Inelastic X-ray scattering studies of phonons propagating along the axial direction of a DNA...

Inelastic X-ray scattering studies of phonons propagating along the axial

direction of a DNA molecule having different counter-ion atmosphere

Yun Liu a, Debora Berti b, Piero Baglioni b, Sow-Hsin Chen a,*, Ahmet Alatas c,

Harald Sinn c, Ayman Said c, Ercan Alp c

a Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USAb Department of Chemistry, University of Florence, 50019 Florence, Italy

c Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA

Abstract

Shear-aligned 40 wt% calf-thymus Na-DNA molecules in aqueous solutions are prepared in their liquid crystalline phases and studied by high

resolution inelastic X-ray scattering (IXS). Measured IXS spectra are analyzed with the generalized three effective eignmode (GTEE) theory. The

phonon dispersion relations along the axial direction of DNA molecules with different MgCl2 concentrations are constructed and compared. It is

found that the sound speed along the axial direction of DNA molecules varies only slightly, but the phonon dampening is greatly affected with the

increase amount of MgCl2 concentration. Using the GTEE theory, we are able to extract the longitudinal viscosity in the hydrodynamic limit from

the Q-dependence of a fitted parameter. We make a comprehensive review of the GTEE theory and discuss detailed analyses of IXS spectra taking

into account finite energy resolution of the instrument.

q 2005 Elsevier Ltd. All rights reserved.

PACS: 61.10.Eq; 87.14.Gg

1. Introduction

Due to the central role it plays in controlling numerous

biological functions, the understanding of structural as well as

dynamical properties of DNA molecules is of paramount

importance. It has been recognized that the dynamics of bio-

macromolecules is intimately related to their biological

functions [1–4]. Therefore, dynamics of DNA molecules has

been studied both theoretically and experimentally by various

methods in the past several decades [4–11]. Scattering

techniques are very powerful for investigating both the static

and dynamic structures of materials. By using inelastic neutron

scattering (INS), it is found that the mean-squared atomic

displacement of hydrated DNA molecules has a sharp rise

when the temperature is above a transition temperature range,

which is between 200 and 230 K [4,5]. Maret and coworkers

first measured the sound speed in dry DNA fibers, films and

hydrated DNA fibers with Brillouin light scattering [6]. They

0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jpcs.2005.09.017

* Corresponding author. Tel.: C1 617 253 3810; fax: C1 617 258 8863.

E-mail address: [email protected] (S.-H. Chen).

found a linear relation between the excitation energy of the

acoustic mode and the transferred wave vector, Q. The sound

speed along the axial direction of dry DNA fibers was observed

to be about 3800 m/s. This sound speed decreases with

increasing DNA hydration level. The wet DNA fibers have a

sound velocity of about 1800 m/s, which is close to the velocity

of longitudinal sound wave of pure water at very low Q value

(about 10K2 nmK1). Grimm and co-workers measured the

acoustic excitations along the axial direction of DNA

molecules with semicrystalline DNA samples made with the

wet-spinning technique at the Q range from about 16–21 nmK1

with a triple-axis instrument [9]. Based on a one dimensional

model employed in their analyses, the acoustic velocity along

the axial direction of DNA molecules was about 2200 m/s

[9,12]. However, due to the instrument limitation, they could

not reach very large Q range in the measurements. Therefore,

the complete dispersion relation was not obtained.

With the development of the high resolution inelastic X-ray

scattering technique (IXS), one can now routinely make a

constant-Q energy scan in very large Q range and very large

energy window. This technique has already been widely used

to measure the acoustic excitations in liquid water, molecular

glasses, liquid metals, and biological samples, such as lipid

Journal of Physics and Chemistry of Solids 66 (2005) 2235–2245

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Y. Liu et al. / Journal of Physics and Chemistry of Solids 66 (2005) 2235–22452236

bilayers, and oriented DNA samples [13–19]. The phonon

dispersion relation along the DNA axial direction in aqueous

solutions with the hexagonal liquid crystalline phase has been

measured with IXS by us in the Q range from about 2–30 nmK1

[19]. By using the generalized three effective eignmode theory

(GTEE), we have obtained the high frequency sound speed of

the value, 3100 m/s, when the transferred wave length

approach the molecular length scale. In this paper, we further

extend our previous study to investigate the effect on the

acoustic excitations along the DNA axial direction due to

different counter-ion concentrations. The phonon damping at

different counter-ion concentrations is compared. The

longitudinal viscosity is extracted. The method, the GTEE

theory, is reviewed comprehensively, and the analysis

procedure is discussed in details.

