Mechanical properties of DNA
under stretching
Why important –
biology: curved/bent DNA important
in packing into nuclei, into viruses,
in regulation of transcription, various
enzymes bend/twist DNA during
replication, transcription, recombination
technology: important for using DNA as tool
to pull, twist objects; to study how
various enzymes that act on DNA work;
to build nanoscale objects using DNA
What we’ll cover:
stretching – low force
concept of entropic spring
freely-jointed chain and
worm-like chain models
high force –> structural change in double helix
B->S form, similarity to phase change
methods used to study
hydrodynamic drag
paramagnetic beads
laser traps
example of how used to study mechanism of
enzyme that works on DNA
Linear polymers and Hooke’s Law
Freely jointed chain (FJC) model
f1
n segments length b joined
at freely rotating joints
Brownian (thermal)
motion randomizes fi
applied force pulls out chain fixed at
contour length L = nb
b
F
x
end
<x>/L = tanh(Fb/kBT) for 1-d model (see Nelson, ch 9.2)
tanh(z) = (ez – e-z)/(ez + e-z)
-> z for z<<1
-> 1 for z>>1
<x>/L = tanh(Fb/kBT)
Low force regime F << kBT/b,
tanh(z)-> z for z<<1
F -> k<x> where k = kBT/Lb
the longer L, the more compliant
the higher T, the less compliant
equipartition theorem: k<x2> = kBT
<x2>1/2 = xrms = (Lb)1/2 = n1/2b n=L/b
xrms independent of T, F at low force
thermal energy randomizes fi
High force regime:
F>>kBT/b,
<x> -> L
Several groups tried to measure b by pulling on DNA
Bustamante (Science 258:1122 (1992)
phage DNA of known L ~30mm
attached at 1 end to glass
other end to r ~ 1mm paramagnetic bead
<x>
att. pt. determined by varying
flow and magnetic field
knowing flow v, Fflow = 6phrv
measuring q, Ftotal = Fflow/cos q
measure <x>
F (pN)
b = 50nm
100
200
<x> (mm)
Problem – poor fit to 3-d FJC model no matter what L or b
x
Worm-like chain model
randomly oriented chain with
“stiffness” defined by:
Persistence length p =
length over which
orientational correlation
falls exponentially to 1/e
t^1
q
^t
2
DNA
s
<cosq(s)>
1
p
s
<x(F)>/L does not have analytic solution,
but in high and low force limits,
Fp/kBT = ¼ (1-<x>/L)-2 – ¼ + <x>/L
at low F, <x>/L << 1, F = ksp<x> where ksp = (3/2)kBT/pL
Bustamante, Science 265:1599 (1994)
FJC
model
WLC
model
WLC model
fits forceextension
data much
better than FJC
WLC model also fits ssDNA if you change p
--------relative
pds @ 50nm (~150 base pairs)
pss @ 1nm
which is “wiggilier”?
ksp = (3/2)kBT/pL
pds = 50nm
pss = 1nm
Is the spring stiffer for ss or ds DNA?
Why does a more flexible DNA chain (ss) act like a
stiffer spring?
Contour length L (= length of fully pulled out chain)
Lss @ .5nm/b * # b
Lds @ .3nm/bp * # bp
Why is the contour length of ds DNA shorter/bp?
(think base stacking in helix…)
Fp/kBT = ¼ (1-<x>/L)-2 – ¼ + <x>/L
L=n*l/bp or l/b
Could you estimate length of ss or ds DNA of
length n in bases (or base pairs) at given F?
--------relative
Why is n-base ssDNA longer at large F but shorter at low F
than n-bp dsDNA?
At F ~ 65pN, dsDNA suddenly begins to stretch
Further pulling
lengthens DNA >L
w/ little increase
F until new, fully
stretched state is
reached (~1.7 L)
Smith et al Science 271:795 (1996)
Stretched “S”-form of DNA probably has base-stacking
interactions disrupted -> change in helix pitch
3.4nm/10bp  5.8nm/10bp
“Cooperativity” of transition suggest S-form segment
spreads along DNA (takes less energy to expand an
S-form region than to initiate one); similar to phase
change ice->water, adding heat doesn’t change
temperature until all ice melted, more pulling work
doesn’t change tension until all DNA converted to S-form.
