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Dynamic frequency-dependent dielectric response of three aqueous biomolecular solutions
Physica A 221 (1995) 419-437
ELSEVIER
Dynamic frequency-dependent dielectric response of three aqueous biomolecular solutions Sankar Department
Chakraborty*,
Brian DeFacio
of Physics and Astronomy, University of Missouri-Columbia,
Columbia, MO 6521 I, USA
Received: 14 February 1995
Abstract A simple two-component Debye-Lorentz model is presented for the frequency dependence of the dielectric permittivity E(W)of aqueous solutions of three biomolecules, Glycine, Myoglobin, and Hemoglobin. A local response function for this model is given for the use of workers in wave propagation and biophysical studies. It is necessary to treat the full inertia tensor of the water, I, # I, # I, # I, to obtain good qualitative agreement with the experimental values of the three biomolecular solutions given. The Hemoglobin solution is shown to exhibit dielectric hysteresis, whereas this effect is absent in Glycine and Myoglobin. The Hemoglobin is found to need a prolate inertial tensor to agree with experiment. Permittivity; Rotational analysis; Liquid diffusion; (General) theoretical biophysics; Solutions of biomolecules
Keywords:
1. Introduction
Water is the most common solvent on earth [l] and is a principal constituent of all living organisms [2]. Approximately seventy percent of the human body is composed of water, with the amount of water in, each type of tissue ranging from over ninety percent of plasma to about thirty percent of fat tissue. Thus, the understanding of biological systems will require additional knowledge of water. Since the biological macromolecules live and function in an aqueous environment, the water molecules play a vital role in determining the structures and properties of the macromolecules. Water does not merely form an inert continuum in which a biomolecule resides. In fact, the double helix DNA structure proposed by Watson and Crick [3] owes its stability partially to the presence of water molecules in the neighborhood, as was observed by Jacobson [4]. Physically, the dielectric properties of water govern both
*Corresponding
author.
0378-4371/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4371(95)00198-O
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/ Physica A 221 (1995) 419-437
the electrostatic forces and the propagation of electromagnetic waves in biological media. From the dielectric studies of water and aqueous solutions of biological macromolecules [2], molecular parameters of the macromolecules such as size, conformation, hydration, and dipole moment in solution can be obtained. These properties will someday contribute to medical and biological applications such as the mechanism of activation of certain anaesthetic agents [S], evaluation of structure of lipoproteins present in blood serum [6], techniques of radiowave or microwave therapy in malignant diseases [7], measurement of membrane thickness [8], diagnosis of senile dementia from dielectric measurements of cerebral spinal fluid (CSF) [9], and many others. There is much left to learn of the physics of biological systems, as Vitiello [lo] has emphasized. Here, only a very small first step toward his ambitious program is taken. Many biologists believe that the water in the neighborhood of biomolecules, which they call “bound water”, behaves differently from ordinary “free” water. The effects of pH, temperature, viscosity, and the complexities due to the dipole-dipole interactions, and boundary conditions make it impossible to give a rigorous theoretical model that can be solved in classical statistical mechanics, so simplifying model assumptions are required. Several macroscopic models have been discussed by Scaife [ 1l] in order to describe experimental data from dielectrics, most of which do not follow the pure Debye or Lorentz models. Three useful models from Scaife are given by empirical formulae; the Cole-Cole formula
‘(O)= ‘,
+
6s -Em Cl
+
(iwzD)“]
for 0 < CI< 1, the Cole-Davidson
E(W) = Em+ [
(1.1)
’
formula
ES - EmI 1 +
(iozD)]P
(1.2)
’
and the Havriliak and Negami formula E(W)=
E,
+
ES - Em
[1 + (ioz&“]a
’
(1.3)
where (~1,j?) E (0, 1) and .sSis the w = 0 dielectric permittivity. A common feature of all three model equations above is that E’(O) -E, is non-negative, in contrast to the Lorentz model at resonance absorption. Dissado and Hill [12] have proposed a cluster model to calculate the dielectric response function. Their model has two power-law regions of frequency dependence governed by two parameters n, m for the clusters as discussed in their work. It has a clear physical basis unlike the empirical formulas in Eqs. (l.l)-( 1.3), and it fits data as well as or better than these. The present work will follow this general approach, albeit in a somewhat different direction.
S. Chakraborty, B. DeFacio J Physica A 221 (I 995) 419-437
421
The pioneering work by Debye [13] on polar molecules early in this century opened dielectrics as an area of research. His model explained the microwave fall off of the real part of the dielectric function and the accompanying absorption peak, but the flat infrared region obtained from the “E,” of his model was not consistent with experiment. Inertia was simply set equal to zero. McConnell Cl43 treated the inertia correctly and showed that it gave additional structures at infrared frequencies. However, this work was unable to explain the experimental data of water. Later, Sparling, Reichl, and Sedlak [15] added both hydrodynamic flow and viscous friction to McConnell’s analysis. However, one limitation of both these approaches is the fact that both are hydrodynamic and, therefore, are only valid at low frequencies and small wave vectors. The problem of calculating the dielectric function of water based on a macroscopic model was first addressed by Rahman and Stillinger [16] in 1971. Their work involved the then-new molecular dynamics which they developed. The water molecule was treated as a rigid symmetric rotor with translational and rotational motions, ignoring the effect of vibrations. A model potential between water atoms with two point negative charges on the oxygen atom and one positive charge on each hydrogen atom gave an inter-molecular molecule interaction, and between two molecules, the potential was chosen such as to include dipole-dipole interaction, repulsion between oxygen atoms and intermediate range interaction leading hydrogen bonds. However, this theory was not used with any dependence of the dielectric function on the frequency and the wave vector. Hundreds of workers have followed their approach with limited success. Recently, the calculations of frequency and wave number of dielectric functions of water and aqueous solutions have been published by several authors [ 17-201 using microscopic theories. Neumann [17] published computer simulations using the MCY (Matsuoka-Clementi-Yoshimine) potential for water, but he obtained values from 25-31 compared to 88 for the dielectric constant of water. Pollack and Alder [18] treated the simple model of the Stockmayer fluid of linear point rotors interacting via dipole-dipole and Lennard-Jones potential. In their computer simulations, they found a collective mode which they called the dipolarons at high frequency and low wave vector. A different approach to the problem was first made by Wei and Patey [19], who used the integral equation formulation of classical statistical mechanics. They treated a collection of ideal molecules as rigid symmetric rotors with electric multipoles to account for the mutual interactions. Then the integral equations of the reference hypernetted chain approximation, RHNC, are solved on the computer to obtain the static dielectric function. The dynamic correlation functions are obtained from their static counterparts by using the Kerr approximation, which they showed to be superior to an approach by Vineyard. S.H. Kim et al. [21] included the full inertia effects and calculated the self-VanHove function in high and low frequency ranges using the Kubo fluctuation-dissipation theorem and the Kerr approximation. They related the self-Van-Hove function to
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the dynamical response function of an external time-dependent potential. Their dynamic response function was found by solving the Boltzmann transport equation for the one-particle distribution function in phase space, in presence of an external time dependent potential. The dielectric function is calculated (in the numberconserving, relaxation-time approximation) from the response function using a local field factor from Wei and Patey. This guarantees the diffusion regime at low frequency and wave vector and interpolates to the high frequency plasma behavior. The experimental input to this model is one relaxation time, TD, the static self-susceptibility xj (o = 0, q = 0),and a static function called the local field factor, Y(q). Physicaly, tD represents incoherent collisions among molecules, 2: is the scale of the o + 0 dielectric permittivity ~(0,0),and the local field factor, Y, represents the coherent long range interactions, which were taken from Wei and Patey [19]. The claim in Kim et al. that xi zs np2/3kT is wrong, due to a simple calculational error. Instead, one needs, xso= 104np2/3kT, which explains why the problem of finding e(0) is so difficult. This model [21] with this correction had a surprisingly good agreement with experiment, although it fails to give a detailed agreement such as that in the curve fit in [22]. It may organize the microscopic water problem for e(w, q) into distinct parts: the static dielectric constant ~(0,0),the microwave relaxation time (or times) TD, and long range coherent effects on a single “hydration” cell. The computer simulations cannot yet calculate xz accurately; their results vary from 31 to 71 for ~(0, 0) compared to the experiment value of 87.7. Oughstun’s group [22] found that two relaxation times are required for a more accurate agreement, and each of which may have its own local field factor. The full asymmetric rotor inertia was treated by Kim et al. and is shown to be necessary in the present study. Here, Kim et al.% model is extended to two free components, which provides a model with a physical basis for E(O), one which is a mixed Lorentz-Debye model. This model will be the starting point of the present study. However, the agreement with experiment was worse than their symmetric rotor model. The splitting of the absorption peak is probably correct. Recently, Oughstun et al. [22] gave a careful, high accuracy curve-fit using a linear combination of the Rocard-Powles extension of the Debye model with two relaxation times combined with the analytic expression for the dielectric function given by four Lorentz oscillator models with four distinct resonance lines. His work agrees in reasonable detail with the best water data [23-261, although it does not explain inconsistensies between the Kramers-Kronig relations for E’and E”. Oughstun and co-workers also calculated a realistic, parametric dielectric function and the propagation of a Gaussian pulse in this medium. The macroscopic model presented here does not contradict McConnell’s statement [14] (on p. 37) because the angular variables (0,&~)of the rotor describe an isolated molecule in a Lorentz sphere in the liquid. The collisions are added in the second stage when this object is embedded in the single particle distribution of a number-conserving, relaxation-time Boltzmann equation by Eq. (2.37). It is remarked that the form of this model is quite different from the one in Ref. [22].