2. Inelastic X-ray experiments

The experiments were performed with a high resolution

inelastic X-ray scattering at the beamline 3-ID-XOR at

Advanced Photon Source (APS), Argonne National Laboratory

in Chicago, USA. The energy of incident X-ray from the

synchrotron source is 21.657 keV with the overall instrument

energy resolution of about 2 meV. The Q-range of the

instrument is from 0 to 30 nmK1. The photon flux at the

sample position is about 6!108 photons/second with 100 mA

current in the storage ring. The spot size of the X-ray beam at

the sample is only 200 mm!150 mm. Therefore, experiments

only need a very small amount of samples, which is very

Fig. 1. This figure demonstrates different structure factors at different relative angles

of the figure represents a single DNA rod. The solid line with arrow is an incident X-

ray. The structure factors of the sample A at these two different orientations are pr

significant for biological experiments that sometimes could not

have large amount samples. The energy resolution function

was measured at QZ10 nmK1 by placing a Plexiglas sample in

the beam line and can be well described by a Pseudo–Voigt

function of the following form [20]

IðuÞ Z I0

2h

pG1 C4

u

G

2 K1

C ð1KhÞ2

G

ln 2

p

1=2

exp K4 ln 2u

G

2 (1)

where I0 is the normalization constant, u the transferred

energy, h the mixing parameter, G can be considered width of

the resolution function.

Three samples are prepared: sample A is 40 wt% calf-

thymus Na-DNA in water; sample B is 40 wt% calf-thymus

Na-DNA in 0.083 M MgCl2; sample C is 40 wt% calf-thymus

Na-DNA in 1.1 M MgCl2. The reason that we specifically call

the DNA molecules, Na-DNA, is that when we purchased DNA

molecules, there are already some sodium salts inside.

The parameters of the resolution function changed slightly

between different beam-times used to perform the experiments.

G and h are, respectively: 1.98 meV, 0.49 for the experiments

on sample A and 2.05 meV, 0.48 for the experiments on sample

B and C. The detailed information of the instrument can be

found in Ref. [20].

Each sample has been aligned and put into the beamline so that

the incident X-ray beam is perpendicular to the aligned

orientation of DNA molecules [19]. Fig. 1 demonstrates different

between an incident X-ray beam and an aligned sample. Each cylinder in the top

ray beam, while the dashed line with arrow shows the direction of a scattered X-

esented in the bottom of the figure.

Y. Liu et al. / Journal of Physics and Chemistry of Solids 66 (2005) 2235–2245 2237

structure factors at different relative angles between an incident

X-ray beam and an aligned sample. Each cylinder in the top panel

represents a single DNA rod. The solid line with the arrow is an

incident X-ray beam, while the dashed line with the arrow shows

the direction of a scattered X-ray. The structure factor of the

sample A at these two different orientations are presented in the

bottom of the figure. D is the distance of neighboring base pairs in

a DNA molecule, and d is the inter-rod distance. When an incident

X-ray is perpendicular to the aligned DNA rods, the scattered

wave vector, ðQ, is along the DNA rods. The IXS thus probes the

structures along the axial direction of DNA molecules. When an

incident X-ray beam is parallel to the DNA molecules, ðQ is

perpendicular to the axial direction. Therefore, the IXS probes the

inter-rod structures.

3. The generalized dynamic structure factor and the

generalized three effective eignmode theory (GTEE)

The GTEE theory was developed by Liao and Chen [21] and

is a natural extension of the three effective eignmode theory

(TEE) originally developed by de Schepper and Cohen [22] to

calculate the dynamic structure factor S(k,u) of one component

liquids in the finite k range. By finite k range, we means that the

wave length of the fluctuations approaches the molecular

length scale. In scattering experiments in liquids where there is

no crystal structure, k is equal to Q. Therefore, in the following,

unless specifically pointed out, we will use the symbol Q

instead of k in all formulas.

