¹, Zagreb, Croatia ² School of Medicine, University of Zagreb, Croatia ³ Institut Ruđer...

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¹ , Zagreb, Croatia ² School of Medicine, University of Zagreb, Croatia ³ Institut Ruđer Bošković, Zagreb, Croatia 4 Dept. of Biological Chemistry, UCLA David Geffen School of Medicine, Los Angeles, USA T.Vuletić T.Vuletić ¹ ¹ , , S.Dolanski S.Dolanski Babi Babi ć ć ² ² , S.Tomić , S.Tomić ¹ ¹ ,D.Vurnek ,D.Vurnek ¹ ¹ , S.Krča , S.Krča ³ ³ , , D.Ivanković D.Ivanković ³ ³ , L.Griparić , L.Griparić 4 4 [email protected] [email protected] ; www.ifs.hr/real_science ; www.ifs.hr/real_science Dielectric Spectroscopy of Dielectric Spectroscopy of Genomic DNA Genomic DNA Solutions Solutions Lyophillized DNA: salmon testes, Lyophillized DNA: salmon testes, Sigma-Aldrich (D1626, Type III); Sigma-Aldrich (D1626, Type III); calf thymus, Rockland calf thymus, Rockland (MB –102-0100) (MB –102-0100) Pure water: MilliPore, Milli-Q, 0.056 Pure water: MilliPore, Milli-Q, 0.056 S/cm S/cm Range of DNA solutions: 0.011 – 18 mg/mL Range of DNA solutions: 0.011 – 18 mg/mL quantified spectro quantified spectro ph ph otom otom e e trically trically at 260 at 260 nm nm SAMPLES & MATERIALS Precision impedance analyzer Precision impedance analyzer Agilent 4294A: 40 Hz-100 MHz Agilent 4294A: 40 Hz-100 MHz C-G, capacitance C-G, capacitance & & real part real part of conductance measured of conductance measured amplitude 20-50 mV amplitude 20-50 mV Agilent BNCs Chamber for complex conductivity of liquid Chamber for complex conductivity of liquid samples – water solutions, conductivity samples – water solutions, conductivity range: 1.5-2000 range: 1.5-2000 S/cm; volume: 50-200 S/cm; volume: 50-200 L L Reproducibility 1%, Long term (2 h) 2% Reproducibility 1%, Long term (2 h) 2% Temperature Temperature control unit control unit Temp. range: Temp. range: 0 0 ° ° to 60 to 60 ° ° C C Stability: Stability: ± ± 10 mK 10 mK Pt chamber steel casing Pt Low-frequency Dielectric Spectroscopy CONDUCTIVITY of DNA-solution vs. temperature and concentration RESULT: Conductivity follows the Conductivity follows the power law with exponent smaller than power law with exponent smaller than 1 for both salmon-DNA and calf-DNA 1 for both salmon-DNA and calf-DNA p p ercolation character of ercolation character of the the conduction path in DNA solutions conduction path in DNA solutions Note: measured conductivity Note: measured conductivity (gray line) of (gray line) of DNA solution was subtracted for 1.5 DNA solution was subtracted for 1.5 S/cm - S/cm - the conductivity of pure water when measured the conductivity of pure water when measured in our chamber. in our chamber. (T) – ion mobility = e/6R(T) (T) – H 2 O viscosity ~ e - H/RT – ionic conductivity = N A e ~ e H/RT H=-18 kJ/mol (Source: CRC Handbook) ionic solutions theory: Note: H, enthalpy is related to energy required for a molecule to escape from its “neighbors” RESULT: DNA solution conductivity shows a temperature dependence typical for ionic solutions Resulting (G DNA -G NaCl , C DNA -C NaCl ) Complex dielectr ic function 1 0 1 1 i HF relaxation process strength = (0) - 0 – central relaxation time symmetric broadening of the relaxation time distribution 1 - 0 ' B S l 0 0 ' ' G G S l Note: C=B/ generalized Debye function FITS to a sum of two generalized Debye functions Worldwide motivation: Worldwide motivation: Transport of electrical Transport of electrical signals in bio-materials signals in bio-materials on a molecular scale on a molecular scale is of fundamental is of fundamental interest in the life interest in the life sciences sciences Our motivation: Our motivation: p p hysical and biological hysical and biological functions of DNA functions of DNA are are strongly affected by its strongly affected by its local environment local environment Our aim: to Our aim: to reveal reveal dynamical and dynamical and conformational properties conformational properties of native DNA of native DNA as a as a function of its