Simple core-shell model for a soft nano particles and virus with analytical solution
TẠP CHÍ KHOA HỌC − SỐ 18/2017
65
SIMPLE COREꢀSHELL MODEL FOR A SOFT NANO PARTICLES
AND VIRUS WITH ANALYTICAL SOLUTION
Phung Thi Huyen1, Luong Thi Theu1, Dinh Thi Thuy2,
Dinh Thi Ha3, Nguyen Ai Viet4
1Hanoi Pedagogical University 2
2Thai Binh University of Medicine and Pharmacy
3Hanoi National University of Education
4Institute of Physics
Abstract: In some recently experiments with virus, their core part are DNA tightly packed
with very high charge density. The contribution of this highly charged part to the
electrical field outside virus now cannot be easily neglected in general case. In this work
we propose a simple coreꢀshell model for this type of soft particles and virus. The soft
particles consider consisted from the two parts: a charged hard core with a high charge
density and a charged outer layer. We assume that the core part is tightly condensed, so
the charge carriers of DNA can be partly bounded and partly moved. With this
consideration, the core part now is very look like the outside solution. The corresponding
PoissonꢀBoltzmann equations for this new model can be solved analytically. These
analytical solutions would be useful in the investigation the problem of virus with
charged core, such as in bacteriophage MS2.
Keywords: Soft nano, virus, coreꢀshell structure, charge density of AND, Poissonꢀ
Boltzmann equation, analytical solutions.
Email: phunghuyen.9xhpu2@gmail.com
Received 20 June 2017
Accepted for publication 10 September 2017
1. INTRODUCTION
In the last years, nanotechnology has a rapid advancement and opened up novel wide
range of applications in life science and material science [1ꢀ3]. Because the complexity of
biological structures and the variation of solvents, despite many effort to theoretical
investigation to understand the properties of soft particles [1, 4ꢀ7], the theoretical models
still face a variety of problematic issues and challenges. Thus, the construction of simple
physics models to explain new observed phenomena and experimental data are important
to the understanding of these complex systems.
66
TRƯỜNG ĐẠI HỌC THỦ ĐÔ Hꢀ NỘI
One of such simple models for soft nano particles was introduced in the works of
Ohshima [5ꢀ8]. The Oshima’s model provides a powerful tool for investigating the
behavior of biocolloidal particles, also viruses and bacteria. In Oshima’s model, the soft
particles are described as a nonꢀpenetrable neutral hard core coated by an ion permeable
polyelectrolyte soft layer with negative constant volume density charge. The electric
potential distribution of this system then is obtained by solving the PoissonꢀBoltzmann
equations. At present, improved Oshima models of soft nano particles are found much
application in the works [9ꢀ14].
In many present investigations, charge of the core part of virus has been rarely taken
into account. In most cases, a core charge is assumed to be neglected, so the electrical
potential outside the core remains unchanged. A theoretical study mentioned the charge of
the virus core in general cases to calculate the nonspecific electrostatic interactions in virus
systems. Recently, experiment data of the case of bacteriophage MS2 [15] have shown that
the ratio between the volume charge density of the core and that of the surface layer is
measured to be half of that found suggesting that the effect of the core charge on the
electrostatic, so electrokinetic properties of the particle should be reꢀexamined.
For explanation this observed phenomenon, a new coreꢀshell model for soft nano
particles was proposed in the work [16] with the consideration that soft particle consists
from two parts: a charged hard core with a volume charge density and a charged outer
layer. Using this model, the contribution of the core parameters, such as the core charge
and the core dielectric constant are studied. The model still complicated and can be solved
by numerical method only.
In this work we propose a simple coreꢀshell model for a soft particles and virus, based
on the assumption the core part is tightly condensed that the charge carriers of DNA can be
partly bounded and partly moved [17]. With this assumption, the core part now is very
look like the outside solution. The corresponding PoissonꢀBoltzmann equations for this
new model can be solved analytically. Our calculations provide the one of the first
theoretical analytical investigations about the effects of temperature and salt concentration
on the electrostatic properties, and could be applied to the case of virus with highly
charged hard cores, such as bacteriophage MS2 [15].