By extending the TEE theory, the GTEE theory takes into

account the multi-spices of atoms inside systems and sets up a

foundation to calculate S(Q,u) for biomaterials. The double

differential cross section of a system with N atoms is [23]

d2s

dQdEZ Nr2

0ð3i3fÞ2 kf

ki

SðQ;EÞ (2)

where EZZu is the energy transferred in the scattering

process, ki and kf are the wave vectors of the incident and

scattered X-rays, respectively, 3i and 3f are the polarization

vectors of X-ray photons before and after the scattering

process, r0, the classical radius of an electron. The generalized

dynamic structure factor S(Q,E) is defined as,

SðQ;EÞ Z1

2pZ

1

N

ðdt eiEt=Z

XN

j;l

hfiðQÞflðQÞeiQrlð0ÞeKiQrjðtÞi (3)

where fi is the form factor of the atom with index i. It will be

convenient to use S(Q,u) in the following, which is equal to

ZSðQ;EÞ.

Introducing now the generalized density fluctuation na(Q,t)

by including the form factor as

naðQ; tÞ Z1ffiffiffiffiffiffiNa

pXNa

jZ1

fjðQÞeKiQrjðtÞ (4)

where Na is the number of ath type of atoms.

Therefore, the generalized intermediate scattering function,

F(Q,t), which is Fourier transform of the dynamic structure

factor, S(Q,u), is

FðQ; tÞ ZXa;b

faðQÞfbðQÞwawbFabðQ; tÞ; (5)

where the partial intermediate scattering function is defined as:

FabðQ; tÞ Z naðQ; 0ÞnbðQ; tÞ

(6)

and waZffiffiffiffiffiffiffiffiffiffiffiNa=N

pthe square root of the number fraction of

atomic type a over the total number of atoms. The generalized

dynamic structure factor, S(Q,u), then can be written as

SðQ;uÞ ZXab

faðQÞfbðQÞwawbSabðQ;uÞ (7)

where the partial dynamic structure factor Sab(Q,u) is the

Fourier transform of Fab(Q,u).

After defining the generalized dynamic structure factor

Sab(Q,u), we can then derive how to calculate it. By

skipping the details of derivations, we will present the

major results, which are directly useful to fit IXS spectra.

The detailed derivations have been given in Ref. [21]. The

normalized generalized dynamic structure factor can be

written as

SðQ;uÞ

SðQÞZ

1

pRe

I

z CHðQÞ

1;1zZiw

; (8)

where I is the 3!3 identity matrix, label 1,1 outside the

curly bracket means the (1,1) element of the matrix, i the

imaginary unit. The generalized hydrodynamic matrix H(Q)

is in the form of

HðQÞ Z

0 ifunðQÞ 0

ifunðQÞ zuðQÞ ifuT ðQÞ

0 ifuT ðQÞ zT ðQÞ

0B@

1CA (9)

where the four Q dependent matrix elements, zT, fuT, zu and

fun, are treated as fitting parameters. Among them, fun(Q)is

the second frequency moment of the dynamic structure

factor and is given in terms of the structure factor, S(Q), as

funðQÞ Z Qv0ðQÞ½ðSðQÞÞK1=2;

v20ðQÞ Z

Xa

f 2a ðQÞw2

av20a;

(10)

where the index a refers to the ath atom, v0aZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBT=ma

p.

Eq. (8) can be evaluated in a more explicit form as

SðQ;uÞ

SðQÞZ

1

pRe

z2Kðzu CzT ÞzCzuzT Cf 2uT

z3Kðzu CzT Þz2 C zuzT Cf 2

uT Cf 2un

zKf 2

unzT

( )zZiu

(11)

Define the polynomial discriminant of the denominator of

Eq. (11)


DZK27f 4unz2

T K4f 2unzT ðzT CzuÞ

3 C18f 3unzT ðzu CzT Þ

f 2un Cf 3

uT CzT zu

CðzT CzuÞ

2 Kf 2unKf 2

uT KzT zu

2

K4 f 2un Cf 2

uT CzT zu

3:

(12)

When DO0, the denominator of Eq. (11) has one real

root Gh, and a pair of conjugate complex roots GsGiUs.