aqueous function of its aqueous environment environment MOTIVATION Experimental characterization Experimental characterization of the counter-ion atmospheres of the counter-ion atmospheres around DNA in solution is around DNA in solution is essential for an understanding essential for an understanding of of DNA physical properties and DNA physical properties and biological functions biological functions Low frequency dielectric Low frequency dielectric spectroscopy (LFDS) to study spectroscopy (LFDS) to study genomic DNA genomic DNA as a function of as a function of electrolyte concentration, electrolyte concentration, counter-ion and pH counter-ion and pH LFDS: powerful tool to probe LFDS: powerful tool to probe charge entities and their charge entities and their background background structure in various structure in various bio-macromolecular structures bio-macromolecular structures R.Das et al.,Phys.Rev.Lett.90 , 188103 ( N.Nandi et al., Chem.Rev.100 , 2013 (2000) M. Sakamoto et al., Biopolymers 18 , 2769 (1979) S.Bone et al., Biochymica et Biophysica Acta 1306 , 93 (1996) • G() and C()=B()/ of DNA solutions are measured • These are subtracted for (G, C) of background (reference) NaCl solution with matching (1-100kHz) conductivity This procedure enables to eliminate the electrode polarization effects, as well as other stray impedance effects. That is, since these influences are nearly the same in DNA and reference solutions, they are reduced by the subtraction. = ’()-i’’() Y()= G()+iB() From complex conductance to complex dielectric function B.Saif et al., Biopolymers 31 , 1171 (1991) DNA IN SOLUTION Coulomb repulsion between PO Coulomb repulsion between PO 4 - groups groups , , DNA DNA is stretched out is stretched out to the rod-like conformation to the rod-like conformation Worm-like Worm-like m m odel odel : c : c hain of N segments of hain of N segments of length a; length a; Contour length L = N · a Contour length L = N · a Rigid over short distance and Rigid over short distance and becomes flexible becomes flexible over large over large distances distances Persistance length L Persistance length L p determines a determines a boundary boundary between the two types of between the two types of b b ehavior ehavior i i n 0.1 M NaCl n 0.1 M NaCl ; ; L L p = = 50 nm : 150 bp length 50 nm : 150 bp length 200 nm une, Molecular Biophysics (Oxford, 2003) Kratky and Porod (1949) Kuhn Results: Two Relaxation Modes in 10 kHz – 10 MHz range HF mode: 10, 1- 0.8 Same features for both salmon and calf DNA LF mode: 100, 1- 0.8 calf DNA salmon DNA : c-independent strong drop at low c 0 : no change at low c levels off at low c ? Electro-kinetics of Electrical Double Layer S.S.Dukhin et al, Adv.Coll. Interface Sci. 13 , 153 (1980) R.W.O’Brian, J. Coll. Interface Sci 113 , 81 (1986). Na+ ions redistributed in the vicinity of DNA Na+ ions redistributed in the vicinity of DNA chain in order to screen chain in order to screen phosphate groups phosphate groups Electrical double layer with thickness Electrical double layer with thickness -1 -1 is is created created Suggestion: Suggestion: Under applied ac field Under applied ac field two types of two types of dielectric dispersion dielectric dispersion two characteristic length two characteristic length scales: scales: -1 -1 - Debye-H - Debye-H ü ü ckel length & contour length of ckel length & contour length of molecule molecule ? ? ? ? ? L L HF,LF HF,LF = ( = ( HF.LF HF.LF D) D) 1/2 1/2 , , from experiments from experiments D=k D=k B T/6 T/6 R, R, D(25°C) = 1.5 ·10 D(25°C) = 1.5 ·10 -9 -9 m m 2 /s /s L L HF HF : : 4 nm – 45 nm 4 nm – 45 nm D D H screening length H screening length ? ? or DNA or DNA mesh size mesh size ? ? L L LF : : 60 60 nm – nm – 750 750 nm nm Persistence Persistence length length ? ? (Source: CRC Handbook) Temperature dependent relaxation M M ode characteristic length ode characteristic length s s appear appear temperature independent temperature independent Since characteristic length L=(D(T)· Since characteristic length