S. Chakraborv,
B. DeFacio 1 Physica A 221 (1995) 419-437
423
Experiments reporting E(W)for water are found in the work of M.R. Querry [24]; data on water is compiled by Thormahlen et al. [25], and Schiebener et al. [26]. There are experiments on the dielectric response function of Glycine solution [2], Myoglobin solution [28], and Hemoglobin solution [27], which have been in the literature for many years. Even the best theories for water today cannot accurately explain the experimental data. A generalization of the Debye model to an ellipsoidal molecule having different relaxation times along the semi-major and semi-minor axes was given by Grant et al. [2],
43
&I =
1 + &,’ E”(W) =
+
Ab 1 + w2r; + em ’
A, * cot, Ab- Wtb + 1 + c!12rZ’ 1 + C&f b
(1.4)
(1.5)
However, this cannot explain the experimental data of E”(O) by Grant et al. especially because of an extra absorption peak around 100 MHz, which is absent from this expression. Since data on the frequency dependence of the permittivity is available in Refs. [2, 24-281, it is interesting to ask how much physical structure is present and whether it can be described by a simple model.
2. The macroscopic model Initially, it was intended to solve a two-component version of the model in Ref. [21] with reasonable interactions between the constituents. The absence of the values of the moments of inertia as a function of the pH of the biomolecules and the lack of any frequency w and wave number k dependence of the permittivity in order to determine the local field factor, made this impossible. Since Kim et al. found inertia in Hz0 to split the absorption peak, a model with II # lZ # Z3 # II for water is required to obtain all of the absorption peaks. The biomolecules were treated with their physical masses and spherical radii of gyrations were chosen which were consistent with their densities. The concentrations of the biomolecules is low enough that the simple version, which follows, is to take two interacting dielectrics. This turned out to give all of the qualitative features of the experiments at this level of analysis, so that more intricate microscopic models cannot be justified for today’s experiments. Experimentalists are challenged to provide the moments of inertia as a function of pH and the real and imaginary parts of E(W)at a variety of concentrations over a range of frequencies, or even better E(W,q). Here, a Lorentz-Debye model for the dielectric function is presented. A Lorentz oscillator model is solved which yields the macroscopic Lorentz equation for unit step function response. Using this response, the frequency dependent polarizability a(w) is
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calculated by applying linear response theory. Then, E(W) is used to obtain the frequency-dependent dielectric permittivity E(W)using Ref. [21]. First, we follow Scaife or McConnell, and consider a macroscopic dielectric material polarized by an external field E0 which is turned on at t = 0, e(t) = &u(t),
(2-l)
where
u(t) =
0,
MO,
3,
t=O,
1,
t>o
(2.2)
is the unit step function. Assuming the field to be uniform, there will be an induced dipole moment m(t) = &H(t)
9
(2.3)
where H(t) is the system unit step-function response. For t < 0, H(t) = 0, because there is no response before the application of the field by the Principle of Causality. For dielectrics, m(t) approaches the equilibrium value lim m(t) = m, = EOcc,, 1-tcu
(2.4)
because of dissipations, which must be non-zero because of the existence of fluctuations. At long times, lim H(t) = ~1,) f-o3
(2.5)
where rx, is the static polarizability of the dielectric. The sudden removal of the external field forces a rapid decay of the dipole moment from its equilibrium value m, to a smaller value. If we remove the field at time tR, then m(t) = m, -E,H(t
- tR) = E,b(r - tR),
(2.6)
which explains the decrease in m(t) over time with b(t) as being an after-effect function that equals zero for t < 0. For an arbitrary external electric field e(t), we can calculate m(t), the induced dipole moment from H(t), provided the fields are not too strong, so that the principle of superposition holds as Htota~(t) = HI (t) + i?(t) = A-H(t),
H2
(4
2
(2.7a) (2.7b)
where La, (t) is the response to the total electric field e1 (t) + e2(t), HI (t), H2(t) being individual responses to e, (t), e2 (t), respectively; A(t) is the response to a field Ae(t), and H(t) is the response to a field e(t).
S. Chakraborry,
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425
1 Physica A 221 (1995) 4I9-437
In case of these linear systems, we consider the external field to be composed of (sufficiently small) piecewise constant steps. The step at time W has an amplitude [de(W)/dW] dW. As a result, we can write f
m(t) = e(0 + )H(t) +
s dW
de(W) -j+w
-
(2.8)
WI,
o+
where e(0 + ) is the amplitude of the initial step. In the above equation, the lower limit of integration, 0+ , implies that the integration begins just after t = 0; e(0 +) removes the difficulty of an infinite derivative in the integrand. Upon integrating Eq. (2.8) by parts, one obtains m(t) = e(t)H(O +) +
I
dz’e(t -7)~
dH(4
(2.9)
0
where H (0 +) = 0 since a physical system cannot respond to a changing stimulus instantaneously. Also, H(t) = 0,
and dH(r) dt=
o ’
for t < 0 by causality, so dH(t)/dt is called the delta function or pulse response. Using causality, we write Eq. (2.9) as
m(t) =
s
dre(t -
7)dz.
dH(4
(2.10)
-m
If we consider the sudden application of a periodic external field at t = 0 co(t) = Eocosotu(t), then f_ m(t) = E,coswt
*_ dr dH (4
s
-
dr dH(r)
cos ox + E. sin wt
d(z)
-
s
dz
.
sin or .
(2.11)
The steady state response to the field co(t) can be obtained by taking values of t much greater than the maximum time-constant of the dielectric. This permits the upper limit in the last integrals (eq. 2.11) to be replaced by infinity, so that the steady-state response becomes m(t) = E. [a’(o) cos cot + cc”(w) sin ot] ,
(2.12)
S. Chakraborty, B. DeFacio / Physica A 221 (1995) 419-437
426
for t > 0 where the real and imaginary parts of the polarizability m
+W
a’(w) =
are given by
dH (t) -COSOt
5 dt
=
dt
s
(2.13a)
dtyLosot,
0
-CO
and
a”(0.J) =
s
dt 9
dt -dH (t) sin wt dt
-
sin cot
dt
(2.13b)
’
Using Eqs. (2.13a) and (2.13b), we can write +CC a (0) =
dH(d
dz exp( - ior) 7
a’(w) - id’(o) =
.
(2.14)
-52
From Eq. (2.13a), co
s
dHW
a’(O)=o~~=
(2.15)
d,-dr=H(co),
0
lim c!(o) = lim
w-to2 s
w-em
dzcosoz
dffb) dz
=
o
,
(2.16)
0
this last result depends on the fact that pll=
dH(r)
o
.
Also, a’(w) = a’( -0))
(2.17a)
a”(w) = -a”( -w) .
