Cargèse, 2002 - 1
DNA AS A POLYELECTROLYTE
Eric Raspaud
Laboratoire de Physique des Solides
Bât. 510,
Université Paris –Sud
91405 Orsay, France
Email : raspaud@lps.u-psud.fr
Cargèse, 2002 - 2
Unlike proteins, each of the DNA
basic units (the base pair) is
composed of two identical ionisable
groups, the phosphate groups.
Projection of the two phosphates on the double-helix axis :
(B-DNA)
Linear charge density :
equivalently,
2 e / 3.4 Å ~ 6 e / nm
1 e / 1.7 Å
One of the most highly charged systems
Cargèse, 2002 - 3
For proteins,
5 / 20 basic units may be ionized :
basic : lysine, arginine, histidine (-NH2+)
acidic : aspartic, glutamic (-COO-)
Amino-acid size ~ 4 Å
Typical Linear Charge density ~ 1 /( 4  4 Å) ~ 0.6 e / nm
For the Cell membrane,
Typical Surface Charge Density ~ 0.1- 1 e / nm2
Cargèse, 2002 - 4
Proteins + (RNA or DNA) complexes :
Nucleosome Core Particle
Tobacco Mosaic Virus
105 Å
Linear Charge density
TMV ~ 2 e / nm
fd-virus ~ 5 e / nm
60 Å
Surface Charge Density ~ 6 e / nm2
Cargèse, 2002 - 5
(Bloomfield, 1999)
Cargèse, 2002 - 6
3
Menu
1_ Brief Introduction on the Ionic dissociation and hydration
1_a) simple salt
1_b) DNA
2_ Simple salts : the Debye-Hückel model
2_a) spatial distribution
2_b) thermodynamic consequences
3_ Polyelectrolyte and the counterions distribution
3_a) the Poisson-Boltzmann equation
3_b) the Manning model
4_ Thermodynamic coefficients and the « free » counterions
4_a) the radius effect
4_b) the length effect
5_ Conformation and Interaction
5_a) Thermal denaturation
5_b) Interaction and chain conformation
6_ Multivalent Counterions
6_a) collapse and phase diagram
6_b) ionic correlation
Cargèse, 2002 - 7
1_ Brief Introduction on the Ionic dissociation and hydration
Cargèse, 2002 - 8
1_a) simple salt
Coulomb’s law (1780)
Thermal Energy,
kT  10-21 Joules (0.6 kcal/mol)
Attraction
force
between
two
elemantary charges e separated by a
distance d
F   e  12
 40  d
2
Roughly
W  kT
and its associated energy :
W   e  1
 40  d
2
In water,   80
For a typical ionic crystal,
W 1018Joules
The salt dissociates
in water
Cargèse, 2002 - 9
In general, one defines a characteristic length
of the ions dissociation/association :
The Bjerrum length lB
kT 
2
e
4 0l B
In water,
lB = 7.14 Å
if d < lB then
ion pairing occurs
For a multivalent salt
zi : zj
|zizj | lB 
Cargèse, 2002 - 10
Similar notion
for the couple
Ion - charged monomer
d
Site – bound :
ion immobilized
and dehydrated
pKa and weak electrolytes
Ex. : acrylate group - COO- and Mg2+
(Oosawa, 1971; Bloomfield, 2000)
strong electrolytes
Ex. : sulfonate group – SO3(Sabbagh & Delsanti, 2000)
Cargèse, 2002 - 11
1_b) DNA
NMR relaxation :
Na+ associated with DNA
are not dehydrated or
immobilized
18 – 23 water
molecules per
nucleotide
(from IR
Spectroscopy)
(Bleam et al., 1980)
Lost of Mn2+ hydration
upon binding to DNA
(Granot & Kearns, 1982)
(Bloomfield, 2000)
For the phosphate group,
For the bases,
Neutral at pH 7
Cargèse, 2002 - 12
Cargèse, 2002 - 13
2_ Simple salts : the Debye-Hückel model (1923)
Solution of strong electrolytes (McQuarrie, 1976) :
Positive and negative charges in a continuum
medium of dielectric constant 0
rj
rij
 i(ri)  
The Coulombic potential acting
upon the ith ion located at ri
ri
or at the arbritary point r :
qj
ji
4   0rij
 (r )   (r')dr'
4  0 rr'
with  (r) the total charge density at r
How does the charges distribution (r) change with respect to
the distance r from a central ion ?
