STRESSINDUCED STRUCTURAL TRANSITIONS IN DNA AND PROTEINS

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STRESS-INDUCED STRUCTURAL TRANSITIONS INDNA AND
PROTEINS
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FORCES AT THE MOLECULAR SCALE

• The smallest force on a molecule is set by the
  Langevin force f n due to thermal agitation. It
  sets the lower limit to force measurements and is
  due to the Brownian fluctuations (of energy kBT
  4 10-21 J 0.6 kcal/molat room temperature) of
  the object of size d (sensor, cell, membrane)
  anchored by the molecule. For a d 2 m diameter
  bead or cell in water (viscosity h 10-3 poise),
  f n (12pkBThd)1/2 10 fN/Hz1/2 (notice that this
  is a noise density, i.e. the faster the
  measurement, the more noisy it is). This can be
  compared to the typical weight of a cell 10
  fN, i.e. every second a cell experiences a
  thermal knock equal to its weight!

• Just above these forces lie the entropic forces
  that result from a reduction of the number of
  possible configurations of the system consisting
  of the molecule. The entropic forces are rather
  weak. Since the typical energies involved are of
  order kBT and the typical lengths are of the
  order of a nanometer, entropic forces are of
  order k T/nm 4 pN (4 10-12 N).

• Noncovalent (e.g. ligand/receptor) bonding forces
  are much stronger. They usually involve
  modifications of the molecular structure on a
  nanometer scale breaking and rearrangement of
  many van der Waals, hydrogen, or ionic bonds and
  stretching of covalent bonds. The energies
  involved are typical bond energies, of the order
  of an electron-volt (1 eV 1.6 10-19 J 24
  kcal/mol). The elastic forces are thus of order
  eV/nm 160 pN.

• Finally the strongest forces encountered at the
  molecular scale are those required to break a
  covalent bond of the order of 1 eV/ A 1600 pN.

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Stress-Induced Structural Transitions
The AFM in its forceextension mode Extension of
a single molecule caused by retraction of the
piezoelectric positioner results in deflection of
the cantilever. This deflection changes the angle
of reflection of a laser beam striking the
cantilever, which is measuredasthe change in
output from a photodetector.
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Forced unfolding of titin
The successive unfolding of titin domains gives
rise to the corresponding peaks and valleys in
the force-extension profile measured with the
atomic force microscope.
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Structure and Function of a Titin Protein
Cartoon of titin I-band function. (Actual I-band
contains 41 Ig domains (Rief et al., 1997).) Ig
domains are explicitly shown, with the PEVK
region depicted as a heavy black line. Horizontal
gray arrows indicate the external force. (a)
Titin I-band resting struc-ture. (b) Titin I-band
with PEVK region extended. (c) Titin I-band with
Ig domain partially unfolded. This figure was
created with VMD (Humphrey et al., 1996).
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Secondary structure of I27 domain
(a) b-sheets are colored differently, with sheet
A B E D ingreen and sheet A G F C in orange. (b)
Schematic view of all b-sheets and backbone
hydrogen bonds (dotted lines) between adjacent
b-strands. This figure was created with VMD
(Humphrey et al., 1996).
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Two-level model for the unfolding of a protein
domain
The domain in its native state (on the left)
unfolds at a rate a0 by hopping over the barrier
(of energy difference DGu ) to unfolding. In the
denatured state (on the right) the refolding rate
b0 is controlled by the energy difference with
the transition state DGf . When a force F is
applied to the system, the free-energy difference
DG between native and unfolded state is skewed
toward unfolding by the work performed by the
force F (xu xf ).
(H. Gaub)
1.5
-75