(2.17b)
Eqs. (2.17a) and (2.17b) then imply that (2.18)
a(o)=a*(-o). 2.1. Dispersion relations
Taking the inverse of Eqs. (2.13a) and (2.13b), m
dH(t) 2 dwa’(m)cosmt = dt n s 0
(2.19a)
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and dH(t)
_
df
2
(2.19b)
= - 7 do a”(w) sin or . rJ
”
0
Combining Eqs. (2.19a, 2.19b) yields U(t) -=-
2
dt
dwa(w)exp(iwt)
TT s
(2.20)
-CC
but dH(t)/dt = 0 for t < 0, so that this equation can be written as a
a(o) =
s
dH(r) dtFexp(-iwt);
(2.21)
0
using Eq. (2.20), we find that a(w)=
s
dtexp(-iwr)
dpa(Aexp -CC
0
(W)
1,
(2.22)
where a(p) is even in p and decreases rapidly enough as p -+ co, so that we get a a’(o) =zP Tc
s
dp
a”(p)w (,u2 -02)’
(2.23a)
0 a
a”(o) =ff’
s
dp
a’(cl)w (m2 -P2) ’
(2.23b)
0
The formulae above are the once-subtracted Kramer+Kronig dispersion relations and express the causality of the system. The real and imaginary part of a causal dielectric function must obey the Kramers-Kronig relation. Stankov [30] has derived an expression for a(O,O) as a function of some unspecified mean molecular kinetic energy and temperature based on the Maxwellian distribution of kinetic energy. Despite extravagant claims, its tables of numbers explain little to us, since it is related to neither a microscopic nor a macroscopic physical model. 2.2. Lorentz Absorption Here, we obtain the unit step-function response, a,(t), by Fourier transforming the Lorentz model of dielectric, where the reader is reminded that the collisions among molecules have not yet been taken into account. This model remains a pure Lorentz dielectric. From a,(t), we find a(o), the polarizability using eq. (2.14).
428
S. Chakraborry
B. DeFacio
1 Physica
A 221 (1995) 419-437
Here, we take E, = 1 for the vacuum response as w + 00. The sudden application of a unit voltage to the circuit at t = 0 builds up a change aL(t) on the capacitor (as - l), so we can write the oscillator differential equation 1 CPU&) &. (0 ~z+Ydt+a,(t)=(&,-l)u(t), ri+, dt
(2.24)
which has the solution o0ev(
q.(t) = (ES- 1) 1 [
-0)
a
1
cos(S2t - <) u(t),
(2.25a)
y=(a,--l)~~exp(-:r)y,(r),
(2.25b)
where (2.26a) (2.26b) (2.26~) and the factor y controls the damping. Earlier in this section we had Eq. (2.21), m a(w)=
s
dH 0) dr- dt exp(-iM,
0
which for the Lorentz model becomes a(w) =
s
dt-
daL (r) dt
exp(-iot).
(2.27)
0
Using Eqs. (2.25a) and (2.27), we can write
m
a(w)=(ES-lb02 dt 2iQ
s
{exp[-i(w
-Q
- ii)t] -exp[-i(o
+ Q -i<)t]}
0
=
_kS
-
1
1
lb,2
(ES -+ lb: =(6$ -co2 2Q
1
(w -5)
-ii)
-(co
+
Q -
ii)
1
iywio)’
but the polarizability
of an ellipsoid along a principal axis is given as
V&orw a(W)={l
(2.28)
- 11
+R[&(O)-11)’
(2.29)
S. Chakrabor& B. DeFacio / Physica A 221 (1995) 419-437
429
where ), is the depolarizing factor. So we can write the relative permittivity (setting I/Q = 1) using Eqs. (2.28) and (2.29) E(U) - 1 (ES- l)& { 1 + ACE(W)- 11= (f& - Cf.?+ iyw,20) ’
(2.30a)
where
i=
,
1,
oblate
0,
prolate,
(2.30b)
spherical.
i f,
Hence, for /z = 0 which is the case for water E(U) -1 =
(6 -
l)w,z
(& - W2)2 + iye&
’
i.e., E’(W) = 1 +
E”(W)=
(ES-1)0&o;
- 02)
[(CD;- Cl?)2 + 02P]
’
(ES- l)&DT
[ (0; - W2)2 + WZP]
’
(2.31a)
(2.31b)
where r = yoi is the damping. For an arbitrary value of A, the response function S[w] is given by E(W)- 1
(ES- 1)002 =:S[w], 1 + ACE(W)- 11= (0,’- co2+ iyo,2w)
(2.32)
i.e.,
E(u)= l-(qo]
sEmI
-1)’
(2.33)
with
sco1 As[w] -1 = F(w)
(2.34)
E(U) = 1 -F(o).
(2.35)
and
Hence, E’(U) = 1 -ReF(o)
,
(2.36a)
and C(O) = Im F(o).
(2.36b)
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S. Chakraborry. B. DeFacio / Physica A 221 (1995) 419-437
Here, we treat the water as asymmetric and the other molecules such as Hemoglobin, Myoglobin and Glycine as prolate symmetric rotors, and the dielectric function for each molecule has two parts: (i) For 1 = 0 (symmetric) for the biomolecule, and (ii) Due to asymmetry i.e., an arbitrary value of A, for the water solvent. Now, the number conserving relaxation-time approximation is incorporated our model; by substitution which now adds collisions among molecules, 0
+
0
+
i/z
into
(2.37)
,
which leads to an additional Debye behavior E(U) - 1
1 + A[s(o) - 11
=Gl&{exp[
-i(o+f-6a-il)r]
)I1,
0
- exp
-i
w+~+G?-ii
t
[(. and finally gives E(U) - 1
l+A[e(w)-ll={(
0.1:-02
(ES- l)& + l/r2 - yo~/r) +
i(yoo~
-24~))
’
(2.38)
which for (A = 0) reduces to E’(W) = 1 +
(ES- l)c&c$
-o2
+ l/t2 -7)
(2.39a)
{(0: -W2 + l/r2 - r/r)2 + (OT - 20/2)2} ’ and (ES- 1)
w;(OT -
242)
sN = {(0,” - o2 + l/T2 - r/r)2 + (WT -202/r}
(2.39b) ’
where r = yoi as shown before.
3. Results and discussion Since the water solution plays an important role in this discussion, Eq. (2.30) with A = 0 is used to calculate the real and imaginary parts of E(O). The index of refraction n(o) = & is shown in Fig. 1. The agreement with experiment is considerably better than that of Kim et al. but is less accurate than Oughstun. The attenuation coefficient is as accurate as Ref. [22] through w * 10r2, but this would fail at higher frequencies where the second relaxation time becomes important. The calculated real and imaginary parts of E(W)for the Glycine solution are shown in Fig. 2. The Cole-Cole plot for this solution is given in Fig. 3. For the dielectric
431
S. Chakraborty. B. DeFacio / Physica A 221 (1995) 419-437
(4 10
0’
6
7
8
9
10
11
(b) 1000. .
.-*
Fig. 1. (a) The real part of the refractive index of water as a function of log frequency. The solid curve is for the calculated values. The dotted curve is for the experimental values from Ref. [23]. (b) The absorption coefficient of water as a function of log frequency. The solid curve is for the calculated values. The dotted curve is for the experimental values from Ref. [23].
response of aqueous solutions of Glycine, where the electric dipole moment is induced: E~,,,~, (0) = O.O75s&0) + 0.925cbA (w) + 0.263 Re [ JsWA(&cS(cu)] sIbtal(~) = 0.075&,(o)
+ 0.925cl;yA(o) + 0.2631m[JEWA(~)~GS(~)],
,
(3.la) (3.lb)
Similarly, for the dielectric response of the Myoglobin aqueous solution: E;,,~,(w)= 0.1&(0)
+ 0.9sLA(c0) + 0.3 Re[J.sWA(~)sy&)],
E&+,(O)= 0.1&s(~)
+ 0.9.sl;yA(u)+ 0.3 Im [J-l
(3.2a) ,
(3.2b)
Here, we see two different regions of dispersion at the respective relaxation frequencies of Myoglobin and water, e.g., the solid line in Fig. 4 has a much larger separation than Glycine. The corresponding Cole-Cole plot is presented in Fig. 5. The dielectric response of the Hemoglobin solution satisfies the equation: &.i(~)
= 0.2&W
+ 0.8aka(~) + 0.4 ReCJmI
E;~,,,(w)= 0.2.$&0) + 0.8al;yA(c0)+ 0.4 Im [,/aWA(~)aH&)]
,
(3.3a)
,
(3.3b)
S. Chakraborty, B. DeFacio / Physica A 221 (1995) 419-437
432
Fig. 2. The real and imaginary
parts of the dielectric response function (E(W)) of 1 M Glycine solution in
water. The solid curve is the calculated values of E’(O); the long dashed curve is the experimental from
Ref. [2];
experimental
the short dashed
curve
is the calculated
values of E”(W); the dotted
curve
values c’(w) is for the
values of E”(O) from Ref. [S].