Cargèse, 2002 - 14
2_a) spatial distribution
r
- Ionic size : s
- cations and anions charges : z+e and z- e
- concentration of the bulk solution : C+0, C-0
z + e C +0 + z - e C - 0 = 0
(Electroneutrality)
(r) is related to the electrostatic potential (r) by the
Poisson’s law and by the Boltzmann’s distribution :
D(r) = -(r )/0
 (r ) = z+e C+0 e-z+e(r)/kT + z-e C-0 e-z-e(r)/kT
with Fi = zie(r) the energy of the charge zie located at the distance r from the central ion
or equivalently the electrostatic work required to bring the charge from the reference state
to the distance r.
Solving the Poisson-Boltzmann equation :
zie(r) << kT
Cargèse, 2002 - 15
(kT/e  25mV)
Expanding the exponential, one gets
the linearized Poisson-Boltzmann or the Debye-Hückel equation :
Df (r) = k2f(r)
with
f(r)= e(r) / kT
k2 = 4 lB (z-2C-0 + z+2C+0 )
1/k : the Debye screening length
Boundary conditions :
r → , f → 0
r = s, E = - df /dr = - z0 lB /s2
(from the Gauss theorem with
z0 the valence of the central ion)
f(r) = A e-kr/ r
a screened coulombic potential
with A = (z0 lB eks)/(1+ks)
Cargèse, 2002 - 16
for a 1:1 salt (NaCl,…)
Csalt
(mM)
1/k (Å)
1
100
10
30
100
10
103
3
s=5Å
f(r)
1/k s
1/k s = 6
1
3
5
7
9
r/s
4
r
Charge
Fraction
of charge
distribution
within
a sphere of
r
radius
r and
(r)
concentric to the
z0= -1 central ion
3
opposite sign
monotonic
increase
2
1
0
1
1
same sign
3
3
5
5
7
7
9
9
« Ionic atmosphere » around the central ion
2_b) thermodynamic consequences
Cargèse, 2002 - 17
Knowing the ion spatial distribution and the electrostatic potential and charging
the ions, one can calculate the electrostatic free energies of the system :
electrostatic energy
Felec / V kT= - (k3/12) t(k s)
Felec < 0 due to the ionic atmosphere
Predict then the thermodynamic coefficients
3_ Polyelectrolyte and the counterions distribution
Cargèse, 2002 - 18
A polyelectrolyte
into an ionic solution
Simple salt
r
r
Apply
similar
calculation ?
-Yes, we can start from the Poisson-Boltzmann equation
- No, the approximation zie(r) << kT is no more valid
Cargèse, 2002 - 19
3_a) the Poisson-Boltzmann equation
The central ion becomes the polyelectrolyte :
(r) its electrostatic potential at a distance r
 (r) the charge density of the surrounded ions
D(r) ~ (r)
 (r) ~ e-ze(r)/kT
D(r) ~ e-ze(r) /kT
In cylindrical geometry,
D(r) = (1/r)  (r  /  r) /  r
a non-linear differential equation
Cargèse, 2002 - 20
 a exact solution if there is no added salt :
(Fuoss et al., 1951); (Zimm & Le Bret, 1983); (Deserno et al., 2000)
Neglected :
* the finite size and spatial correlations of the small ions
* the non-uniformity of the dielectric constant (water)
The chain :
an infinitely long
cylinder of radius r0
The monomeric unit :
Size b and valence zm
The linear charge density : zme/b
For B – DNA,
r0 = 10 Å
2r0
a dinucleotide
b = 3.4 Å and zm = -2
-2e /3.4
Or equivalently :
-1e /1.7
with b = 1.7 Å and zm = -1
Cargèse, 2002 - 21
Number of charges per Bjerrum length lB :
xM = lB / b the Manning parameter
For B – DNA, b = 1.7 Å :
xM = 4.2
Without added salt :
Only the counterions are present
b
zc e Cctotal + zm e Cmtotal = 0
2r0
Counterions
Monomers
zc = + 1
monovalent
(Na+, K+,…)
zm = - 1
a nucleotide
(Phosphate-)
Counterions distribution ?