Thus Us is the phonon excitation energy, Gs the phonon

Fig. 2. The typical fittings of IXS energy spectra at QZ6.5 nmK1 are shown from th

results from the sample A, B, and C, respectively. Each spectrum is fitted with the G

total area of each spectrum. The thin solid line is the fitted dynamic structure facto

calculated by convolving the thin solid line with the energy resolution function of

damping. Eq. (11) can then be written in a hydrodynamic

form as the sum of three Lorentzian terms,

SðQ;uÞ

SðQÞZ

1

pA0

Gh

u2 CG2h

CAs

Gs CbðuCUSÞ

ðuCUSÞ2 CG2

s

CAs

GsKbðuKUSÞ

ðuKUSÞ2 CG2

s

(13)

ree different samples. From the top to the bottom, the figures correspond to the

TEE theory. The dots with error bars are experimental results normalized by the

r S(Q,u)/S(Q) multiplied by the detailed balance factor. The thick solid line is

the instrument.


where

MðxÞZx2Kðzu CzT ÞxCzuzT Cf 2uT ;

N ZMðGsKiUsÞ

2iUsðGhKðGsKiUsÞÞ;

A0 ZMðGhÞ

ðGsKGhÞ2 CU2

s

;

As ZReðNÞ;

b ZKImðNÞ

ReðNÞ:

(14)

When D!0, the denominator of Eq. (11) has three

unequal real roots, G1, G2, and G3. Therefore, all peaks are

centered at uZ0. There are no phonons in this case. Hence

Eq. (11) can be expressed as

SðQ;uÞ

SðqÞZ

1

p

X3

iZ1

Ai

Gi

u2 CG2i

; (15)

where

MðxÞZx2Kðzu CzT ÞxCzuzT Cf 2uT ;

A1 ZMðG1Þ

ðG1KG2ÞðG1KG3Þ;

A2 ZMðG2Þ

ðG2KG3ÞðG2KG1Þ

A3 ZMðG3Þ

ðG3KG1ÞðG3KG2Þ

(16)

When DZ0, the denominator of Eq. (11) has only real

roots and at least two are equal. In this case, there is also

no phonon. Since D is not zero in almost all the cases, we

will not write it in the explicit form here.

In the hydrodynamic limit (Q/0), the four matrix elements

can be expressed in thermal dynamical variables and transport

coefficients as

funðQÞ ZQcsffiffiffi

gp ;

zuðQÞ Z fQ2;

fuT ðQÞ Z Qcs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiðgK1Þ

g

s;

zT ðQÞ Z gDT Q2;

(17)

where cs is the adiabatic speed of sound, f the longitudinal

viscosity, gZcp/cv the ratio of specific heat at constant

pressure and volume, DT the thermal diffusivity.

When DO0, i.e. there is phonon in the spectrum, Gh, Gh, Us,

and Gs can be solved in the hydrodynamic limit up to the order

of O(Q2),

Gh Z DT Q2

Us Z csQ

Gs Zðf=2ÞC ðgK1Þ

2DT

Q2

(18)

In the fitting, we calculate the normalized generalized

dynamic structure factor, S(Q,u)/S(Q), with Eq. (11), and treat

zu, zT, fun, and fuT as fitting parameters. Denote the detailed

balance factor as SD(u). A theoretically calculated IXS

spectrum is obtained by convolving SD(u)S(Q,u)/S(Q) with

the energy resolution function and is compared with the

measured spectrum normalized by its own area. Among the

four fitting parameters, fun can be estimated directly from an IXS

spectrum and the corresponding energy resolution function [19].

This estimation can serve as the initial trial of the fittings.

4. Results and discussions

In this section, we apply the GTEE theory to analyze the

IXS spectra. Although the dispersion relations of sample A and

sample B have been previous published [19]. But the detailed

analyses of IXS spectra are not presented. We will present the

detailed analyses here and compare them with the results from

sample C. The counter-ion effects due to different concen-

trations of MgCl2 on the phonon damping and the longitudinal

viscosity are studied.

Fig. 2 shows the typical fittings of IXS spectra at QZ6.5 nmK1 for all three samples. The thin solid line is the

dynamic structure factor S(Q,u) calculated from the fitted

parameters multiplied by the detailed balance factor. The

dashed line is the energy resolution function. The thick

solid line is the convolution of the energy resolution

function with the thin solid line. Only the lower part of

each spectrum is plotted in order to amplify the acoustic

excitation features. From the top to the bottom, each panel

corresponds to the spectrum of sample A, B, and C,

respectively. At this Q value, the collective excitations can

be very clearly seen from the spectra. The asymmetric

property of each spectrum is due to the detailed balance

factor. From the thin solid line, we can directly see that the

phonon damping is much stronger in sample B and sample

C than that of sample A.

Fig. 3 shows the typical fittings of IXS spectra at QZ16.0 nmK1 for all three samples. The symbols and

arrangements of figures are the same as that of Fig. 2.