2. OSHIMA MODEL FOR SOFTꢀPARTICLES
In the figure 1 we present our coreꢀshell model for nano soft particles. We consider a
soft particle with radius b immersed in an electrolyte solution. The soft particle is assumed
to contain a hard core of radius a coated by an ionꢀpenetrable surface charge layer of
TẠP CHÍ KHOA HỌC − SỐ 18/2017
67
polyelectrolyte with thickness (b − a). Identified with the Ohshima model, the volume
charge density of the soft shell is ZNe, where e is an electron charge, Z and N are the
valence and the charge density of the polyelectrolyte ions, respectively.
density
The theoretical model of a soft particle including a hard core with the charge
ρcore and the dielectric constant εcore, and an ionꢀpenetrable surface layer of
polyelectrolyte
solution with the charge
coated around. The soft particle is immersed in an electrolyte
density ρel and the permittivity ε
r (see in Fig. 1).
The electric potential distribution obeys the Poissonꢀ Boltzmann equations [6, 15]
ρel
εrε0
ꢀψ = −
ꢀψ = −
ꢀψ = −
,
b
a
≤
≤
≤
r<∞
ρel + ZNe
,
r<b
r<a
(1)
εrε0
ρcore
εcoreε0
,
0
Fig 1. The theoretical coreꢀshell model of soft nano particles with a hard core charge.
the charge
density ρel
Here ε0 are the permittivity of vacuum,
Boltzmann distribution:
distribution
is the
(2)
M
zieψ
ρ (r) = z en exp −
,
∑
el
i
i
kBT
i=1
where M, zi, ni are the number ion types, the i th ionic valance and the ion concentration in
solution, respectively. Considering a simple case that the solution only contains a
monovalent salt M = 2 and zi = {− z, z}, we get:
68
TRƯỜNG ĐẠI HỌC THỦ ĐÔ Hꢀ NỘI
zeψ
kBT
ρ (r) = −2nzesinh
.
(3)
el
In the case of a low potential, the charge density in the electrolyte solution is given by:
2nz2e2
(4)
ρel (r) =
ψ,
kBT
Substituting Eq. (4) into Eq. (1) provides:
d2ψ 2dψ
+
= κ2ψ,
b ≤ r < ∞
a ≤ r < b
dr2
rdr
2
d ψ 2dψ
ZNe
(5)
+
= κ2 ψ −
,
dr2
rdr
κ2εrε0
d2ψ 2dψ
ρcore
εcoreε0
+
= −
,
0 ≤ r < a
dr2
rdr
where κ2 = 2z2e2n / εrε0kBT is the DebyeꢀHuckel parameter.
The spherical PoissonꢀBoltzmann equation (5) does not have a general analytical
solution and can be numerically solved only.
3. NEW SIMPLE COREꢀSHELL MODEL FOR SOFT NANO PARTICLES
In this part we propose a new model for soft nano particles and the virus. This simple
model can be solved analytically. Due to the tidily packed effect, we hypothesis that chare
of DNA in the virus core is quasiꢀbounded or can move quasiꢀfreely [17] like the charge in
solvent, then in the expression (5) the third equation has the same form of first equation.
The electric potential distribution now satisfies new Poissonꢀ Boltzmann equations
d2ψ 2dψ
+
= κ2ψ,
b ≤ r < ∞
a ≤ r < b
dr2
rdr
2
d ψ 2dψ
ZNe
+
= κ2 ψ −
,
(6)
dr2
rdr
κ2εrε0
d2ψ 2dψ
+
= −κ2coreψ,
0 ≤ r < a
dr2
rdr
2
κcore = ρcore / εcoreε0
where
is the DebyeꢀHuckel parameter of core.
TẠP CHÍ KHOA HỌC − SỐ 18/2017
69
The general solution of Eq. (6) gives us:
e−kr
r
ekr
r
ψ(r) = A
+ B
,
b ≤ r ≤ ∞
a ≤ r ≤ b
1
1
e−kr
ekr
ZNe
k2εrε0
ψ(r) = A2
+ B2
+
,
(7)
r
r
corer
corer
e−k
ek
ψcore = A
+ B3
,
0 ≤ r ≤ a
3
r
r
The coefficients A1, A2, A3, B1, B2, and B3 in Eq. (7) can be found by applying the
following boundary conditions:
ψ(∞) = 0, ψ(0) ≠ ∞,
(8)
ψ(a− ) = ψ(a+ ), ψ(b− ) = ψ(b+ ),
εcoreε0ψ'(a− ) = εrε0ψ'(a+ ), ψ(b− ) = ψ(b+ ),
(9)
(10)
The founding of the solution of system of equations (7ꢀ10) is very difficult in general
cases. We try to solve this problem in the next section.