The Cell model : each chain enclosed in a coaxial cylindric cell
Cargèse, 2002 - 22
- ionic distribution :
the whole solution
=
a single cylindric cell
- concentration effect :
DNA dilution
=
R increase
2r0
2R
Cmtotal = Cctotal =
1/bR2
Cargèse, 2002 - 23
Returning to the Poisson-Boltzmann equation,
D(r) = - (r) /0
with
(r) the counterions charge density at a
radial distance r from the
chain
(r) = zce C(R) e-zce (r) /kT
Assumption : (r = R)=0
Df(r)= k2 ef(r)
with f(r)= - zce(r) / kT
The screening length 1/k becomes :
k2 = 4lB zc2C(R)
[ for the simple salt, k2 = 4 lB (z-2C-0 + z+2C+0 ) ]
0
r
r0
R
Cargèse, 2002 - 24
Boundary conditions :
r0 = 10 Å
R = 50 Å (40 g/l DNA)
1/ k  25 Å
a whole cell is neutral,
for r = R
f(r)/r = 0
the chain is charged with xM = lB / b = 4.2,
for r = r0 f(r)/r = - 2 xM /r0
xM =
5
4.2
4
f(r)
3
f(r) = -2 ln( [kr /g2] cos (g ln[r/RM])) !!!
with g and RM two numerical constants
dependent on xM , r0 , R
1.5
2
0.8
1
r
0
0
10
r0
20
30
40
50
R
Cargèse, 2002 - 25
Fraction of counterions within a cylinder of radius r
R = 50 Å
1
xM =
0.5
4.2
1.5
Counterions
accumulation
0.8
0
1
2
3
4
5
r/r0
0
r0
R
r
Monotonic increase for xM < 1
Cargèse, 2002 - 26
R = 250 Å
1
(1.7 g/l DNA)
similar variation
xM =
4.2
0.5
Similar to simple salt;
Debye – Hückel or linearised
Poisson-Boltzmann equation
1.5
ze(r) << kT or f(r) <<1
Counterions
accumulation
0.8
0
1
2
5
10
20
r/r0
Cargèse, 2002 - 27
1
Using the linearised equation,
Df(r)= k2f(r)
f(r)= - [2 xM / kr0 K1(kr0) ] K0(kr)
0.5
with f(kr <0.5) ln(kr)
0.8
0
1
2
5
10
xM = 0.8
20
For xM < 1, the electrostatic potential (r) is
lower than kT and no counterions accumulation
or condensation occurs. Counterions may be
treated like simple salts.
For DNA xM= 4.2 > 1, a strong counterion
condensation occurs near the chain; only the
counterions far from the chain are submitted to a
low electrostatic potential and may be treated
like simple salts.
1
10
Cargèse, 2002 - 28
Monte-Carlo simulation (Le Bret & Zimm, 1984) :
C (r) (M)
- regular spacing of
the phosphate charge
on a cylinder
local concentration
of countreions
-charges set along the
double-helix
0
r0
r
R
Experimentally :
X-Ray scattering (Chang et al., 1990)
Heavy counterions (Thallium+)
Cargèse, 2002 - 29
Neutron scattering (van der Maarel et al., 1992)
Contrast matching
3_b) the Manning model and the limiting laws
(Manning, 1969; 1978)
Cargèse, 2002 - 30
-An infinite line charge with a charge density xM
- Simple salt is present in excess over polyelectrolyte
Two-states model for xM > 1:
« free » ions treated in the
Debye-Hückel approximation
« condensed » ions reducing the monomer charge
to an effective charge qeff.