Fig. 3. The Cole-Cole
Plot for a 1 M Glycine
solution in water.
where the first term accounts for symmetric Hemoglobin, the second term is for asymmetric water, and the last term accounts for the uncorrelated interaction of lowest order between water and Hemoglobin molecules following Eyring and Jhon [29]. Upon inspection of the dielectric response curve (upper solid line) in Fig. 6, we see that for the Hemoglobin solution, the real part of E~~,~,(o),has two regions of dispersion - one around 1 MHz and the other around 10 GHz. The first one is due to the relaxation of the large Hemoglobin molecules whereas the other one is due to the Debye relaxation of the water molecules. Unlike the Hemoglobin and Myoglobin solutions, the Glycine solution does not have two diseersion regions, but only one. In Fig. 2, the solid line shows that the relaxation times of water and Glycine are not very different, since the molecular weight of Glycine is 75, it is non-polar. This is to be compared to x 15000 for Myoglobin and x 68000 for Hemoglobin, i.e., they relax almost together. This is one difference from the aqueous solutions of very large biomolecules. The overlapping of relaxation regions is a characteristic of solution of biomolecules whose mass and intertia is close to that of water (or the solvent).
S. Chakraborty,
B. DeFacio
/ Physica A 221 (1995) 419-437
433
Fig. 4. The real and imaginary parts of the dielectric response function (E(W))of 10% Myoglobin solution in water. The solid curve is the claculated values of E’(W);the dotted curve is the experimental values of E’(O) from Ref. [2]; the dashed curve is the calculated values of E”(W).
Fig. 5. The Cole-Cole Plot for a 10% Myoglobin solution in water.
0
2
4
6
a
10
12
Fig. 6. The real and imaginary parts of the dielectric response function (E(O))of 20% Hemoglobin solution in water. The upper solid curve is the calculated values of E’(W);the long dashed curve is the experimental values of E’(u) from Ref. [2]; the lower solid curve is the calculated values of C”(w);the dotted curve is for the experimental values of E”(O) from Ref. [2].
434
S. Chakraborty B. DeFacio / Physica A 221 (1995) 419-437
0.. 0
25
50
75
100
125
150
171
Fig. 7. The Cole-Cole Plot for a 20% Hemoglobin solution in water. The dielectric hysteresis is the closed curve. It occurs in the frequency range lo6 Hz < v < 10” Hz of Fig. 6 where the splitting of the absorption peak due to asymmetric rotor inertia of water appears.
The frequency dispersion can be explained in different ways: (a) The dipole moment of the protein molecules experience an orientational force by the applied electric field. The Brownian motion, due to collisions with water dipoles, tends to randomize this alignment via collisions among water molecules, and the net magnitude of the polarization depends on these two effects. If an ac field of increasing frequency is applied, the permittivity of the solution does not change until the frequency is of the order of the fluctuation rate of the motion of the molecules. Beyond that frequency, the molecules do not have time to reorient between different directions, thus the polarization vanishes and eIotal(~) drops to the vacuum value E, Z 1. (b) If the pH is close to PK of the acid/base sites, protons continually bind and disassociate, causing the dipole moment p to fluctuate in time. It is not (p) but the fluctuation of p that contributes to the dispersion of E’(W).The dispersion is proportional to $. Therefore, it is possible to have dispersion even for molecules with (p) = 0, provided there are fluctuations in p, which explains the dispersion of Glycine solution because Glycine is non-polar. (c) There is a cloud H+ ions around proteins from spontaneous water which could be polarized by application of a field.
dissociation of
Consider the imaginary part of the dielectric response c;bta,@) for Glycine, Myoglobin and Hemoglobin solutions, Figs. 2, 4, and 6 - short-dashed lines in Figs. 2, 4 and lower solid line in Fig. 6. We see that for Glycine and Myoglobin solutions, E&,~,(w)has two absorption peaks, whereas for the Hemoglobin solution, the absorption has three peaks. The full water inertia tensor is required to obtain a third peak, and the Hemoglobin cannot be spherically symmetric. At these peaks, energy is absorbed by molecules from the field at their relaxation frequencies. For the Hemoglobin solution, there exists a small peak in the absorption around 100 MHz which is
S. Chakraboiv,
B. DeFacio / Physica A 221 (1995) 419-437
Fig. 8. The absolute value of the refractive index, Hemoglobin
1n(w)1( x
435
Re n(w)), as a function of log frequency, of 20%
solution.
due to interactions between Hemoglobin and water. Also, the absorption peaks of the aqueous solutions shift toward lower frequency from water alone as was shown by S.H. Kim et al. [21]. This may be due to some interaction between water and the solute macromolecules which is absent in this model. 3.1 Cole-Cole Plots A Cole-Cole plot is a plot of E”(O) vs. E’(O) in the form E”(O) or E’(O) - 1 = 0; - 09 . The Cole-Cole plots for the Glycine solution, the Myoglobin solution, and the Hemoglobin solution are shown in Figs. 3, 5 and 7, respectively. The Cole-Cole plot for the Hemoglobin solution crossed itself reflected dielectric hysteresis. The amount of energy absorbed is related to the area enclosed by the curve. For Glycine, it is a semicircle with a small lobe, and there is no dielectric hysteresis for Glycine, possibly because Glycine is non-polar.
4. Conclusion Using the Lorentz-Debye model, we gave simple expressions for E’(O), E”(O) for Glycine, Myoglobin, and Hemoglobin solutions, which were compared with the experimental results from Grant et al. [a]. Ours model is a macroscopic LorentzDebye model which follows from the work of Kim et al. [21]. A local frequencydomain response function is given for the dielectric permittivity for wave propagation studies in these interesting materials. Still, this simple model can explain the major features of the dielectric response curves (Figs. l-8) for macromolecule solutions
436
S. Chakraborry, B. DeFacio 1 Physica A 221 (1995) 419-437
without conflicting with the limited available experimental results [2]. Comparing experimental results in Figs. 1, 2, 4 and 6, we see that our model is quite consistent except for the Hemoglobin solution dispersion curve between 1 MHz and 100 MHz. Again, the experimental results on water by Grant et al. [2] differ from the later and better values by Jackson [23], M.R. Querry [24], Oughstun [22] and S.H. Kim et al. [21]. Also, the relaxation frequencies of Myoglobin and Hemoglobin are not accurately known. The pH is a major factor affecting the dipole moment of macromolecules. Temperature is also a factor that affects the dielectric response [2,30]. Also, a detailed model involving the inertia effect of macromolecules should be more accurate, but calculation of inertia of macromolecules is not easy, All of these factors can cause differences between this theoretical model and the experimental data. It is a pleasant surprise that such a simple model works this well. The formulas given in Eqs. (3.la, b), (3.2a, b), and (3.3a, b) with suitable values of the concentrations and with Eq. (2.30) for the components can be used for other biomolecular solutions to study wave propagation in these media.
Acknowledgements Dr. Lee Carlson, Department of Physics and Astronomy, University of MissouriColumbia and Dr. S.H. Kim, Department of Physics, University of Minnesota are thanked for their useful conversations and Dr. Carlson is thanked for his help with the computer programming. This work has been partially supported by grants AFOSR 91 NM203,94 NM387 and an O.M. Stewart Scholarship for one of us (SC).
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J. McConnell,
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L.E. Reich] and J.E. Sedlak,
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M. Neumann, J. Chem. Phys. 82 (1985) 5663; Mol. Phys. 57 (1986) 97. E.O. Pollack and B.J. Alder, Physica A 102 (1980) I; Phys. Rev. Lett. 46 (1981) 950. D. Wei and G.N. Patey, J. Chem. Phys. 91 (1989) 7113; 93 (1990) 1399. F.O. Raineri, H. Resat and H.L. Freidman, J. Chem. Phys. 96 (1992) 3068. S.H. Kim. G. Vignale and B. DeFacio, Phys. Rev. A 46 (1992) 7548-7560; Phys. Rev. E 48 (1993) 3172-3175; Phys. Rev. E 50 (1994). K. Oughstun, J.E.K. Laurens and C.M. Balictis, in: Ultra-Wideband, Short-Pulse Electromagnetics, H. Bertoni et al., Editors (Plenum Press, New York, 1993), 223. J.D. Jackson, Classical Electrodynamics, 2nd Ed. (John Wiley, New York, 1975). M.R. Querry et al., J. Opt. Sot. Am. 59 (1969) 1299. 1. Thormahlen et al., J. Phys. Chem. Ref. Data 19 (1985) 933. P. Schiebener et al., J. Phys. Chem. Ref. Data 19 (1990) 677. B.E. Pennock and H.P. Schwan, J. Phys. Chem. 76 (1969) 2600. E.H. Grant et al. Biochem. J. 139 (1974) 375. H. Eyring and MS. Jhon, Significant Liquid Structures (Wiley, New York, 1969). A. Stankov, Physica A 211 (1994) 420-426,427-430, 43 l-435.