• 108Hz is the relaxation time inverse xu0.3 nm
  xf15 nm, F20pN

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Unfolding of Titin Immunoglobulin Domains by
Steered Molecular Dynamics Simulation
The intermediate stages of pulling simulations.
The protein domain I27 (residues 188) is drawn
in cartoon representation with the two b-sheets
presented in different colors, and water
molecules are drawn in line representation. (a)
Region I, preburst, at extension 10 Å. (b) Region
II, immediately after the major burst, at
extension 17 Å. (c) Region III, postburst, at
extension 150 Å. (d) Region IV, fully extended
domain, at extension 285 Å. The bar at the lower
left corner of each figure represents 10 Å. This
figure was created with VMD (Humphrey et al.,
1996).
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Force extension profile of SMD simulations for I27
Force extension profile of SMD simulations for
I27 with a pulling speed of 0.5 Å/ps. The
extension domain is divided into four regions I,
preburst II, major burst III, postburst IV,
pulling of fully extended chain.
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Single Molecule Force Spectroscopy of
SpectrinRepeats Low Unfolding Force in Helix
Bundles
immunoglobulin
spectrin
0.8 mm/s
0.08 mm/s
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Unfolding Proteins by External Forces Computer
Simulations
Structures and topology of the protein studied.
(A) Fn3 domain (1TEN) and an Ig domain (1TIT).
(B) a-Spectrin domain (1AJ3) and the four-helix
bundle protein acyl-coenzyme A-binding protein
(2ABD). The N- and C-terminal atoms are
represented by red and green spheres,
respectively the same colors are used to
represent b-strands belonging to the N- and
C-terminal sheets of the b-sandwich proteins. In
the schematic topology, triangles represent
b-strands, and circles represent a-helices (and
also a turn of 310-helix in the Ig domain).
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End-to-end Distance vs. Time
Unfolding reaction coordinate rNC (in Å) as a
function of time (in picoseconds). Summary of the
results for (A) the two b-sandwich domains and
(B) the two a-helical domains. The external force
applied to unfold the domains is also plotted as
a function of rNC the force values are averaged
over all trajectories and over 3-Å intervals of
rNC.
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Pathway for the Unfolding of the 1TIT Domain
(A) X-ray structure, rNC 45 Å BD
temperature-induced unfolding pathway (B) 0.6
ns, rNC 34 Å, (C) 1.5 ns, rNC 37 Å (D) 4.8
ns, rNC 38 Å. EH forced unfolding pathway (E)
rNC 54 Å (F) rNC 75 Å (G) rNC 126 Å (H) rNC
138 Å. The N- and C-terminal atoms are
represented by red and green spheres,
respectively nonnative are in grey.
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Pathway for the Unfolding of the 1AJ3 Domain
(A) X-ray structure, rNC 42 Å (B) rNC 50 Å
(C) rNC 70 Å (D) rNC 100 Å (E) rNC 130 Å
(E) rNC 142 Å (F) rNC 160 Å (G) rNC 250
Å. E is a stable intermediate found in one set
of unfolding pathways.
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Pathway for the Unfolding of the 2ABD Domain
(A) X-ray structure, rNC 28.4 Å (B) rNC 30
Å (C) rNC 50 Å (D) rNC 80 Å (E) rNC 110
Å (F) rNC 150 Å (G) rNC 200 Å (H) rNC
250 Å.
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FORCE-INDUCED TRANSITION TO S-DNA
Force versus relative extension curves of single
DNA molecules. The dots correspond to several
experiments performed over a wide range of
forces. The force was measured using the Brownian
fluctuation technique. The full line curve is a
best fit to the WLC model for forces smaller than
5 pN. The dashed curve is the result of the
freely jointed chain (FJC) model with the same
persistence length (it is clearly a worse
description of the behavior of DNA under stress
than the WLC model). At high forces, the molecule
first elongates slightly, as would any material
in its elastic regime. Above 70 pN, the length
abruptly increases, corresponding to the
appearance of a new structure called S-DNA.
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Structural Transition of DNA under stress
The new structures of DNA obtained in numerical
simulations when pulling on the molecule (R.
Lavery). Left usual B-DNA structure, middle if
the molecule is pulled by its 5 ends, it keeps a
double helical structure with inclined bases.
Right if the DNA is pulled by its 3
extremities, the final structure resembles a
ladder.
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DNA under Torsion
Left Schematic view of the buckling transition
for a twisted rubber tube (dotted line) or a DNA
molecule (solid line). When nc,b turns have been
added, the system abruptly exchanges twisting
energy for bending energy and plectonemes begin
to form. The plectonemes grow linearly with
subsequent twisting, and the torque remains
constant thereafter. In the case of DNA the same
picture holds, except that thermal fluctuations
round off the transition that takes place at sc,b
. Right The torque acting on the DNA (dashed
curve) increases linearly until sc,b and remains
essentially constant thereafter. The short-dash
curve represents the ratio of writhe to twist
note that the writhe is never zero and increases
rapidly as sgtsc,b . Finally, the solid line
measures the fraction of plectonemes in DNA
stable supercoiled structures only appear after
the torsional buckling transition has been passed.
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Topological Properties of Coiled DNA

• twist (Tw) the number of helical turns along the
  molecule.
• For a torsionally unconstrained B-DNA, Tw Tw0
  N/h, where N is the number of bases and h 10.4
  is the number of bases per turn of the helix.

• writhe (Wr) a measure of the coiling of the DNA
  axis about itself, like a twisted phone cord that
  forms interwound structures in order to relieve
  its torque.