Cargèse, 2002 - 31
Calculation of the effective charge qeff. :
-The condensed ions decrease the repulsions between
the chain charges :
gel. = (q eff. 2 /40)  e-krij/ rij
By summing over all pairs of effective charges of the
chain, the electrostatic contribution to the free energy per
monomer becomes for kb << 1:
gel. = - (q eff. 2 /40b) ln kb
Set q the number of condensed ions per monomer,
the effective charge per monomer becomes:
q eff. = zme + zce q = zme [1+ (zc/zm)q]
with zczm<0
gel. / kT = - zm2 [1+ (zc/zm)q]2 xM ln kb
Cargèse, 2002 - 32
- The mixing entropy decreases the number of
condensed ions :
Concentration of the free salt :
Cs
As q ions are confined in Vp,
the lost of mixing energy is
gmix../ kT = q ln (Clocal/Cs)
Volume of the region where ions are said to be
condensed : Vp
The ionic local concentration : Clocal = 103 q /Vp
Cargèse, 2002 - 33
Minimization of the free energy g el. + g mix. with respect to q :
1 - ln(Vp Cs/103) - zczm [1+ (zc/zm)q ] xM ln (kb)2 =0
Focus on the salt concentration Cs dependence with k2 ~ Cs
- ln Cs ~ zczm [1+ (zc/zm)q ] xM ln Cs
The only solution to cancel the Cs dependence (in particular when Cs
0) :
-1 = zczm [1+ (zc/zm)q ] xM
For  zm  =  zc  = 1
q = 1- 1/ xM
radial extension of the
condensation region
b
(Å)
B-DNA
Singlestranded
DNA
1.7
4
xM
4.2
1.8
salt
zc : 1
zcq
Clocal
(M)
Cargèse, 2002 - 34
Dx
(Å)
1
0.76
1.17
7.4
2
0.88
0.39
11.2
4
0.94
0.12
16.9
1
0.44
0.21
12.5
4
0.86
0.021
32.2
76 % of the phosphates
have a condensed counterion
or the effective charge is
reduced by 1/4
For a long DNA chain (of size L) in a monovalent salt,
its structural charge is proportional to L/b
and
its effective charge to (L/b) (1-q) = (L/b)/xM = L/lB !
The DNA chain behaves like a chain where the distance between
charged monomers is equal to lB
Cargèse, 2002 - 35
4_ Thermodynamic coefficients and the « free » counterions
« free » in the Manning sense :
For xM < 1, all the counterions are « free »
For xM > 1, only 1/ xM counterions are « free »
But they are submitted to the electrostatic potential created by the DNA chains;
Similar to the ionic atmosphere of the Debye-Hückel interations for simple salts
 thermodynamically « free » : where ions are only submitted to the thermal agitation
Cargèse, 2002 - 36
- For the Poisson-Boltzmann cell model without added salt,
C(R)
Only the counterions far from the chain
or located at r = R
are thermodynamically « free »
(r = R) = 0
- For DNA in excess of salt, only the ions far
from the chain are thermodynamically « free »
For xM > 1 and the limiting law,
Only 1/ 2xM counterions are thermodynamically « free »
Cargèse, 2002 - 37
The Donnan effect :
Semi-permeable membrane
At the thermodynamical equilibrium,
the ions chemical potentials
from both sides are equal :
m(Cc Na+ , Cs ) = m( Cs0 )
reservoir
Na-DNA
+
Salt Cs
Salt Cs0
(NaCl)
Cs = C ClCs0 = C Na+0 = C Cl-0
C Na+ = Cs + Cc Na+
Counterions
coming from Na-DNA
Cargèse, 2002 - 38
4_a) the radius effect
In an excess of salt (NaCl) with diluted Na-DNA,
Cs0 - Cs’  ( 1/ 2xM ) Cc Na+ /2
The salt exclusion factor :
G  lim (Cs0 - Cs )/ CDNA
and
2G
for Cc Na+ or CDNA
1/ 2xM
Information on the fraction of thermodynamically « free » counterions
0
Cargèse, 2002 - 39
Experimentally,
the salt concentration in the DNA
compartment Cs is determined by
selective electrodes
Simple salt limit 2G =1
1
2G
0.8
0.6
-Theoretically,
0.4
by solving the P-B equation and
averaging C Na+ (r) and C Cl-(r)
1/ 2xM
2G  (1/ 2xM ) f( kr0 )
with
f(kr0
0.2
0
kr0
1
0.1
Cs0
10 mM
0.1 M 1 M
0)  1 the limiting law
(Strauss et al., 1966-67; Shack et al., 1952)
Cargèse, 2002 - 40
The osmotic pressure  :
=gh
h
Na-DNA +
In excess of Na-DNA compared to NaCl,
Cs’
0
Salt Cs
the salt is excluded
 / kT = fosm Cc Na+
Salt
Cs0
with fosm the osmotic coefficient
Returning to the P-B equation in salt-free conditions,
fosm = C(R) / Cc Na+ ;
fosm = ( 1+g(kr0 ) ) / 2xM with g(kr0 )
0
Used by T. Odijk for the free energy of DNA in bacteriophage (1998)
Cargèse, 2002 - 41
2G
fosm
- Strong deviation from the limiting
law due to the finite radius r0
1
0.8
0.6
0.4
1/ 2xM
1/k
0.2
0
kr0
0.1
1
r0
CDNA
1 g/l
100 g/l
(Auer & Alexandrowicz, 1969); (Raspaud et al., 2000)
-Theories and Experiments are
qualitatively in good agreement.