ELSEVIER
Dynamic frequency-dependent dielectric response of three aqueous biomolecular solutions Sankar Department
Chakraborty*,
Brian DeFacio
of Physics and Astronomy, University of Missouri-Columbia,
Columbia, MO 6521 I, USA
Received: 14 February 1995
Abstract A simple two-component Debye-Lorentz model is presented for the frequency dependence of the dielectric permittivity E(W)of aqueous solutions of three biomolecules, Glycine, Myoglobin, and Hemoglobin. A local response function for this model is given for the use of workers in wave propagation and biophysical studies. It is necessary to treat the full inertia tensor of the water, I, # I, # I, # I, to obtain good qualitative agreement with the experimental values of the three biomolecular solutions given. The Hemoglobin solution is shown to exhibit dielectric hysteresis, whereas this effect is absent in Glycine and Myoglobin. The Hemoglobin is found to need a prolate inertial tensor to agree with experiment. Permittivity; Rotational analysis; Liquid diffusion; (General) theoretical biophysics; Solutions of biomolecules
Keywords:
1. Introduction
Water is the most common solvent on earth [l] and is a principal constituent of all living organisms [2]. Approximately seventy percent of the human body is composed of water, with the amount of water in, each type of tissue ranging from over ninety percent of plasma to about thirty percent of fat tissue. Thus, the understanding of biological systems will require additional knowledge of water. Since the biological macromolecules live and function in an aqueous environment, the water molecules play a vital role in determining the structures and properties of the macromolecules. Water does not merely form an inert continuum in which a biomolecule resides. In fact, the double helix DNA structure proposed by Watson and Crick [3] owes its stability partially to the presence of water molecules in the neighborhood, as was observed by Jacobson [4]. Physically, the dielectric properties of water govern both
*Corresponding
author.
0378-4371/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4371(95)00198-O
420
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B. DeFacio
/ Physica A 221 (1995) 419-437
the electrostatic forces and the propagation of electromagnetic waves in biological media. From the dielectric studies of water and aqueous solutions of biological macromolecules [2], molecular parameters of the macromolecules such as size, conformation, hydration, and dipole moment in solution can be obtained. These properties will someday contribute to medical and biological applications such as the mechanism of activation of certain anaesthetic agents [S], evaluation of structure of lipoproteins present in blood serum [6], techniques of radiowave or microwave therapy in malignant diseases [7], measurement of membrane thickness [8], diagnosis of senile dementia from dielectric measurements of cerebral spinal fluid (CSF) [9], and many others. There is much left to learn of the physics of biological systems, as Vitiello [lo] has emphasized. Here, only a very small first step toward his ambitious program is taken. Many biologists believe that the water in the neighborhood of biomolecules, which they call “bound water”, behaves differently from ordinary “free” water. The effects of pH, temperature, viscosity, and the complexities due to the dipole-dipole interactions, and boundary conditions make it impossible to give a rigorous theoretical model that can be solved in classical statistical mechanics, so simplifying model assumptions are required. Several macroscopic models have been discussed by Scaife [ 1l] in order to describe experimental data from dielectrics, most of which do not follow the pure Debye or Lorentz models. Three useful models from Scaife are given by empirical formulae; the Cole-Cole formula
‘(O)= ‘,
+
6s -Em Cl
+
(iwzD)“]
for 0 < CI< 1, the Cole-Davidson
E(W) = Em+ [
(1.1)
’
formula
ES - EmI 1 +
(iozD)]P
(1.2)
’
and the Havriliak and Negami formula E(W)=
E,
+
ES - Em
[1 + (ioz&“]a
’
(1.3)
where (~1,j?) E (0, 1) and .sSis the w = 0 dielectric permittivity. A common feature of all three model equations above is that E’(O) -E, is non-negative, in contrast to the Lorentz model at resonance absorption. Dissado and Hill [12] have proposed a cluster model to calculate the dielectric response function. Their model has two power-law regions of frequency dependence governed by two parameters n, m for the clusters as discussed in their work. It has a clear physical basis unlike the empirical formulas in Eqs. (l.l)-( 1.3), and it fits data as well as or better than these. The present work will follow this general approach, albeit in a somewhat different direction.
S. Chakraborty, B. DeFacio J Physica A 221 (I 995) 419-437
421
The pioneering work by Debye [13] on polar molecules early in this century opened dielectrics as an area of research. His model explained the microwave fall off of the real part of the dielectric function and the accompanying absorption peak, but the flat infrared region obtained from the “E,” of his model was not consistent with experiment. Inertia was simply set equal to zero. McConnell Cl43 treated the inertia correctly and showed that it gave additional structures at infrared frequencies. However, this work was unable to explain the experimental data of water. Later, Sparling, Reichl, and Sedlak [15] added both hydrodynamic flow and viscous friction to McConnell’s analysis. However, one limitation of both these approaches is the fact that both are hydrodynamic and, therefore, are only valid at low frequencies and small wave vectors. The problem of calculating the dielectric function of water based on a macroscopic model was first addressed by Rahman and Stillinger [16] in 1971. Their work involved the then-new molecular dynamics which they developed. The water molecule was treated as a rigid symmetric rotor with translational and rotational motions, ignoring the effect of vibrations. A model potential between water atoms with two point negative charges on the oxygen atom and one positive charge on each hydrogen atom gave an inter-molecular molecule interaction, and between two molecules, the potential was chosen such as to include dipole-dipole interaction, repulsion between oxygen atoms and intermediate range interaction leading hydrogen bonds. However, this theory was not used with any dependence of the dielectric function on the frequency and the wave vector. Hundreds of workers have followed their approach with limited success. Recently, the calculations of frequency and wave number of dielectric functions of water and aqueous solutions have been published by several authors [ 17-201 using microscopic theories. Neumann [17] published computer simulations using the MCY (Matsuoka-Clementi-Yoshimine) potential for water, but he obtained values from 25-31 compared to 88 for the dielectric constant of water. Pollack and Alder [18] treated the simple model of the Stockmayer fluid of linear point rotors interacting via dipole-dipole and Lennard-Jones potential. In their computer simulations, they found a collective mode which they called the dipolarons at high frequency and low wave vector. A different approach to the problem was first made by Wei and Patey [19], who used the integral equation formulation of classical statistical mechanics. They treated a collection of ideal molecules as rigid symmetric rotors with electric multipoles to account for the mutual interactions. Then the integral equations of the reference hypernetted chain approximation, RHNC, are solved on the computer to obtain the static dielectric function. The dynamic correlation functions are obtained from their static counterparts by using the Kerr approximation, which they showed to be superior to an approach by Vineyard. S.H. Kim et al. [21] included the full inertia effects and calculated the self-VanHove function in high and low frequency ranges using the Kubo fluctuation-dissipation theorem and the Kerr approximation. They related the self-Van-Hove function to
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Physica A 221 (1995) 419-437
the dynamical response function of an external time-dependent potential. Their dynamic response function was found by solving the Boltzmann transport equation for the one-particle distribution function in phase space, in presence of an external time dependent potential. The dielectric function is calculated (in the numberconserving, relaxation-time approximation) from the response function using a local field factor from Wei and Patey. This guarantees the diffusion regime at low frequency and wave vector and interpolates to the high frequency plasma behavior. The experimental input to this model is one relaxation time, TD, the static self-susceptibility xj (o = 0, q = 0),and a static function called the local field factor, Y(q). Physicaly, tD represents incoherent collisions among molecules, 2: is the scale of the o + 0 dielectric permittivity ~(0,0),and the local field factor, Y, represents the coherent long range interactions, which were taken from Wei and Patey [19]. The claim in Kim et al. that xi zs np2/3kT is wrong, due to a simple calculational error. Instead, one needs, xso= 104np2/3kT, which explains why the problem of finding e(0) is so difficult. This model [21] with this correction had a surprisingly good agreement with experiment, although it fails to give a detailed agreement such as that in the curve fit in [22]. It may organize the microscopic water problem for e(w, q) into distinct parts: the static dielectric constant ~(0,0),the microwave relaxation time (or times) TD, and long range coherent effects on a single “hydration” cell. The computer simulations cannot yet calculate xz accurately; their results vary from 31 to 71 for ~(0, 0) compared to the experiment value of 87.7. Oughstun’s group [22] found that two relaxation times are required for a more accurate agreement, and each of which may have its own local field factor. The full asymmetric rotor inertia was treated by Kim et al. and is shown to be necessary in the present study. Here, Kim et al.% model is extended to two free components, which provides a model with a physical basis for E(O), one which is a mixed Lorentz-Debye model. This model will be the starting point of the present study. However, the agreement with experiment was worse than their symmetric rotor model. The splitting of the absorption peak is probably correct. Recently, Oughstun et al. [22] gave a careful, high accuracy curve-fit using a linear combination of the Rocard-Powles extension of the Debye model with two relaxation times combined with the analytic expression for the dielectric function given by four Lorentz oscillator models with four distinct resonance lines. His work agrees in reasonable detail with the best water data [23-261, although it does not explain inconsistensies between the Kramers-Kronig relations for E’and E”. Oughstun and co-workers also calculated a realistic, parametric dielectric function and the propagation of a Gaussian pulse in this medium. The macroscopic model presented here does not contradict McConnell’s statement [14] (on p. 37) because the angular variables (0,&~)of the rotor describe an isolated molecule in a Lorentz sphere in the liquid. The collisions are added in the second stage when this object is embedded in the single particle distribution of a number-conserving, relaxation-time Boltzmann equation by Eq. (2.37). It is remarked that the form of this model is quite different from the one in Ref. [22].