L=(D(T)· (T)) (T)) 1/2 1/2 Thus, Thus, (T)~ 1/D(T) ~ 1/(T (T)~ 1/D(T) ~ 1/(T · · e e - H/RT H/RT ) ) Therefore, Therefore, should be FIT to e should be FIT to e H/RT H/RT /T /T Energy scale Energy scale s s of of the the mode mode s s are are quite similar t quite similar t o o energy scale of ionic conductivity energy scale of ionic conductivity : : H= -18 kJ/mol H= -18 kJ/mol H= - H= - 20 20±2 kJ/mol kJ/mol Conclusion Conclusion - - Origin of dielectric dispersion Origin of dielectric dispersion in DNA solutions in DNA solutions DNA chain: DNA chain: Random sequence of Random sequence of segments placed in segments placed in counter-ion counter-ion atmosphere atmosphere . With . With ac field ac field applied, appear applied, appear broad broad relaxation relaxation modes due to oscillating modes due to oscillating counter-ions at different length counter-ions at different length and time scales and time scales Modes: Modes: 1) Contour length; f 1) Contour length; f 0 < 1 kHz < 1 kHz M. Sakamoto et al., Biopolymers 18 , 2769 (1979) S.Takashima, J.Phys.Chem.70 , 1372 (1966) L -1 Na + , Cl - L p L HF - - - - - - - - - 2) LF mode: 2) LF mode: 1 kHz < f 1 kHz < f 0 < 70 < 70 kHz kHz Persistence Persistence length: length: distance distance bound by potential bound by potential barriers due to barriers due to variation of local variation of local conformation conformation As expected L As expected L p ~ ~ I I -1/2 -1/2 when salt is added when salt is added 3) HF mode: 3) HF mode: 0.1 kHz < f 0.1 kHz < f 0 < 15 < 15 MHz MHz Mesh size: Mesh size: DNA DNA chains form a loose mesh chains form a loose mesh defining a characteristic defining a characteristic length for relaxation– length for relaxation– attribution is strongly attribution is strongly supported by L supported by L HF HF independence independence of added salt I. of added salt I. L L HF HF ~ ~ c c 1/2-1/3 1/2-1/3 , indicates , indicates dimensionality of the web dimensionality of the web between 2 & 3. between 2 & 3. HF Mode Characteristic Length: DNA mesh size Inherent Inherent (I (I NaCl NaCl =0) =0) Na Na + + ions only: ions only: /c /c ~ ~ L L HF HF 2 in accord with in accord with Mandel-Manning Mandel-Manning model model Added salt ions (I Added salt ions (I NaCl NaCl ≠0) do not ≠0) do not contribute to relaxation. contribute to relaxation. On the contrary, they increase On the contrary, they increase screening and strongly reduce Na screening and strongly reduce Na + ions ions active in HF relaxation active in HF relaxation L L HF HF is DNA concentration dependent, but added salt is DNA concentration dependent, but added salt independent independent L L HF HF can not be can not be -1 -1 ~ ~ I I -1/2 -1/2 , Debye-H , Debye-H ü ü ckel length ckel length L L HF HF given by mesh size given by mesh size , , ie. ie. average distance average distance between DNA chains in solution (this length scale between DNA chains in solution (this length scale does not vary with added salt, I does not vary with added salt, I NaCl NaCl ≠0) ≠0) M.N. Spiteri et al., Phys.Rev.Lett.77 , 5218 (1996) LF Mode Characteristic Length: Persistence Length Similar effect of inherent and added Similar effect of inherent and added Na+ ions Na+ ions All ions contribute to screening All ions contribute to screening L L LF LF ~ ~ I I -1/2 -1/2 implying L implying L LF LF ~ ~ -1 -1 as expected for as expected for persistence length persistence length Important Important difference in L difference in L p of salmon and calf DNA of salmon and calf DNA at low concentration at low concentration Certainly Certainly LF LF ~ ~ L L 2 and we found and we found /c /c ~ ~ L L LF LF 2 Both relaxation parameters should be Both relaxation parameters should be proportional to characteristic length, L proportional to characteristic length, L 2 according to : according to : M.Mandel, Ann.NY Acad.Sci. 303 , 74 (1977) G.S.Manning, Biophys.Chem. 9 , 65 (1978) P.G.de Gennes et al.,J.Phys. (Paris), 37 , 1461 (1976) M.N. Spiteri et al., Phys.Rev.Lett.77 , 5218 (1996)