4. ANALYTICAL SOLUTION OF THE MODEL
In this part we solve the system of equations (7ꢀ10) and derive the coefficients A1, A2,
A3, B1, B2, and B3 in explicit analytical forms.
At infinity the electrical potential must be zero, we can put
, and using the
B1 = 0
above boundary we get a linear system of equations for five variable A1, A2, A3, B2, and B3
e−ka
a
e−ka
a
eka
a
ZNe
k2εrε0
A
= A2
+ B2
+
,
1
e−ka e−ka
e−ka e−ka
eka eka
A −k
−
= A2 −k
−
+ B2 k
−
,
1
a
a2
a
a2
a
a2
coreb
e−k
coreb
e−kb
ekb
ZNe
ek
A2
b
+ B2
+
= A
+ B3
,
(11)
3
b
k2εrε0
b
b
coreb
coreb
b
b
e−kb e−kb
ekb ekb
e−k
e−k
ek
ek
core
core
A2 −k
−
+ B2 k
−
= A −kcore
−
+ B kcore
−
,
3
3
b
b2
b
b2
b
b2
b
b2
Taking the case of symmetrical solution we can put B3= ꢀ A3, now we have a linear
system of 4 equations for 4 variable A1, A2, A3, and B2
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TRƯỜNG ĐẠI HỌC THỦ ĐÔ Hꢀ NỘI
e−ka
a
e−ka
a
eka
a
ZNe
k2εrε0
A
= A2
+ B2
+
,
1
e−ka e−ka
e−ka e−ka
eka eka
A −k
−
= A2 −k
−
+ B2 k
−
,
1
a
a2
a
a2
a
a2
coreb
coreb
e−kb
ekb
ZNe
e−k
ek
A2
b
+ B2
+
= A
+ B3
,
(12)
3
b
k2εrε0
b
b
kcore
kcore
coreb
e−kb e−kb
ekb ekb
e
b + e−k
b
e
b − e−k
core
b
A2 −k
−
+ B2 k
−
= A −k
−
,
3
core
b
b2
b
b2
b2
Above linear system of equations can be solved analytically. For easier to see that, we
A → x1, A2 → x2, B2 → x3, A3 → x
kcore = k
replace
3 , and
C . We take the matrix
1
form of this linear system of equations:
ꢀX = BX,
(13)
where ∆ is the (4x4) matrix
e− ka
a
e− ka
a
eka
a
−
0
0
e− ka e− ka
e− ka e− ka
eka
a2
eka
−k
−
k
+
− k
a2
a
a 2
a
,
a
ꢀ =
e− kb
b
ekb
b
ek b − e− k b
c
c
0
0
b
− kcb
e− kb e− kb
ekb ekb
e
+ ek b
ek b − e− k b
c
c
c
c
−k
−
k
−
k
−
b
b2
b
b2
b
b2
X and B are the 4ꢀvectors:
ZNe
k2εrε0
0
x
1
x2
X =
,
.