Quantitatively…..
Cargèse, 2002 - 42
q/qeff
2
(Nicolai & Mandel, 1989)
(Stigter, 1978)
(Liu et al., 1998)
(Griffey, unpublished)
(Padmanabhan et al., 1988)
(Braulin et al., 1987)
q/qeff= 5.9
4.2
2.1
2.5
To be compared with xM = 4.2
- Permeability of the
double-helix
- Specific interactions…
- different sensibilities of
the different technics
Cargèse, 2002 - 43
4_b) the length effect
Counterions evaporation
(Alexander et al.,1984)
For charged spheres, ion chemical potential :
End effects
far from the spheres, mainly entropic kT ln C
on its surface, mainly enthalpic (electrostatic)
-qeffe2/40a
1/k
qeff ~ - ln C
L
The effective charge increases with dilution
Counterions evaporation
Cargèse, 2002 - 44
fraction of condensed counterions
qeff ~ - ln C
1/k
1/k
(Gonzalez-Mozuelos & Olvera de la Cruz, 1995)
dilution
Returning to DNA,
Cargèse, 2002 - 45
Importance of end effects
for short oligonucleotide helices : Simulations of Olmsted et al. (1991)
1/k = 73 Å
Cs = 1.8 mM
2G
1
Nucleotides
behave
like simple
salt
denaturated
DNA
k Lend effects
2
1.5
native DNA
2G 
(1/ 2xM ) f( kr0 )
1
0.5
1/k

( N = number of
nucleotides per chain)
0
0
5
10
15
Cs (mM)
N = 25
N = 10
5 base-pairs
Cargèse, 2002 - 46
N  72
The local
concentration
at the DNA
surface
N = 24
Ends effect
becomes negligeable
on average
but evaporation still
occurs at the ends.
N = 12
1/k
(Olmsted et al., 1989)
5_ Conformation and Interaction
Cargèse, 2002 - 47
5_a) Thermal denaturation
Calculation of the free energy using the charge renormalisation
(Manning, 1972)
Nonelectrostatic +
term
Interaction energy
of the chain with its
ionic atmosphere
- (np/xM ) kT ln k
+
Entropic term
of the salt, of +…
the « free » ions
+ (np/xM ) kT
ln Cs
Addition
of salt
Destabilizes
Ionic atmosphere
(ns salt + np/xM « free » ions)
(salt in excess but at low concentration)
L/lB or np/xM
charged phosphate
(np/xM ) denaturated > (np/xM )native DNA
the double-helix
Stabilizes
Cargèse, 2002 - 48
- (np/xM ) kT ln k
+ (np/xM ) kT
ln Cs
k ~ Cs 1/2
+ (np/2xM ) kT
ln Cs
The dominant term
is entropic
Finally, the electrostatic free energy Gele= H –TS > 0
and dominated by the entropy S
Entropic cost to condense or confine the counterions
 Gain in electrostatic free energy to denaturate DNA
D Gele = Gelesingle-stranded – Geledouble-stranded < 0
Cargèse, 2002 - 49
D Gnon-ele + D Gele = 0
At the melting temperature Tm,
After manipulation :
d Tm /d log Cs  A [ (1/2xM )denaturated-(1/2xM )native]
with A = k Tm2/DHm
100
80
Tm (°C)
60
40
20
CNa+ (M)
0
-4
10
-3
10
-2
10
-1
10
T4 DNA (Record, 1975)
Cargèse, 2002 - 50
From experimental Tm and calculated D Gele
DGnon-ele
per
nucleotide
* Tm = k log Cs + 0.41 XGC + 81.5
(Schildkraut & Lifson, 1965)