Transcript of ¹, Zagreb, Croatia ² School of Medicine, University of Zagreb, Croatia ³ Institut Ruđer...

Page 1: ¹, Zagreb, Croatia ² School of Medicine, University of Zagreb, Croatia ³ Institut Ruđer Bošković, Zagreb, Croatia 4 Dept. of Biological Chemistry, UCLA.

¹ , Zagreb, Croatia² School of Medicine, University of Zagreb, Croatia³ Institut Ruđer Bošković, Zagreb, Croatia4 Dept. of Biological Chemistry, UCLA David Geffen School of Medicine, Los Angeles, USA

T.Vuletić T.Vuletić ¹¹,, S.Dolanski Babi S.Dolanski Babićć ²², , S.Tomić S.Tomić ¹¹,D.Vurnek ,D.Vurnek ¹¹, S.Krča, S.Krča³³, , D.IvankovićD.Ivanković³³, L.Griparić, L.Griparić44

[email protected]@ifs.hr ; www.ifs.hr/real_science ; www.ifs.hr/real_science

Dielectric Spectroscopy ofDielectric Spectroscopy of Genomic DNA SolutionsGenomic DNA Solutions

Lyophillized DNA: salmon testes, Lyophillized DNA: salmon testes, Sigma-Aldrich (D1626, Type III); Sigma-Aldrich (D1626, Type III); calf thymus, Rockland (MB –102-0100) calf thymus, Rockland (MB –102-0100)Pure water: MilliPore, Milli-Q, 0.056 Pure water: MilliPore, Milli-Q, 0.056 S/cmS/cm

Range of DNA solutions: 0.011 – 18 mg/mLRange of DNA solutions: 0.011 – 18 mg/mLquantified spectroquantified spectrophphotomotomeetrically trically at 260 at 260 nmnm

SAMPLES & MATERIALS

Precision impedance analyzerPrecision impedance analyzer

Agilent 4294A: 40 Hz-100 MHzAgilent 4294A: 40 Hz-100 MHz

C-G, capacitance C-G, capacitance && real part real part of conductance measured of conductance measured amplitude 20-50 mVamplitude 20-50 mV

AgilentBNCs

Chamber for complex conductivity of liquid Chamber for complex conductivity of liquid samples – water solutions, conductivity samples – water solutions, conductivity range: 1.5-2000range: 1.5-2000S/cm; volume: 50-200 S/cm; volume: 50-200 LL

Reproducibility 1%, Long term (2 h) 2%Reproducibility 1%, Long term (2 h) 2%

Temperature Temperature

control unitcontrol unit

Temp. range: Temp. range:

00°° to 60 to 60°°CC

Stability: Stability:

±±10 mK10 mK

Pt

cham

ber

steelcasing

Pt

Low-frequency Dielectric Spectroscopy

CONDUCTIVITY of DNA-solution vs. temperature and concentrationRESULT: Conductivity follows the power Conductivity follows the power

law with exponent smaller than 1 for law with exponent smaller than 1 for both salmon-DNA and calf-DNAboth salmon-DNA and calf-DNA p percolation character ofercolation character of the the conduction path in DNA solutionsconduction path in DNA solutionsNote: measured conductivity Note: measured conductivity (gray line) of DNA (gray line) of DNA solution was subtracted for 1.5 solution was subtracted for 1.5 S/cm - the S/cm - the conductivity of pure water when measured in conductivity of pure water when measured in our chamber.our chamber.

(T) – ion mobility = e/6R(T)

(T) – H2O viscosity ~ e-H/RT

– ionic conductivity = NAe~ eH/RT

H=-18 kJ/mol (Source: CRC Handbook)

ionic solutions theory:

Note: H, enthalpy is related to energy required for a molecule to escape from its “neighbors”

RESULT: DNA solution conductivity shows a temperature dependence typical for ionic solutions

Resulting (GDNA-GNaCl, CDNA-CNaCl)

Complex dielectric function

1

01

1

iHF∞

relaxation process strength = (0) - ∞

0 – central relaxation time

symmetric broadening of the relaxation time distribution 1 -

0

'B

S

l

0

0''GG

S

l

Note: C=B/

generalized Debye function

FITS to a sum of two generalized Debye functions

Worldwide motivation: Worldwide motivation: Transport of electrical Transport of electrical signals in bio-materials on signals in bio-materials on a molecular scale a molecular scale is of fundamental interest is of fundamental interest in the life sciencesin the life sciences

Our motivation: pOur motivation: physical hysical and biological functions of and biological functions of DNADNA are strongly affected are strongly affected by its local environmentby its local environment

