case (I) was equal to that of the
equivalent sphere (II): deqv = σ
3√
1+3l∗/2− l∗3/2. Example (II) might be interpreted as the usual hard
sphere model of diameter deqv, with 2KB of sites B, i.e., the same number as on the two fused spheres. In
such an interpretation, the A–A contributions to the physical properties can be neglected. Describing the
aggregation of fused spheres of diameter deqv within the limiting conditions l∗ → 0 andD(l∗) → 1
2 , where
ηeff = η/2 =πρd 3
eqv
/12 =πρdd 3
eqv
/6, led us to themodel examined in reference [8]. Other parameters and
relations between examples (I) and (II) are listed in table 3.
In figure 5, we show phase diagrams for the variants (I) and (II) described above, at three l∗ values
and for different numbers of sites B. As observed before [41], an increase of the number of sites B shifts
the critical density toward higher values. What is more interesting here is the effect of the separation
distance parameter l∗ on the phase behavior. For a sufficiently small l∗, i.e., l∗ = 0.2 — see figure 5 (a),
the difference between the phase diagrams for two versions of the model becomes negligible, regardless
of the KB value. If centers of fused spheres are located at larger distance, l∗ = 0.6 — see figure 5 (b),
the difference becomes more pronounced: both critical temperature and density are lowered. Deviations
23801-9
M. Kastelic, Yu.V. Kalyuzhnyi, V. Vlachy
Table 3. Pair potential parameters and relations used in investigation of effects of protein’s shape.
two fused spheres at L (I) equivalent sphere (II)
diameter: σ deqv =σ 3p
1+1.5l∗−0.5l∗3
ηeff : D(l∗)η πρdd 3
eqv
/6
εBB: ε ε
dB: σ/2 deqv/2
aBB: 0.1σ 0.1σ
number of sites B: KB (per building block) 2KB
become the strongest for the limiting example of two spheres fused in contact, that is for l∗ = 1.0, cf.
figure 5 (c). In this case, the larger number of sites (larger area available for interaction) additionally
affects the liquid-liquid phase diagram. The shift toward lower critical densities (or packing fractions) is
consistent with experimental studies of the Y-shaped antibodies [12, 13].
4. Conclusions
Proteins come in many shapes, from ellipsoidal to Y-like and are never perfectly spherical as treated
by most theoretical models. Further, even in dilute solutions they have a tendency to form dimers and
can be represented by two fused spheres. For dense systems close to precipitation, the actual geometry
of the protein molecules is important; the inter-particle interactions are directional and of short-range.
In the present study, we modify the first-order thermodynamic perturbation theory for associating fluids
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0 0.1 0.2 0.3 0.4 0.5 0.6
k
B
T
/ε
ηeff
KB = 2 (4)
KB = 3 (6)
KB = 4 (8)
(a)
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0 0.1 0.2 0.3 0.4 0.5 0.6
k
B
T
/ε
ηeff
KB = 2 (4)
KB = 3 (6)
KB = 4 (8)
(b)
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0 0.1 0.2 0.3 0.4 0.5 0.6
k
B
T
/ε
ηeff
KB = 2 (4)
KB = 3 (6)
KB = 4 (8)
(c)
Figure 5. (Color online). Phase diagrams for different l∗ values: 0.2 (a), 0.6 (b), and 1.0 (c), color notation
is the same as in figure 4. Model of two fused spheres (I) is denoted by dashed lines and the limitingmodel
of equivalent sphere (II) by solid lines. The results apply to three different KB values, written without (I)
and within brackets (II), respectively.
23801-10
Fluid of fused spheres
to be applicable to the models allowing hard-sphere particles to inter-penetrate. These particles can fur-
ther aggregate. We confront theoretical predictions for thermodynamic properties of the proposedmodel
with predictions of the correspondingMonte Carlo simulations. We obtain an excellent agreement for the
pressure and fair agreement for the excess internal energy. Next, we use this model to predict the liquid-
liquid phase diagram for protein solutions. We are interested in the effects of protein shape on the phase
coexistence curve.We show that the fused hard-spheremodel reduces the critical density of the system in
comparison with the same quantity calculated for the hard-sphere model. This finding is consistent with
experimental observations for Y-shaped antibodies. Using the cloud-point temperature measurements,
we currently investigate the influence of various salts on the stability of lactoferrin solutions in water.
The latter protein has a shape of two fused spheres, and the hard-sphere model is not a good represen-
tation of it. Theoretical approach developed in this paper will be used to analyze experimental data for
lactoferrin and some other proteins of non-spherical geometry.
Acknowledgements
This study was supported by the Slovenian Research Agency fund (P1-0201), NIH research grant
(GM063592), and by the Young Researchers Program (M.K.) of the Republic of Slovenia.
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Плин зi спаяних сфер як модель розчину протеїнiв
М. Кастелiч1,Ю.В. Калюжний2, В. Влахi1
1 Факультет хiмiї i хiмiчної технологiї, Унiверситет Любляни, вул. Вечна, 113, 1000 Любляна, Словенiя
2 Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
В цiй роботi ми дослiджуємо термодинамiку плину з “молекулами”, представленими двома спаяними
твердими сферами, якi декорованi вузлами з притягувальними потенцiалами типу квадратної ями. Вза-
ємодiя мiж цими вузлами є короткодiюча i спричиняє асоцiацiю мiж частинками спаяних сфер. Модель
може бути використана для дослiдження несферичних (чи димеризованих) протеїнiв у розчинi. Термо-
динамiчнi величини системи розраховуються за допомогою модифiкацiї термодинамiчної теорiї збурень
Вертгайма, i результати порiвнюються з новими симуляцiями методом Монте Карло при iзобарично-
iзотермiчних умовах. Зокрема, нас цiкавить фазове розшарування рiдина-рiдина в таких системах. Мо-
дельний плин використовується для оцiнки ефекту форми молекул, що змiнюється вiд сферичної до
бiльш видовженої (двi спаянi сфери). Результати вказують,що ефект несферичної форми має зменшува-
ти критичну густину i температуру. Це узгоджується з експериментальними спостереженнями для антитiл
iз несферичною формою.
Ключовi слова: несферичнi протеїни, перехiд рiдина-рiдина, напрямляюча сила, агрегацiя,
термодинамiчна теорiя збурень
23801-12
http://dx.doi.org/10.1063/1.466971
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Introduction
Model, theory, and simulations
Model
Theory
Association parameters AA and BB
Cavity correlation function yhs
Other thermodynamic properties
N,P,T Monte Carlo simulation
Results and discussion
Thermodynamic properties: Theory against Monte Carlo simulations
Effect of protein's shape on the liquid-liquid phase diagram
Conclusions
|