From the top to the bottom, each panel corresponds to the

spectrum from the sample A, B, and C. Although we could

not directly observe side peaks from the spectrum of sample

A, the detailed analysis from our model shows that there is

still acoustic excitations in sample A, while in sample B and

sample C, our model cannot identify the phonon peaks. This

becomes more clear when we plot the decomposed modes

from the fittings in Fig. 5.


Fig. 4 shows the typical fittings from all three samples at QZ25.0 nmK1. All the symbols and figure arrangements are the

same as that of Fig. 3. In this figure, we show that the acoustic

excitation still exists at very large Q value, which correspond to

the second Brillouin zone in a crystal system. It thus implies that

a DNA molecule behaves like a one-dimensional crystal. The

phonon damping is very strong at this large Q value.

–40 –30 –20 –10

–40 –30 –20 –10

–40 –30 –20 –10

0

0.5

1

1.5

2

S(Q

,ω)/

S(Q

) (s

econ

d)

0

0.5

1

1.5

2

S(Q

,ω)/

S(Q

) (s

econ

d)

0

0.5

1

1.5

2x 10 –14

x 10 –14

x 10 –14

S(Q

,ω)/

S(Q

) (s

econ

d)

ω(

Q=16 nm–1

Q=16 nm–1

Q=16 nm–1

Fig. 3. The typical fittings of IXS energy spectra with the GTEE theory at QZ16.0

figures correspond to the results from the sample A, B, and C, respectively. The do

spectrum. The thin solid line is the fitted dynamic structure factor S(Q,u)/S(Q) m

convolving the thin solid line with the energy resolution function of the instrumen

In Fig. 5, we show the decomposed three modes calculated

from the fitted parameters from different samples and different

Q values.

It recently comes to our attention that the acoustic phonons

in water-stabilized DNA molecules have been measured by

INS (Grimm and co-workers) [9], and by using Brillouin

scattering (Maret and co-workers) [6].

0 10 20 30 40

0 10 20 30 40

0 10 20 30 40

meV)

Sample A

Sample B

Sample C

nmK1 are shown from three different samples. From the top to the bottom, the

ts with error bars are experimental results normalized by the total area of each

ultiplied by the detailed balance factor. The thick solid line is calculated by

t.


Grimm’s measurements covered the Q range from about

16–21 nmK1. Maret’s measurement focused on very small Q

values (10K2 nmK1). Here we would like to compare our

results of sample A with their measurements on wet samples,

which are hydrated by equilibrating to controlled humidity. We

expect that the results of their wet samples should be similar

with our sample A, which does not add additional MgCl2 salts.

Fig. 4. The typical fittings of IXS energy spectra at QZ25.0 nmK1 are shown from t

results from the sample A, B, and C, respectively. The dots with error bars are experi

is the fitted dynamic structure factor S(Q,u)/S(Q) multiplied by the detailed balance f

energy resolution function of the instrument.

Because Grimm’s data is not available for us, we directly

extracted the results from a figure of their published paper.

Both the results from our previous paper and those of Grimm’s

are summarized in Fig. 6. Open circles show the phonon

dispersion relation of sample A obtained by us, while the star

and cross symbols are Grimm’s results. The star symbols were

calculated by fitting the INS spectra with one-dimension liquid

hree different samples. From the top to the bottom, the figures correspond to the

mental results normalized by the total area of each spectrum. The thin solid line

actor. The thick solid line is calculated by convolving the thin solid line with the

–40 –20 0 20 400

1

2

3

4

5

6x 10–15

S(Q

,ω)/

S(Q

) (s

econ

d)

–40 –20 0 20 400

1

2

3

4

5

6

x 10–15

S(Q

,ω)/

S(Q

) (s

econ

d)

–40 –20 0 20 400

1

2

3

4

5

6x 10–15

S(Q

,ω)/

S(Q

) (s

econ

d)

ω(meV)

–40 –20 0 20 40

0

2

4

6

8

10

12x 10–15

–40 –20 0 20 40

0

2

4

6

8

10

x 10–15

–40 –20 0 20 40

0

2

4

6

8

10x 10–15

ω(meV)

–40 –20 0 20 400

0.5

1

1.5

2

2.5

3

3.5

x 10–14

–40 –20 0 20 400

0.5

1

1.5

2

2.5

x 10–14

–40 –20 0 20 400

0.5

1

1.5

2

2.5x 10–14

ω(meV)