Our aim: to Our aim: to revealreveal dynamical and dynamical and conformational propertiesconformational propertiesof native DNAof native DNA as a function as a function of its aqueous of its aqueous environmentenvironment

MOTIVATION Experimental characterization Experimental characterization of the counter-ion atmospheres of the counter-ion atmospheres around DNA in solution is around DNA in solution is essential for an understanding of essential for an understanding of DNA physical properties and DNA physical properties and biological functionsbiological functions

Low frequency dielectric Low frequency dielectric spectroscopy (LFDS) to study spectroscopy (LFDS) to study genomic DNAgenomic DNA as a function of as a function of electrolyte concentration, electrolyte concentration, counter-ion and pH counter-ion and pH

LFDS: powerful tool to probe LFDS: powerful tool to probe charge entities and their charge entities and their backgroundbackground structure in various structure in various bio-macromolecular structures bio-macromolecular structures

R.Das et al.,Phys.Rev.Lett.90, 188103 (2003)

N.Nandi et al., Chem.Rev.100, 2013 (2000)M. Sakamoto et al., Biopolymers 18, 2769 (1979)S.Bone et al., Biochymica et Biophysica Acta 1306, 93 (1996)

• G() and C()=B()/ of DNA solutions are measured• These are subtracted for (G, C) of background (reference) NaCl solution with matching (1-100kHz) conductivity This procedure enables to eliminate the electrode polarization effects, as well as other stray impedance effects. That is, since these influences are nearly the same in DNA and reference solutions, they are reduced by the subtraction.

=’()-i’’()Y()= G()+iB()

From complex conductance to complex dielectric function

B.Saif et al., Biopolymers 31, 1171 (1991)

DNA IN SOLUTION Coulomb repulsion between POCoulomb repulsion between PO44

-- groups groups, , DNA DNA is stretched outis stretched out to the rod-like conformationto the rod-like conformation Worm-like Worm-like mmodelodel: c: chain of N segments of hain of N segments of length a;length a; Contour length L = N · aContour length L = N · a

Rigid over short distance and Rigid over short distance and becomes flexiblebecomes flexible over largeover large distancesdistances Persistance length L Persistance length Lpp determines a determines a boundaryboundary between the two types of between the two types of bbehaviorehavior iin 0.1 M NaCln 0.1 M NaCl; ; LLpp = = 50 nm : 150 bp length50 nm : 150 bp length

200 nm

M. Daune, Molecular Biophysics (Oxford, 2003)

Kratky and Porod (1949) Kuhn

Results: Two Relaxation Modes in 10 kHz – 10 MHz range

HF mode: 10, 1- 0.8Same features for both salmon and calf DNA

LF mode: 100, 1- 0.8 calf DNA salmon DNA

: c-independent strong drop at low c0 : no change at low c levels off at low c

?

Electro-kinetics of Electrical Double Layer

S.S.Dukhin et al, Adv.Coll. Interface Sci. 13, 153 (1980)

R.W.O’Brian, J. Coll. Interface Sci 113, 81 (1986).

Na+ ions redistributed in the vicinity of DNA chain in Na+ ions redistributed in the vicinity of DNA chain in order to screenorder to screen phosphate groupsphosphate groups Electrical double layer with thickness Electrical double layer with thickness -1-1 is created is created Suggestion: Suggestion: Under applied ac fieldUnder applied ac field two types of two types of dielectric dispersion dielectric dispersion two characteristic length two characteristic length scales: scales: -1-1 - Debye-H- Debye-Hüückel length & contour length of ckel length & contour length of moleculemolecule

????? LLHF,LFHF,LF= (= (HF.LFHF.LFD)D)1/21/2 , , from experiments from experiments D=kD=kBBT/6T/6R,R, D(25°C) = 1.5 ·10D(25°C) = 1.5 ·10-9 -9 mm22/s/s

LLHFHF:: 4 nm – 45 nm 4 nm – 45 nmDDH screening lengthH screening length?? or DNA mesh size or DNA mesh size??

LLLLFF:: 6060 nm – nm – 750750 nm nm Persistence Persistence lengthlength??