(14)
B =
x
3
ZNe
k2εrε
−
x4
0
0
The solutions of the matrix equation (13) can be obtained as follows:
detꢀ
1 , x2 =
detꢀ
detꢀ
detꢀ
detꢀ4
detꢀ
x1 =
2 , x3 =
detꢀ
3 , x4 =
detꢀ
,
where the matrix determinants are:
TẠP CHÍ KHOA HỌC − SỐ 18/2017
71
e−ka
a
e−ka
a
eka
a
−
0
0
e−ka e−ka
e−ka e−ka eka
eka
a
−k
−
k
+
− k
a
a2
a
a2
a2
det ꢀ =
e−kb
ekb
b
ek b −e−k b
c
c
0
0
b
b
e−kb e−kb
ekb ekb
e
−k b + ek b
ek b − e−k b
c
c
c
c
c
−k
−
k
−
k
−
b
b2
b
b2
b
b2
e−ka
= 4
sinhk(b − a) −coshk(b − a) k cosh k b + ksinh(k b),
(15)
[
]
(
)
C
C
C
a3b2
ZNe
k2εrε0
e−ka
a
eka
a
−
0
e−ka e−ka eka
eka
a
0
k
+
− k
0
a
e−kb
a2
a2
det ꢀ1 =
ekb
b
ek b − e−k b
c
c
ZNe
k2εrε0
−
b
b
e−kb e−kb
ekb ekb
e
−k b + ek b
ek b − e−k b
c
c
c
c
0
−k
−
k
−
k
−
c
b
b2
b
b2
b
b2
4ZNe
1
ab2
1
=
k
cosh k(b − a) kC cosh(kCb) − sinh(kCb)
k2εrε
a
0
1
ab2
1
1
1
+
sinhk(b − a) kC cosh(kCb) − k2 sinh(kCb) +
kC cosh(kCb) − sinh(kCb)
,
3
a
a b
b
(16)
e−ka
a
ZNe
eka
a
0
0
k 2εrε0
e−ka e−ka
eka
a2
eka
a
−k
−
0
− k
a
a2
det ꢀ2 =
ekb
b
ek b − e−k b
c
c
ZNe
k 2εrε0
0
0
−
b
−kcb
ekb ekb
e
+ ek b
ek b − e−k b
c
c
c
c
0
k
−
k
−
b
b2
b
b2
k (b−a)
2ZNe e
2
a3b2
1
=
ka +1 k cosh k b − k sinh(k b) +
kC cosh(kCb) − sinh(kCb)
,
(
)
(
)
(
)
C
C
C
k2εrε0 a2b2
b
(17)
72
TRƯỜNG ĐẠI HỌC THỦ ĐÔ Hꢀ NỘI
e−ka
a
e−ka
a
ZNe
−
0
0
k2εrε0
e−ka e−ka
e−ka e−ka
−k
−
k
+
0
a
a2
a
a2
det ꢀ3 =
e−kb
ZNe
k2εrε0
ek b − e−k b
c
c
0
0
−
b
b
e−kb e−kb
e
−k b + ek b
ek b − e−k b
c
c
c
c
c
−k
−
0
k
−
b
b2
b
b2
−k (b+a)
2ZNe e
1
=
ka +1
sinh k b − k cosh(k b) ,
(18)
(
)
(
)
C
C
C
k2εrε
a2b2
b
0
e−ka
a
e−ka
a
eka
a
ZNe
k2εrε0
−
e−ka e−ka
e−ka e−ka eka
eka
−k
−
k
+
− k
0
a
a2
a
e−kb
a2
a2
a
det ꢀ4 =
ekb
b
ZNe
k2εrε0
0
0
−
b
e−kb e−kb
ekb ekb
−k
−
k
−
0
b
b2
b
b2
e−ka
e−ka e−ka
2ZNe
(19)
=
k
ka +1 + k
+
cosh k(b − a) −sinh k(b − a) ,
(
)
(
)
k2εrε0 a2b2
a3b a3b2
1
1
With: m = kC cosh(kCb) − sinh(kCb) , n = kC cosh(kCb) − k2 sinh(kCb) ,
b
a
h = sinh k(b − a) − cosh k(b − a) .
Therefore, the coefficients A1, A2, A3, and B2 are
4ZNe
1
1
n
m
a3b
k
cosh k(b − a) kC cosh(kCb) − sinh(kCb) +
sinhk(b −a) +
k2εrε
ab2
a
ab2
0
A =
, (20)
(21)
1
e−ka
a3b2
4h
k cosh k b + k sinh(k b)
(
)
C
C
C
k(b−a)
2ZNe e
2
ka +1 k cosh k b − k sinh(k b) +
m
(
)
(
)
(
)
C
C
C
k2εrε0 a2b2
a3b2
A2 =
,
e−ka
4h
k cosh k b + k sinh(k b)
(
)
C
C
C
a3b2
TẠP CHÍ KHOA HỌC − SỐ 18/2017
73
e−ka
e−ka e−ka
ZNe
k
ka +1 − h k
+
(
)
k2εrε0 a2b2
a3b a3b2
A = −B3 =
,
(22)
3
e−ka
2h
k cosh k b + k sinh(k b)
(
)
C
C
C
a3b2
ZNe
k2εrε0
−kb
ka +1 (−m)
e
(
)
B2 =
,
(23)
2h
k cosh k b + k sinh(k b)
(
)
C
C
C
a
Finally the electrical potential of virus in our new model can be founded in the explicit
analytical forms:
e−kr
r
e−kr
r
ψ(r) = A
,
b ≤ r ≤ ∞
ZNe
1
ekr
r
(24)
ψ(r) = A2
+ B2
+
,
a ≤ r ≤ b
k2εrε0
2
ψ
core (r) = A sinh k r ,
0 ≤ r ≤ a
(
)
3
core
r
where the set of coefficients A1, A2, A3, and B2 are now well defined by the physical
parameters of the virus and solution environment as above.