Q=6.5nm–1 Q=16nm–1 Q=25nm–1

Sample A

Sample B

Sample C

Fig. 5. The decomposed three modes calculated from fitted results are shown. From the top to the bottom rows, the figures correspond to the sample A, sample B, and

sample C. From the left to the right columns, the figures correspond to QZ6.5 nmK1, 16.0 nmK1, and 25.0 nmK1.


picture model [12,24]. The cross symbols were calculated

results according to their model. The solid line is drawn to

guide eyes. Notice that in our curve, the phonon disappears at a

small range centered at QZ18.7 nmK1, where the Billouin

zone center lies if a DNA molecule is considered as a one-

dimensional crystal. This is because the energy resolution of

IXS spectra is not good enough so that small energy acoustic

excitations are masked by the broad resolution functions.

According to the results of Fig. 6, there are qualitative

agreement between these two experimental results.

To extract the sound speed, different models are applied

and give different results. The so-called ‘sound speed’ from

Grimm’s paper is about 2180 m/s. The sound speed we

obtain from the acoustic excitations at small Q values is

about 3100 m/s. Because of the small Q range in the

measurements of Grimm and co-workers due to the

limitation of the instrument, they extracted sound speed at

relatively large Q values. In our experiments, we could

access much lower Q values while keeping the large range

of energy scan. Therefore, we can directly extract sound

speed by assuming the linear relation between excitation

energy and Q values at very small Q. This method has been

widely used to identify the sound speed of water, liquid

metals, and molecular glasses [13,15,16].

Fig. 6. The phonon dispersion relation of sample A (open circles) is shown

together with the results from Ref. [9] (star and cross symbols). The star

symbols represent the results by fitting INS spectra with their model, while the

cross symbols are calculated results. The solid line is drawn to guide eyes.


The sound speed of wet DNA samples measured by Maret

and co-workers is 1800 m/s, while our results is about

3100 m/s. We attribute this difference to different properties

of collective motions of water molecules at different Q ranges.

0 5 10 1

0

5

10

15

0 5 100

50

100

150

200

250

S(Q

)

0 5 100

50

100

150

200

250

S(Q

)Ω

s (m

eV)

~2780m/s

Fig. 7. The top panel shows the phonon dispersion relations along the axial direction

the bottom panel show the calculated structure factors in absolute scale compared

The generalized dynamic structure factor S(Q,u) has the

contributions from all atoms. When the DNA samples are

hydrated with water, S(Q,u) consists of the partial dynamic

structure factor from both water molecules, DNA molecules,

and the cross terms, and should reflect the dynamics of this

mixed systems. It is now clear that the sound speed of pure

water is different at different Q-range. At very small Q value,

its sound speed is about 1500 m/s, while the sound speed

becomes 2900 m/s when QO2 nmK1 [14]. Therefore, the

increase of sound speed of DNA molecules should be attributed

to the increase of sound speed of water molecules. If we

consider the sound speed of a dry DNA molecule is Q

independent, the sound speed of dry DNA molecules is

3800 m/s for different Q ranges, which is always larger than the

sound speed of bulk water. When water molecules attach to the

DNA molecules during the hydration, they will affect each

other so that the overall sound speed is severely changed

compared with both bulk water and pure dry DNA. The

increase of sound speed of water at our Q range will then

increase the overall sound speed of the wet DNA molecules,

which is much larger than that observed by Maret et al at much

smaller Q values.

The top panel of Fig. 7 shows phonon dispersion relation of

40 wt% calf-thymus Na-DNA in 1.1 M MgCl2 (sample C)

together with the dispersion relation of 40 wt% calf-thymus Na-

5 20 25 30

15 20 25 30

15 20 25 30

Q(1/nm)

Na-DNA in 0.083M MgCl2

Na-DNA in 1.1M MgCl2

of DNA molecule with different MgCl2 concentrations. The middle panel and

with the measured structure factors up to a scale constant.

Fig. 9. This figure shows the Q-dependence of the ratio between the phonon

damping, G5, and the phonon frequency, U5 at different samples. The solid lines

are drawn to guide eyes.