kT
* d Tm /d log Cs ~ D (1/2xM )
~ D (2G )
(Olmsted et al., 1991)
(Korolev et al., 1998; see also Rouzina & Bloomfield, 1999)
D G non-ele > 0
G non-eledenaturated > G non-elenative
Cargèse, 2002 - 51
5_b) Interaction and chain conformation
Intermolecular interactions :
between short DNA L = Lp = 50 nm (150 bp)
For neutral rods,
excluded volume v0
 L2 d
d = 2r0
For DNA rods,
e- kr
(Brenner & Parsegian, 1972)
v  L2deff
with an effective diameter
deff = 2r0 + (1/k) f( xM , k r0
At low salt , deff >> d and
v >> v0
In the high salt limit, deff = d and
v = v0
)
Cargèse, 2002 - 52
v  L2deff
v
20
isotropic
15
50 v0
C i-
10
a
5
0
-3
10
anisotropic
-2
10
-1
10
0
10
CNaCl (M)
v0
fi-a ~ deff / L
Ci – a ~ 1 / v
(Nicolai & Mandel, 1989)
CDNA increases with Cs
Cargèse, 2002 - 53
Intramolecular interactions :
The persistence length Lp : the bending flexibility
Bending makes the phosphate charges lie closer
from one another : electrostatic contribution to Lp
Lp = l0 + lele
- Using the Debye-Hückel approximation and the limiting law (k Lp >>1),
lele = lOSF = lB / 4 (kb)2
with
b
lB
(Odijk, 1977; Skolnick & Fixman, 1977)
- Using the P-B equation, lele = lOSF
f(k r0 )
(Fixman, 1982; Le Bret, 1982)
Cargèse, 2002 - 54
Lp (nm)
200
150
Lp = l0 + lOSF
Lp = l0 + lOSF f(k r0 )
100
6360 bp (Maret & Weill, 1983)
l0 = 50 nm
50
0
0.01
T2 DNA (Harrington, 1978)
587 bp (Hagerman, 1981)
0.1
1
k r0 10
T7 DNA (Sobel & Harpst, 1991)
CNaCl
0.01 0.1
1M
linear Col E1 plasmid
(Borochov et al., 1981)
Cargèse, 2002 - 55
Rg : radius of gyration
Rg (nm) of T7 DNA (40 103 bp)
800
700
Including :
600
- excluded volume
with deff
500
- persistence length
with lOSF f(k r0 ), l0 = 40 nm
(Stigter, 1985)
Experimental data
(Sobel & Harpst, 1991)
Random coil
400
10-3
10-2
10-1
1
101
CNaCl
6_ Multivalent Counterions
6_a) collapse and phase diagram
b
(Å)
B-DNA
Singlestranded
DNA
1.7
4
xM
4.2
1.8
salt
zc : 1
zcq
1
Cargèse, 2002 - 56
In the Manning approach,
Dx
Clocal
(M)
(Å)
0.76
1.17
7.4
2
0.88
0.39
11.2
4
0.94
0.12
16.9
1
0.44
0.21
12.5
4
0.86
0.021
32.2
The charge fraction zcq of
condensed ions increases
like
Experimentally, electrophoretic mobility m (cm2/V.s. × 10-4) :
Linear plasmid in 50 mM monovalent salt
+ 0.2 mM of zc-valent
(Ma & Bloomfield, 1994)
zcq = 1- 1/ zcxM
the effective charge is
reduced by 94 %
m
zc
1.47
1
1.31
2
0.92
3
Cargèse, 2002 - 57
Low monovalent salt (1 mM) + Cobalt Hexamine zc = 3 on l DNA
Light scattering (Widom & Baldwin, 1980-83)
Toroidal globule
Diffusion coefficient
Scattered
Intensity
RH  40 nm
Cobalt Hexamine Concentration
with spermidine (zc = 3),
RH  50 nm
(Wilson & Bloomfield, 1979)
Cargèse, 2002 - 58
For higher DNA concentrations :
aggregation
My first clichés !