(Source: CRC Handbook)

Temperature dependent relaxation

MMode characteristic lengthode characteristic lengthss appearappear temperature independent temperature independent Since characteristic length L=(D(T)·Since characteristic length L=(D(T)·(T))(T))1/21/2

Thus, Thus, (T)~ 1/D(T) ~ 1/(T(T)~ 1/D(T) ~ 1/(T··ee--H/RTH/RT)) Therefore, Therefore, should be FIT to e should be FIT to eH/RTH/RT/T/T

Energy scaleEnergy scaless of of thethe mode modess areare quite similar tquite similar to energy o energy scale of ionic conductivityscale of ionic conductivity:: H= -18 kJ/molH= -18 kJ/mol

H= -H= -2020±±22 kJ/mol kJ/mol

ConclusionConclusion - - Origin of dielectric dispersion in DNA solutionsOrigin of dielectric dispersion in DNA solutionsDNA chain:DNA chain: Random sequence of Random sequence of segments placed insegments placed in counter-ion counter-ion atmosphereatmosphere. With. With ac field ac field applied, applied, appearappear broad broad relaxation modes relaxation modes due to oscillating counter-ions at due to oscillating counter-ions at different length and time scalesdifferent length and time scales

Modes:Modes:1) Contour length; f1) Contour length; f00 < 1 kHz < 1 kHz

M. Sakamoto et al., Biopolymers 18, 2769 (1979)S.Takashima, J.Phys.Chem.70, 1372 (1966)

L

-1

Na+, Cl-

Lp

LHF

--

---

- - -

-

2) LF mode: 2) LF mode: 1 kHz < f1 kHz < f00 < 70 kHz < 70 kHz Persistence length: Persistence length: distance bound by distance bound by potential barriers potential barriers due to variation of due to variation of local conformationlocal conformationAs expected LAs expected Lpp ~~ I I-1/2-1/2 when salt is addedwhen salt is added

3) HF mode: 3) HF mode: 0.1 kHz < f0.1 kHz < f00 < 15 MHz < 15 MHz Mesh size:Mesh size: DNA chains form a DNA chains form a loose mesh defining a loose mesh defining a characteristic length for characteristic length for relaxation– attribution is relaxation– attribution is strongly supported by Lstrongly supported by LHFHF independence of added salt I.independence of added salt I.LLHFHF ~~ c c1/2-1/31/2-1/3, indicates , indicates dimensionality of the web dimensionality of the web between 2 & 3.between 2 & 3.

HF Mode Characteristic Length: DNA mesh size

Inherent Inherent (I(INaClNaCl=0) Na=0) Na+ +

ions only:ions only: /c /c ~~ L LHFHF

22 in accord within accord withMandel-Mandel-Manning Manning modelmodel

Added salt ions (IAdded salt ions (INaClNaCl≠0) do not ≠0) do not contribute to relaxation. contribute to relaxation. On the contrary, they increase On the contrary, they increase screening and strongly reduce Nascreening and strongly reduce Na++ ions ions active in HF relaxationactive in HF relaxation

LLHF HF is DNA concentration dependent, but added salt is DNA concentration dependent, but added salt independent independent LLHFHF can not be can not be -1 -1 ~~ I I-1/2 -1/2 , Debye-H, Debye-Hüückel length ckel length LLHFHF given by mesh sizegiven by mesh size, , ie.ie. average distance average distance between DNA chains in solution (this length scale between DNA chains in solution (this length scale does not vary with added salt, Idoes not vary with added salt, INaClNaCl≠0)≠0)M.N. Spiteri et al., Phys.Rev.Lett.77, 5218 (1996)

LF Mode Characteristic Length: Persistence Length

Similar effect of inherent and added Similar effect of inherent and added Na+ ionsNa+ ions All ions contribute to screeningAll ions contribute to screening LLLFLF ~~ I I-1/2-1/2 implying L implying LLFLF ~~ -1-1 as expected for as expected for persistence lengthpersistence length

Important Important difference in Ldifference in Lpp of salmon and calf DNA of salmon and calf DNA at low concentration at low concentration

Certainly Certainly LFLF ~~ L L22 and we found and we found /c /c ~~ L LLFLF22

Both relaxation parameters should be Both relaxation parameters should be proportional to characteristic length, Lproportional to characteristic length, L22 according to :according to : M.Mandel, Ann.NY Acad.Sci. 303,

74 (1977)G.S.Manning, Biophys.Chem. 9,

65 (1978)

P.G.de Gennes et al.,J.Phys.(Paris), 37, 1461 (1976)M.N. Spiteri et al., Phys.Rev.Lett.77, 5218 (1996)