5. CONCLUTIONS
In many present investigations using the Oshima model for soft nano particles, the
core charge distribution has been rarely taken into account. In most cases, a core part is
neutral or core charge is assumed to be zero, so the electrical potential outside particles
remains unchanged. In recently experiments with virus, the core part are the tightly
confined DNA with very high charge density. The contribution of this high charged part to
the electrical field outside virus now cannot be easily omitted in general and have to more
detail investigation.
In this work we propose a simple coreꢀshell model for soft particles. The soft particles
consider consisted from the two parts: a charged hard core with a high charge density and a
charged outer layer. We assume that the core part is tightly condensed, so the charge
carriers of DNA can be partly bounded and partly moved. With this consideration, the core
part now is very look like the outside solution. The corresponding PoissonꢀBoltzmann
equations for this new model can be solved analytically.
We believe that using the obtained analytical solutions from our model with
improvement by numerical calculation on PC could explain the identical properties of
untreatedꢀMS2 and RNAꢀfree MS2 reported in works [15].
74
TRƯỜNG ĐẠI HỌC THỦ ĐÔ Hꢀ NỘI
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MÔ HÌNH LÕIꢀVꢁ ĐƠN GIꢂN CHO CÁC HꢃT NANO MꢄM VÀ
VIRUT VꢅI LꢆI GIꢂI GIꢂI TÍCH
Tóm tꢁt: Trong mꢂt sꢃ thí nghiꢄm hiꢄn nay vꢅi virut, phꢆn lõi cꢇa chúng có thꢈ là các
ADN cuꢂn chꢉt có mꢊt ñꢂ ñiꢄn tích rꢋt cao. Đóng góp cꢇa lõi tích ñiꢄn cao này vào ñiꢄn
thꢈ quanh virut là không thꢈ dꢌ dàng bꢍ qua trong các trưꢎng hꢏp tꢐng quát. Trong bài
báo này, chúng tôi ñꢑ xuꢋt mꢂt mô hình lõi – vꢍ ñơn giꢒn cho các hꢓt nano mꢑm và virut.
Hꢓt nano mꢑm ñưꢏc giꢒ thuyꢔt gꢕm 2 thành phꢆn: mꢂt lõi cꢖng tích ñiꢄn vꢅi mꢊt ñꢂ ñiꢄn
tích cao và mꢂt lꢅp vꢍ tích ñiꢄn. Chúng tôi giꢒ thiꢔt rꢗng phꢆn lõi ñã ñưꢏc cuꢂn chꢉt, vì
vꢊy các hꢓt tꢒi ñiꢄn cꢇa AND có thꢈ là bán cꢆm tù hoꢉc bán tꢘ do. Vꢅi giꢒ thuyꢔt này,
phꢆn lõi sꢙ giꢃng vꢅi lꢅp bên ngoài virut. Phương trình PoissonꢀBoltzmann tương ꢖng
vꢅi mô hình mꢅi này có thꢈ giꢒi ñưꢏc và cho lꢎi giꢒi dưꢅi dꢓng giꢒi tích tưꢎng minh. Các
lꢎi giꢒi dưꢅi dꢓng giꢒi tích tưꢎng minh này sꢙ có ích trong viꢄc nghiên cꢖu các virut có
lõi tích ñiꢄn, ví dꢚ như thꢘc khuꢛn thꢈ MS2.
Tꢜ khóa: Hꢓt nano mꢑm,Virut, Cꢋu trúc lõiꢀvꢍ, Mꢊt ñꢂ ñiꢄn tích ADN, Phương trình
PoissonꢀBoltzmann, Lꢎi giꢒi giꢒi tích.
10 trang
18/04/2022
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