S. Chakraborv,
B. DeFacio 1 Physica A 221 (1995) 419-437
423
Experiments reporting E(W)for water are found in the work of M.R. Querry [24]; data on water is compiled by Thormahlen et al. [25], and Schiebener et al. [26]. There are experiments on the dielectric response function of Glycine solution [2], Myoglobin solution [28], and Hemoglobin solution [27], which have been in the literature for many years. Even the best theories for water today cannot accurately explain the experimental data. A generalization of the Debye model to an ellipsoidal molecule having different relaxation times along the semi-major and semi-minor axes was given by Grant et al. [2],
43
&I =
1 + &,’ E”(W) =
+
Ab 1 + w2r; + em ’
A, * cot, Ab- Wtb + 1 + c!12rZ’ 1 + C&f b
(1.4)
(1.5)
However, this cannot explain the experimental data of E”(O) by Grant et al. especially because of an extra absorption peak around 100 MHz, which is absent from this expression. Since data on the frequency dependence of the permittivity is available in Refs. [2, 24-281, it is interesting to ask how much physical structure is present and whether it can be described by a simple model.
2. The macroscopic model Initially, it was intended to solve a two-component version of the model in Ref. [21] with reasonable interactions between the constituents. The absence of the values of the moments of inertia as a function of the pH of the biomolecules and the lack of any frequency w and wave number k dependence of the permittivity in order to determine the local field factor, made this impossible. Since Kim et al. found inertia in Hz0 to split the absorption peak, a model with II # lZ # Z3 # II for water is required to obtain all of the absorption peaks. The biomolecules were treated with their physical masses and spherical radii of gyrations were chosen which were consistent with their densities. The concentrations of the biomolecules is low enough that the simple version, which follows, is to take two interacting dielectrics. This turned out to give all of the qualitative features of the experiments at this level of analysis, so that more intricate microscopic models cannot be justified for today’s experiments. Experimentalists are challenged to provide the moments of inertia as a function of pH and the real and imaginary parts of E(W)at a variety of concentrations over a range of frequencies, or even better E(W,q). Here, a Lorentz-Debye model for the dielectric function is presented. A Lorentz oscillator model is solved which yields the macroscopic Lorentz equation for unit step function response. Using this response, the frequency dependent polarizability a(w) is
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B. DeFacio / Physica A 221 (1995) 4/9-437
calculated by applying linear response theory. Then, E(W) is used to obtain the frequency-dependent dielectric permittivity E(W)using Ref. [21]. First, we follow Scaife or McConnell, and consider a macroscopic dielectric material polarized by an external field E0 which is turned on at t = 0, e(t) = &u(t),
(2-l)
where
u(t) =
0,
MO,
3,
t=O,
1,
t>o
(2.2)
is the unit step function. Assuming the field to be uniform, there will be an induced dipole moment m(t) = &H(t)
9
(2.3)
where H(t) is the system unit step-function response. For t < 0, H(t) = 0, because there is no response before the application of the field by the Principle of Causality. For dielectrics, m(t) approaches the equilibrium value lim m(t) = m, = EOcc,, 1-tcu
(2.4)
because of dissipations, which must be non-zero because of the existence of fluctuations. At long times, lim H(t) = ~1,) f-o3
(2.5)
where rx, is the static polarizability of the dielectric. The sudden removal of the external field forces a rapid decay of the dipole moment from its equilibrium value m, to a smaller value. If we remove the field at time tR, then m(t) = m, -E,H(t
- tR) = E,b(r - tR),
(2.6)
which explains the decrease in m(t) over time with b(t) as being an after-effect function that equals zero for t < 0. For an arbitrary external electric field e(t), we can calculate m(t), the induced dipole moment from H(t), provided the fields are not too strong, so that the principle of superposition holds as Htota~(t) = HI (t) + i?(t) = A-H(t),
H2
(4
2
(2.7a) (2.7b)
where La, (t) is the response to the total electric field e1 (t) + e2(t), HI (t), H2(t) being individual responses to e, (t), e2 (t), respectively; A(t) is the response to a field Ae(t), and H(t) is the response to a field e(t).
S. Chakraborry,
B. DeFacio
425
1 Physica A 221 (1995) 4I9-437
In case of these linear systems, we consider the external field to be composed of (sufficiently small) piecewise constant steps. The step at time W has an amplitude [de(W)/dW] dW. As a result, we can write f
m(t) = e(0 + )H(t) +
s dW
de(W) -j+w
-
(2.8)
WI,
o+
where e(0 + ) is the amplitude of the initial step. In the above equation, the lower limit of integration, 0+ , implies that the integration begins just after t = 0; e(0 +) removes the difficulty of an infinite derivative in the integrand. Upon integrating Eq. (2.8) by parts, one obtains m(t) = e(t)H(O +) +
I
dz’e(t -7)~
dH(4
(2.9)
0
where H (0 +) = 0 since a physical system cannot respond to a changing stimulus instantaneously. Also, H(t) = 0,
and dH(r) dt=
o ’
for t < 0 by causality, so dH(t)/dt is called the delta function or pulse response. Using causality, we write Eq. (2.9) as
m(t) =
s
dre(t -
7)dz.
dH(4
(2.10)
-m
If we consider the sudden application of a periodic external field at t = 0 co(t) = Eocosotu(t), then f_ m(t) = E,coswt
*_ dr dH (4
s
-
dr dH(r)
cos ox + E. sin wt
d(z)
-
s
dz
.
sin or .
(2.11)
The steady state response to the field co(t) can be obtained by taking values of t much greater than the maximum time-constant of the dielectric. This permits the upper limit in the last integrals (eq. 2.11) to be replaced by infinity, so that the steady-state response becomes m(t) = E. [a’(o) cos cot + cc”(w) sin ot] ,
(2.12)
S. Chakraborty, B. DeFacio / Physica A 221 (1995) 419-437
426
for t > 0 where the real and imaginary parts of the polarizability m
+W
a’(w) =
are given by
dH (t) -COSOt
5 dt
=
dt
s
(2.13a)
dtyLosot,
0
-CO
and
a”(0.J) =
s
dt 9
dt -dH (t) sin wt dt
-
sin cot
dt
(2.13b)
’
Using Eqs. (2.13a) and (2.13b), we can write +CC a (0) =
dH(d
dz exp( - ior) 7
a’(w) - id’(o) =
.