DNA in 0.083 M MgCl2 (sample B). The open circles represent

the results of sample C, while the star symbols represent the

results of sample B. These two phonon dispersion relation

curves are almost the same. The sound speed of sample B is

about 2761 m/s and the sound speed of sample C is about

2197 m/s. In the top panel of Fig. 7, the sound speed is indicated

as the average of the sound speed of those of sample B and

sample C. Therefore, the sound speed is almost unchanged when

the concentration of MgCl2 increases from 0.083 to 1.1 M, while

the increase of MgCl2 concentrations from 0 to 0.083 M

decreases the sound speed from 3100 to 2761 m/s. Another

major difference compared to the phonon dispersion relation of

sample A is that there are a gap in the phonon dispersion relation

curve of sample B and C between 12 and 22.5 nmK1, where the

phonon disappears. We consider that this is due to the fact that

the addition of divalent counter-ions increases the phonon

damping. Therefore, phonons become over-dampened and

cannot be extracted from the spectra.

We can also calculate the absolute value of the structure

factor. Since fun is known from the fitted results, if we know

how to calculate v0(Q), we can directly calculate the structure

factor through Eq. (10). In order to calculate the generalized

thermal velocity v0(Q), the contributions from all atoms should

be taken into consideration. As an approximation, we just

simply assume that the major contributions of v0(Q) is mainly

from P atoms since it has the largest atomic number. The

calculated results with this method are shown in the middle

panel and bottom panel of Fig. 7, where the middle panel is the

results of sample B and the bottom panel is the results of

sample C. The solid lines are measured structure factors scaled

by a constant since they are in an arbitrary unit. The meaning of

different symbols is the same as that in the top panel. Although

the agreement is not very good, after considering the rough

approximation of v0, the calculated results agree with the

measured results fairly.

Fig. 8 shows the Q dependence of one of the fitted

parameters, zu. The open circle, star, and diamond symbols

Fig. 8. The Q-dependence of one fitted parameter, zu, is shown at different

conditions. The symbols are results from fitting IXS spectra with the GTEE

theory. The solid lines are the best fitting with zuZfQ2 for different samples.

represent the results of sample A, B, and C, respectively.

According to the hydrodynamic limit results, zuZfQ2, where

fZ(4/3hCz)/r is the longitudinal viscosity. h and z are the

shear and bulk viscosity, and r is the density. The solid lines

are best fitting with the function zuZfQ2. From the fitting, f is

0.11, 0.22, and 0.30 meV nm2 for sample A, B, and C,

respectively. It thus indicates that the increase of counter-ion

concentrations increases the viscosity.

Fig. 9 shows the ratio of phonon damping and the phonon

frequency as a function of Q for all three different samples. The

solid circle, star, and solid diamond symbols represent results

from sample A, B, and C, respectively. The solid lines are drawn

to guide eyes. It clearly demonstrates that the increase of MgCl2concentration increases the phonon damping a lot. This effect

can be linked to the increase of viscosity in the presence of

MgCl2 concentration. In the hydrodynamic limit,

Gs=UsZQð1=2fC1=2ðgK1ÞDT Þ=cs. Therefore, if cs does not

change too much, the increase of viscosity naturally leads to

larger damping results. The increase of stronger damping is thus

directly related to the increase of the longitudinal viscosity.

5. Conclusion

In this paper, we have reviewed the GTEE theory and

successfully apply it to analyze the IXS spectra of aligned

DNA samples. With the fitted parameters, we could calculate

the dynamic structure factor, phonon excitation energy, and

phonon damping. By extending the hydrodynamic limit

relation into higher Q range, we can also extract the sound

speed and the longitudinal viscosity information. The sound

speed of 40 wt% calf-thymus Na-DNA in water is about

3100 m/s. When adding 0.083 M MgCl2 into it, the sound

speed is changed to about 2780 m/s. However, the increase of

MgCl2 concentration from 0.083 to 1.1 M only change the

sound speed very slightly. During a Q range from 12 to

22.5 nmK1, the phonon disappears in sample B and C. We


attribute the reason to the addition of MgCl2 salt, which

increases the phonon damping and causes phonons to be over-

dampened. The damping effect due to different counter-ion

concentrations is also shown by plotting Q dependence of the

ratio between the phonon damping and the phonon excitation

energy. From the Q dependence of one fitted parameter, zu, we

also extract the longitudinal viscosity, which shows that the

viscosity increases when MgCl2 concentration becomes larger.

This is consistent with the result that the phonon damping is

stronger in the presence of higher MgCl2 concentrations.

Acknowledgements

Research at MIT is supported by the Basic Sciences

Division, Material Research Program of US DOE, DE-FG02-

90ER45429. Use of the Advanced Photon Source was

supported by the U. S. Department of Energy, Office of

Science, Office of Basic Energy Sciences, under Contract No.

W-31-109-Eng-38.

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