If the DNA molecule is torsionally constrained,
the linking number, Lk Tw Wr, is a
topological invariant
The relative difference in linking number between
the supercoiled and relaxed forms of DNA is
called the degree of supercoiling, s s (Lk -
Lk0)/Lk0 DLk/Lk0
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Calculating Linking Number of DNAs
The schematic diagram to calculate link number in
our simulations. (a) For a linear supercoiled DNA
chain with one end attached to a microscope slide
and the other attached to a magnetic bead, when
the orientation of the bead is fixed and the DNA
chain is forbidden to go round the bead, the
number of times for two strands to intertwine,
and the linking number of the linear DNA (Lkl),
is a topological constant. (b) The DNA double
helix is stretched to a fully extended form while
the orientation of bead remains unchanged. The
link number of the linear DNA chain is equal to
the twist number, i.e., Lkl Twl .(c) Three
long, flat ribbons are connected to the two ends
of the linear twisted DNA of (b). The link number
of the new double helix circle is equal to that
of the linear DNA chain, i.e., Lkc Twc Twl
Lkl because the writhe of the rectangle loop is
0. (d) The DNA circle in (c) can be deformed into
a new circle, one part of which has the same
steric structure as the linear supercoiled DNA
chain in (a). So, by adding three straight
ribbons, the link number of linear double-helix
DNA can be obtained by calculating the link
number of the new DNA circle, i.e., Lkl Lkc
Tw Wr.
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Theoretical DNA Model
The embeddings of two backbones are defined by
r1(s) and r2(s) and they are connected by the
hydrogen-bond-director b(s) by r2(s) r1(s)
2Rb(s).
The bending energy of the DNA chain is
The relations among the tangential vectors (t1
d r1 /ds, t2 d r2 /ds, and t) are
q is the intersection angle between tangent
vector of backbones t1(2) and DNA central axis t.
Thus,
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Relative extension of a DNA molecule versus the
degree of supercoiling s for various stretching
forces. For the three curves obtained at low
forces, the behavior is symmetrical under s
-s. The shortening corresponds to the formation
of plectonemes upon writhing. When the force is
increased above 0.5 pN, the curve becomes
asymmetric supercoils still form for positive
coiling while local denaturation relieves the
torsional stress for negative s. At forces larger
than 3 pN, no plectonemes are observed even on
positively supercoiled DNA again, the torsional
stress is absorbed not by writhe but in local
structural changes of the molecule.
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Structure of P-DNA deduced from molecular
modeling (R. Lavery). Space filling models of a
(dG)18 .(dC)18 fragment in B-DNA (left) and P-DNA
(right) conformations. These models were created
with the JUMNA program, by imposing twisting
constraints on helically symmetric DNA with
regularly repeating base sequences.
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Monte Carlo Implementation of Supercoiled
Double-Stranded DNA
The conformation of DNA molecule of N straight
cylinder segments is specified by the space
positions of vertices of its central axis, ri
(x(i), y(i), z(i)) in three-dimensional Cartesian
coordinate system, and the folding angle of the
sugar-phosphate backbones around the central
axis, qi , i 1,2,...,N.
The length of the i-th segment satisfies
where lt gt0 means the thermal average for a
relaxed DNA molecule and nbp is the amount of
basepairs.
The configuration of discrete DNA chain in the
model.
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The van der Waals Interaction Potential
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Trial Motions of the DNA Chain during Monte Carlo
Simulations
(a) The folding angle in i-th segment qi is
changed into qi l1 . All segments between i-th
vertex and the free end are translated by the
distance of Dsi - Dsi. (b) A portion of the
chain is rotated by an angle of l2 around the
axis connecting the two ends of rotated chain.
(c) The segments from a randomly chosen vertex to
the free end are rotated by an angle l3 around an
arbitrary orientation axis that passes the chosen
vertex. The current conformation of the DNA
central axis is shown by solid lines and the
trial conformation by dashed lines.
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Physical Quantities of the DNA Chain
The total energy of DNA is
The averaged extension is
The averaged torsion is
The writhe of DNA is
The twist of DNA is
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Force versus Relative Extension Curves
Force versus relative extension curves for
negatively (a, b) and positively (c, d)
supercoiling DNA molecule. (a) and (c) are the
results of Monte Carlo simulations, and the
horizontal bars of points denote the statistic
error of relative extension in our simulations.
(b) and (d) are the experimental data (Strick et
al., 1998). The solid curves serve as guides for
the eye.
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Relative Extension versus Supercoiling Degree
Relative extension versus supercoiling degree of
DNA polymer for three typical stretch forces.
Open symbols denote the experimental data (Strick
et al., 1998) and solid symbols the results of
Monte Carlo simulationg. The vertical bars of the
solid symbols signify the statistic error of the
simulations. The solid lines connect the solid
points to guide the eye.