(2.14)
-52
From Eq. (2.13a), co
s
dHW
a’(O)=o~~=
(2.15)
d,-dr=H(co),
0
lim c!(o) = lim
w-to2 s
w-em
dzcosoz
dffb) dz
=
o
,
(2.16)
0
this last result depends on the fact that pll=
dH(r)
o
.
Also, a’(w) = a’( -0))
(2.17a)
a”(w) = -a”( -w) .
(2.17b)
Eqs. (2.17a) and (2.17b) then imply that (2.18)
a(o)=a*(-o). 2.1. Dispersion relations
Taking the inverse of Eqs. (2.13a) and (2.13b), m
dH(t) 2 dwa’(m)cosmt = dt n s 0
(2.19a)
427
S. Chakraborty, B. DeFacio / Physica A 221 (1995) 419-437
and dH(t)
_
df
2
(2.19b)
= - 7 do a”(w) sin or . rJ
”
0
Combining Eqs. (2.19a, 2.19b) yields U(t) -=-
2
dt
dwa(w)exp(iwt)
TT s
(2.20)
-CC
but dH(t)/dt = 0 for t < 0, so that this equation can be written as a
a(o) =
s
dH(r) dtFexp(-iwt);
(2.21)
0
using Eq. (2.20), we find that a(w)=
s
dtexp(-iwr)
dpa(Aexp -CC
0
(W)
1,
(2.22)
where a(p) is even in p and decreases rapidly enough as p -+ co, so that we get a a’(o) =zP Tc
s
dp
a”(p)w (,u2 -02)’
(2.23a)
0 a
a”(o) =ff’
s
dp
a’(cl)w (m2 -P2) ’
(2.23b)
0
The formulae above are the once-subtracted Kramer+Kronig dispersion relations and express the causality of the system. The real and imaginary part of a causal dielectric function must obey the Kramers-Kronig relation. Stankov [30] has derived an expression for a(O,O) as a function of some unspecified mean molecular kinetic energy and temperature based on the Maxwellian distribution of kinetic energy. Despite extravagant claims, its tables of numbers explain little to us, since it is related to neither a microscopic nor a macroscopic physical model. 2.2. Lorentz Absorption Here, we obtain the unit step-function response, a,(t), by Fourier transforming the Lorentz model of dielectric, where the reader is reminded that the collisions among molecules have not yet been taken into account. This model remains a pure Lorentz dielectric. From a,(t), we find a(o), the polarizability using eq. (2.14).
428
S. Chakraborry
B. DeFacio
1 Physica
A 221 (1995) 419-437
Here, we take E, = 1 for the vacuum response as w + 00. The sudden application of a unit voltage to the circuit at t = 0 builds up a change aL(t) on the capacitor (as - l), so we can write the oscillator differential equation 1 CPU&) &. (0 ~z+Ydt+a,(t)=(&,-l)u(t), ri+, dt
(2.24)
which has the solution o0ev(
q.(t) = (ES- 1) 1 [
-0)
a
1
cos(S2t - <) u(t),
(2.25a)
y=(a,--l)~~exp(-:r)y,(r),
(2.25b)
where (2.26a) (2.26b) (2.26~) and the factor y controls the damping. Earlier in this section we had Eq. (2.21), m a(w)=
s
dH 0) dr- dt exp(-iM,
0
which for the Lorentz model becomes a(w) =
s
dt-
daL (r) dt
exp(-iot).
(2.27)
0
Using Eqs. (2.25a) and (2.27), we can write
m
a(w)=(ES-lb02 dt 2iQ
s
{exp[-i(w
-Q
- ii)t] -exp[-i(o
+ Q -i<)t]}
0
=
_kS
-
1
1
lb,2
(ES -+ lb: =(6$ -co2 2Q
1
(w -5)
-ii)
-(co
+
Q -
ii)
1
iywio)’
but the polarizability
of an ellipsoid along a principal axis is given as
V&orw a(W)={l
(2.28)
- 11
+R[&(O)-11)’
(2.29)
S. Chakrabor& B. DeFacio / Physica A 221 (1995) 419-437
429
where ), is the depolarizing factor. So we can write the relative permittivity (setting I/Q = 1) using Eqs. (2.28) and (2.29) E(U) - 1 (ES- l)& { 1 + ACE(W)- 11= (f& - Cf.?+ iyw,20) ’
(2.30a)
where
i=
,
1,
oblate
0,
prolate,
(2.30b)
spherical.
i f,
Hence, for /z = 0 which is the case for water E(U) -1 =
(6 -
l)w,z
(& - W2)2 + iye&
’
i.e., E’(W) = 1 +
E”(W)=
(ES-1)0&o;
- 02)
[(CD;- Cl?)2 + 02P]
’
(ES- l)&DT
[ (0; - W2)2 + WZP]
’
(2.31a)
(2.31b)
where r = yoi is the damping. For an arbitrary value of A, the response function S[w] is given by E(W)- 1
(ES- 1)002 =:S[w], 1 + ACE(W)- 11= (0,’- co2+ iyo,2w)
(2.32)
i.e.,
E(u)= l-(qo]
sEmI
-1)’
(2.33)
with
sco1 As[w] -1 = F(w)
(2.34)
E(U) = 1 -F(o).
(2.35)
and
Hence, E’(U) = 1 -ReF(o)
,
(2.36a)
and C(O) = Im F(o).
(2.36b)
430
S. Chakraborry. B. DeFacio / Physica A 221 (1995) 419-437
Here, we treat the water as asymmetric and the other molecules such as Hemoglobin, Myoglobin and Glycine as prolate symmetric rotors, and the dielectric function for each molecule has two parts: (i) For 1 = 0 (symmetric) for the biomolecule, and (ii) Due to asymmetry i.e., an arbitrary value of A, for the water solvent. Now, the number conserving relaxation-time approximation is incorporated our model; by substitution which now adds collisions among molecules, 0
+
0
+
i/z
into
(2.37)
,
which leads to an additional Debye behavior E(U) - 1
1 + A[s(o) - 11
=Gl&{exp[
-i(o+f-6a-il)r]
)I1,
0
- exp
-i
w+~+G?-ii
t
[(. and finally gives E(U) - 1
l+A[e(w)-ll={(
0.1:-02
(ES- l)& + l/r2 - yo~/r) +
i(yoo~
-24~))
’
(2.38)
which for (A = 0) reduces to E’(W) = 1 +
(ES- l)c&c$
-o2
+ l/t2 -7)
(2.39a)
{(0: -W2 + l/r2 - r/r)2 + (OT - 20/2)2} ’ and (ES- 1)
w;(OT -
242)
sN = {(0,” - o2 + l/T2 - r/r)2 + (WT -202/r}
(2.39b) ’
where r = yoi as shown before.
3. Results and discussion Since the water solution plays an important role in this discussion, Eq. (2.30) with A = 0 is used to calculate the real and imaginary parts of E(O). The index of refraction n(o) = & is shown in Fig. 1. The agreement with experiment is considerably better than that of Kim et al. but is less accurate than Oughstun. The attenuation coefficient is as accurate as Ref. [22] through w * 10r2, but this would fail at higher frequencies where the second relaxation time becomes important. The calculated real and imaginary parts of E(W)for the Glycine solution are shown in Fig. 2. The Cole-Cole plot for this solution is given in Fig. 3. For the dielectric
431
S. Chakraborty. B. DeFacio / Physica A 221 (1995) 419-437
(4 10
0’
6
7
8
9
10
11
(b) 1000. .