Cryo-electron microscopy
Cargèse, 2002 - 59
Phase diagram
For
DNA redissolution 1
In excess of
spermine
-Nucleosome core
particles
- H1-depleted
chromatine fragments
10-2
Spermine
Concentration (M)
- short single-stranded
DNA
10-4
DNA precipitation
10-6
10-6
10-4
10-2
1
DNA concentration (M)
(Raspaud et al., 1998-99)
Cargèse, 2002 - 60
Attractive interaction : in violation of the basic P-B theory
- a simple charge (quasi)-neutralization with a non-Coulombic attractive force ?
doesn’t explain the redissolution in excess of multivalent salt
Cargèse, 2002 - 61
- a salt bridge model : « ion-bridging » model ?
(Olvera de la Cruz et al., 1995)
Intermolecular interactions in dilute solutions with an excluded volume v :
electrostatic repulsion
between like-charged chains
short-range attraction
The reduced effective charge
: - qeff
xM
1
1 / zc
Cmultivalent salt
-
Cspermine (mM)
-
local
charge
inversion
CDNA (mM)
6_b) ionic correlation
Fraction of counterions
within a cylinder of radius r
(from the P-B equation)
Cargèse, 2002 - 62
1
zc = 4
0.8
zc = 1
0.6
0.4
0.2
0
1
2
5
10
20
r/r0
Strong charge accumulation
near the surface ( one layer)
Few Angstroms
Cargèse, 2002 - 63
Coulomb energy between neighboring ions
Coupling parameter :
Wc = (zce)2/40 az
az
W = Wc / kT
For multivalent ions, zc > 1 and W > 1 strong coupling
Important longitudinal correlations between ions
« strongly correlated liquid »
(Rouzina & Bloomfield, 1996; Grosberg et al., 2002 and references therein)
Cargèse, 2002 - 64
Two-dimensional lattice of
the multivalent counterions
around the surface
Correlation Energy c
per ion < 0 (attractive)
If n : local concentration of
the correlated liquid
c  – W c
c(2n) < c(n )
Attraction between DNA
- qeff
- In the dissolved state,
xM
Cargèse, 2002 - 65
Free energy per DNA molecule with qeff
- In the condensed state, using c
+
log
Energy
of DNA
condensation
per nucleotide
+
-
log
- c = 0.07 kT
Phase diagram
Cmultivalent salt
Cobalt-hexamine
Cmultivalent salt
(Rau & Parsegian, 1992)
log
CDNA
(Nguyen et al., 2000 ; Nguyen & Shklovskii, 2001)
Cargèse, 2002 - 66
About the charge inversion,
Short DNA
fragments
+
T4 DNA
Spermine concentration (zc = + 4)
-
+ 4 Nucleosome
+
+3
-
zc = + 2
Core
Particles
Spermidine concentration (zc = + 3)
(Yamasaki et al., 2001)
(Raspaud et al., 1999; De Frutos et al., 2001)
Cargèse, 2002 - 67
- specific effects : polyamine is a polycation
at long length scale, behaves like a zc-valent ions
similar shape of the phase diagram
at short length scale (az), each of the zc charged groups
may be considered as a monovalent ion
attenuated correlation effect
(Lyubartsev & Nordenskiöld, 1997)
- the
electrophoretic mobility and its associated zeta potential
+
rz
-
Slipping surface located at rz
during the electrophoretic migration
Comparison between
rz and the ionic size s
r/s
rz = s rz = 2s
(Deserno et al., 2001)
Cargèse, 2002 - 68
- mixture of mono and zc-valent cations condensed onto the DNA
attenuated correlation effect
- non-ideal behavior of the “free” multivalent salt
Debye-Hückel ionic atmosphere and the ionic size effect
reduced and may suppressed the DNA overcharging
(Solis & Olvera de la Cruz, 2001)
Cargèse, 2002 - 69
As a conclusion,
DNA IS A POLYELECTROLYTE
which behaves like others charged systems (for instance, synthetic materials)
and which behavior may be understood in a first approach by electrostatics notion.
Salt may regulate - DNA-DNA interactions and their conformational changes
without specific effects
- DNA - charged proteins interactions
Course limited to Simple Naked DNA study with ions,
more complex behavior for Chromatin,…
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Cargèse, 2002 - 70
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