Stretching experiments used laser trapNobel prize
Highly focused laser pulls
object with higher index of
refraction towards brightest
part of laser beam (x=0); small
displacement x -> restoring
force ~ -kx. Given trap strength
k, observing x, one can infer F
Mechanism: light E-field polarizes object with diff.
dielectric constant -> attractive dipole force
E
-->
--
++
in gradient E, polarized object
feels net force
Alternative explanation – photons carry momentum;
bending ray changes
photon momentum;
momentum conservation => object feels
opposing force; if
beam asymmetric,
force from brightest
region dominates
Newman and Block, Rev Sci Instr 75:2787 (2004)
Quadrant photodetector reports bead
displacement Dx from
trap center, i.e. reports
F given trap stiffness
ksp since F = ksp
Moving laser trap
stretches DNA
Trap position reports
DNA end-to-end
length
What is length of mixed ds-ssDNA?
Numerator = observed D length (compared to all ds)
Denominator = max D length if all ss compared to all ds
Ratio = fractional D in length ~Nss/Ntot
What enzyme did Bustamante et al add to
this system?
Watching DNA polymerase act in real time
Enzyme + dNTP added to ds/ss tether
Data collected every 0.125s; how fast does enz. move?
Bottom curve = slope averaged over sliding 3s windows
How might you interpret the “bumps”?
Where on velocity trace is enz. active?
Why doesn’t velocity -> 0 between bumps?
Why is “off time” (1/koff) the aver. time enz. is on?
Can you estimate koff, kon from this data?
Complicated scheme of E + D <->ED where E can bind
as polymerase (p), then bind dNTP, add base (n->n+1)
or as exonuclease (x) then remove a base (n->n-1),
or convert between p and x configurations
Rate, binding constants
from literature, “bulk” expts.
You could compare your single-molecule kon, koff
to data from bulk expts; this might strengthen
your interpretation but does not advance the field
What is biological role of exonuclease function?
What happens to misincorporation rate if you mutate
(eliminate) exo function?
Effect of tension (F) on enzyme velocity
Why are error bars bigger ~6pN?
Why might velocity decrease as tension increases?
Complicated model for enzyme pulling a few (n) bases of
template ss into configuration of ds; this requires work
W(n) against tension; velocity ~e-W(n)/kT; how do models
of different n’s fit the data?
n=1
n=2
n=3
Does data strongly support n = 1, 2, or 3?
Above stall force ~40pN, only exo activity (this is how
they converted ds tethers to partially ss!)
What does inset show?
Is conversion reversible?
How would you interpret
“bumps” in exo velocity?
Unfortunately, obs. koff, kon’s suggest bumps can’t be enz.
falling off, rebinding, but involve pol <-> exo conversions
What can single-molecule expts. show that would
be very hard to learn from bulk expts.?
Are enzyme molecules heterogeneous or all the same?
Is enzyme rate sequence-dependent?
Is enzyme rate slowed by tension? This could inform
detailed models of how enzyme works
What makes enzyme interconvert between pol and exo
conformations?
Summary
laser traps/magnets/tethered bead expt’l. system:
allow application of pN forces
measurement of pN forces and DNA/RNA lengths
with near nm precision
WLC model predicts DNA mechanical properties accurately
(extension as function of force, twist and
buckling as function of torque)
Clever experimental systems -> real-time observation
of single enzymes/assemblies at work, potentially
elucidating mechanistic details
Lots of other examples of single-molecule studies:
RNA polymerases that partially melt dsDNA
and make RNA copies
Motors that pack DNA into virus particles
Helicases that unwind ds DNA/RNA
Topoisomerases that nick, religate DNA, relieving
torsional strain and topological entanglement
Ribosomes that copy RNA into protein
These studies combine nano-scale biology and
engineering -> new discipline
For now, mostly research applications…
Understanding nanoscale biosystems provide
insight, tools potentially applicable to
non-biological nanosystems
Example – experimental test of basic physics prediction
of relation between work W done on non-equilibrium
system and free energy change DG at equil.
W > DG (due to dissipation) classical eqn
<e-W/kT> = eDG
Jarzynski prediction 1999
slow
W = area betw
nfold
curves
efold
Science vol 296
fast
p1832, 2002
Next week – DNA sequencing
why the interest
first “next generation” method
Homework problems on DNA mechanics
Midterm due by end of weekend