.-*
Fig. 1. (a) The real part of the refractive index of water as a function of log frequency. The solid curve is for the calculated values. The dotted curve is for the experimental values from Ref. [23]. (b) The absorption coefficient of water as a function of log frequency. The solid curve is for the calculated values. The dotted curve is for the experimental values from Ref. [23].
response of aqueous solutions of Glycine, where the electric dipole moment is induced: E~,,,~, (0) = O.O75s&0) + 0.925cbA (w) + 0.263 Re [ JsWA(&cS(cu)] sIbtal(~) = 0.075&,(o)
+ 0.925cl;yA(o) + 0.2631m[JEWA(~)~GS(~)],
,
(3.la) (3.lb)
Similarly, for the dielectric response of the Myoglobin aqueous solution: E;,,~,(w)= 0.1&(0)
+ 0.9sLA(c0) + 0.3 Re[J.sWA(~)sy&)],
E&+,(O)= 0.1&s(~)
+ 0.9.sl;yA(u)+ 0.3 Im [J-l
(3.2a) ,
(3.2b)
Here, we see two different regions of dispersion at the respective relaxation frequencies of Myoglobin and water, e.g., the solid line in Fig. 4 has a much larger separation than Glycine. The corresponding Cole-Cole plot is presented in Fig. 5. The dielectric response of the Hemoglobin solution satisfies the equation: &.i(~)
= 0.2&W
+ 0.8aka(~) + 0.4 ReCJmI
E;~,,,(w)= 0.2.$&0) + 0.8al;yA(c0)+ 0.4 Im [,/aWA(~)aH&)]
,
(3.3a)
,
(3.3b)
S. Chakraborty, B. DeFacio / Physica A 221 (1995) 419-437
432
Fig. 2. The real and imaginary
parts of the dielectric response function (E(W)) of 1 M Glycine solution in
water. The solid curve is the calculated values of E’(O); the long dashed curve is the experimental from
Ref. [2];
experimental
the short dashed
curve
is the calculated
values of E”(W); the dotted
curve
values c’(w) is for the
values of E”(O) from Ref. [S].
Fig. 3. The Cole-Cole
Plot for a 1 M Glycine
solution in water.
where the first term accounts for symmetric Hemoglobin, the second term is for asymmetric water, and the last term accounts for the uncorrelated interaction of lowest order between water and Hemoglobin molecules following Eyring and Jhon [29]. Upon inspection of the dielectric response curve (upper solid line) in Fig. 6, we see that for the Hemoglobin solution, the real part of E~~,~,(o),has two regions of dispersion - one around 1 MHz and the other around 10 GHz. The first one is due to the relaxation of the large Hemoglobin molecules whereas the other one is due to the Debye relaxation of the water molecules. Unlike the Hemoglobin and Myoglobin solutions, the Glycine solution does not have two diseersion regions, but only one. In Fig. 2, the solid line shows that the relaxation times of water and Glycine are not very different, since the molecular weight of Glycine is 75, it is non-polar. This is to be compared to x 15000 for Myoglobin and x 68000 for Hemoglobin, i.e., they relax almost together. This is one difference from the aqueous solutions of very large biomolecules. The overlapping of relaxation regions is a characteristic of solution of biomolecules whose mass and intertia is close to that of water (or the solvent).
S. Chakraborty,
B. DeFacio
/ Physica A 221 (1995) 419-437
433
Fig. 4. The real and imaginary parts of the dielectric response function (E(W))of 10% Myoglobin solution in water. The solid curve is the claculated values of E’(W);the dotted curve is the experimental values of E’(O) from Ref. [2]; the dashed curve is the calculated values of E”(W).
Fig. 5. The Cole-Cole Plot for a 10% Myoglobin solution in water.
0
2
4
6
a
10
12
Fig. 6. The real and imaginary parts of the dielectric response function (E(O))of 20% Hemoglobin solution in water. The upper solid curve is the calculated values of E’(W);the long dashed curve is the experimental values of E’(u) from Ref. [2]; the lower solid curve is the calculated values of C”(w);the dotted curve is for the experimental values of E”(O) from Ref. [2].
434
S. Chakraborty B. DeFacio / Physica A 221 (1995) 419-437
0.. 0
25
50
75
100
125
150
171
Fig. 7. The Cole-Cole Plot for a 20% Hemoglobin solution in water. The dielectric hysteresis is the closed curve. It occurs in the frequency range lo6 Hz < v < 10” Hz of Fig. 6 where the splitting of the absorption peak due to asymmetric rotor inertia of water appears.
The frequency dispersion can be explained in different ways: (a) The dipole moment of the protein molecules experience an orientational force by the applied electric field. The Brownian motion, due to collisions with water dipoles, tends to randomize this alignment via collisions among water molecules, and the net magnitude of the polarization depends on these two effects. If an ac field of increasing frequency is applied, the permittivity of the solution does not change until the frequency is of the order of the fluctuation rate of the motion of the molecules. Beyond that frequency, the molecules do not have time to reorient between different directions, thus the polarization vanishes and eIotal(~) drops to the vacuum value E, Z 1. (b) If the pH is close to PK of the acid/base sites, protons continually bind and disassociate, causing the dipole moment p to fluctuate in time. It is not (p) but the fluctuation of p that contributes to the dispersion of E’(W).The dispersion is proportional to $. Therefore, it is possible to have dispersion even for molecules with (p) = 0, provided there are fluctuations in p, which explains the dispersion of Glycine solution because Glycine is non-polar. (c) There is a cloud H+ ions around proteins from spontaneous water which could be polarized by application of a field.
dissociation of
Consider the imaginary part of the dielectric response c;bta,@) for Glycine, Myoglobin and Hemoglobin solutions, Figs. 2, 4, and 6 - short-dashed lines in Figs. 2, 4 and lower solid line in Fig. 6. We see that for Glycine and Myoglobin solutions, E&,~,(w)has two absorption peaks, whereas for the Hemoglobin solution, the absorption has three peaks. The full water inertia tensor is required to obtain a third peak, and the Hemoglobin cannot be spherically symmetric. At these peaks, energy is absorbed by molecules from the field at their relaxation frequencies. For the Hemoglobin solution, there exists a small peak in the absorption around 100 MHz which is
S. Chakraboiv,
B. DeFacio / Physica A 221 (1995) 419-437
Fig. 8. The absolute value of the refractive index, Hemoglobin
1n(w)1( x
435
Re n(w)), as a function of log frequency, of 20%
solution.
due to interactions between Hemoglobin and water. Also, the absorption peaks of the aqueous solutions shift toward lower frequency from water alone as was shown by S.H. Kim et al. [21]. This may be due to some interaction between water and the solute macromolecules which is absent in this model. 3.1 Cole-Cole Plots A Cole-Cole plot is a plot of E”(O) vs. E’(O) in the form E”(O) or E’(O) - 1 = 0; - 09 . The Cole-Cole plots for the Glycine solution, the Myoglobin solution, and the Hemoglobin solution are shown in Figs. 3, 5 and 7, respectively. The Cole-Cole plot for the Hemoglobin solution crossed itself reflected dielectric hysteresis. The amount of energy absorbed is related to the area enclosed by the curve. For Glycine, it is a semicircle with a small lobe, and there is no dielectric hysteresis for Glycine, possibly because Glycine is non-polar.
4. Conclusion Using the Lorentz-Debye model, we gave simple expressions for E’(O), E”(O) for Glycine, Myoglobin, and Hemoglobin solutions, which were compared with the experimental results from Grant et al. [a]. Ours model is a macroscopic LorentzDebye model which follows from the work of Kim et al. [21]. A local frequencydomain response function is given for the dielectric permittivity for wave propagation studies in these interesting materials. Still, this simple model can explain the major features of the dielectric response curves (Figs. l-8) for macromolecule solutions
436
S. Chakraborry, B. DeFacio 1 Physica A 221 (1995) 419-437
without conflicting with the limited available experimental results [2]. Comparing experimental results in Figs. 1, 2, 4 and 6, we see that our model is quite consistent except for the Hemoglobin solution dispersion curve between 1 MHz and 100 MHz. Again, the experimental results on water by Grant et al. [2] differ from the later and better values by Jackson [23], M.R. Querry [24], Oughstun [22] and S.H. Kim et al. [21]. Also, the relaxation frequencies of Myoglobin and Hemoglobin are not accurately known. The pH is a major factor affecting the dipole moment of macromolecules. Temperature is also a factor that affects the dielectric response [2,30]. Also, a detailed model involving the inertia effect of macromolecules should be more accurate, but calculation of inertia of macromolecules is not easy, All of these factors can cause differences between this theoretical model and the experimental data. It is a pleasant surprise that such a simple model works this well. The formulas given in Eqs. (3.la, b), (3.2a, b), and (3.3a, b) with suitable values of the concentrations and with Eq. (2.30) for the components can be used for other biomolecular solutions to study wave propagation in these media.
Acknowledgements Dr. Lee Carlson, Department of Physics and Astronomy, University of MissouriColumbia and Dr. S.H. Kim, Department of Physics, University of Minnesota are thanked for their useful conversations and Dr. Carlson is thanked for his help with the computer programming. This work has been partially supported by grants AFOSR 91 NM203,94 NM387 and an O.M. Stewart Scholarship for one of us (SC).
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