Sensitivity of DNA Looping to
Sequence-dependent Stiffness
Sachin Goyal1, Todd Lillian2, David Wilson2, Edgar Meyhofer2,
Jens-Christian Meiners2 and Noel Perkins2.
1Woods Hole Oceanographic Institution, Woods Hole, MA, USA,
2University of Michigan, Ann Arbor, MI, USA.
Protein-mediated DNA looping plays an important role in gene regulation. Empirically
it is known that sequence-dependent mechanical effect such as intrinsic bends or softer
regions in the substrate DNA affect loop formation, but quantitative models are lacking.
We employ a continuum rod model to simulate protein-mediated DNA looping as a
means to explore how the sequence maps to the overall structural properties of the
duplex. The model includes sequence-dependent intrinsic curvature, chirality, and
stiffness. We address the fundamental question of how sequence-dependent stiffness
influences the looping of DNA bound to regulatory proteins like the lactose repressor.
We report two major findings: First, any non-uniform stiffness tends to lower the
energetic cost of looping. Second, the deformation tends to localize in ‘softer’
regions which in turn affects the loop topology as characterized by twist and
writhe. The model also offers the capability to calibrate and benchmark experimental
measurements of sequence-dependent stiffness.
1
Protein-Mediated Looping of DNA
Schematic of Lac Operon (E. coli):
5’ to 3’
RNAP
O3
CAP
O1
LacZ
LacY
LacA
3 structural genes
DNA loop
computed from
“rod” model
Introduction:
O3
•
The genes LacZ, LacY and LacA are
repressed when the Lac-R protein binds to
“operator” sites O3 and O1 of DNA and the
intervening DNA is deformed into a loop.
•
DNA looping is a common mechanism in
gene regulation.
•
Mechanics of loop formation and how it
influences gene regulation is an area of
research.
O1
Known crystal structure
of Lactose-Repressor
protein bound to O3 and
O1 sites of DNA
[Lewis et al., 1995]
2
Protein-Mediated Looping of DNA
Lac Operon (E. coli)
Mechano-Chemistry:
+
Free DNA
(Genes “on”)
LacR Protein
Gene Repression Level
Free Energy Budget
E
-


LacR-DNA Complex
E
Free Energy
(Genes “off”)
  E 
 exp 

rateunlooping
 kT 
ratelooping
Strain energy U of DNA loop
Entropy [2574-Plat, Wilson et al.]
Dominant component
for sub-persistence
length DNA
Protein Flexibility
Other (e.g. surface binding, electrostatics)
3
Protein-Mediated Looping of DNA
Lac Operon (E. coli)
Mechano-Chemistry:
+
Free DNA
Known
[Gabrielian et al., 1996]
Initial conditions,
structural properties
(material law)
•
•
•
Stiffness
Intrinsic curvature
Chirality (righthandedness)
-

LacR protein
LacR-DNA complex
Known
Unknown ?
[Lewis et al., 1995]
Boundary conditions
Rod Model
E
Free energy
Unknown ?
Loop properties
ai (s, t )
•
•
[Goyal et al. 2005]
•
R ( s, t )
Energy
Topology
(twist and writhe)
Reaction moment &
force on protein
4
Rod Model
(Captures stiffness in two-axes bending and torsion)
s
Cross-section fixed
reference frame
ai (s)
Material Law:
q( s)  fn( ( s), s)
Restoring Moment
where:
ai
   ai
s
Curvature and Torsion
5
Formulation of Nonlinear Rod Dynamics
[Goyal et al., 2005]
Field Variables: {v, ω, f, κ}
(internal
force)
(internal
moment)
f
v (velocity)
q

(angular
velocity)
Free Body Diagram:
Field Equations:
(curvature
& twist)
Linear Momentum Equation:
f
 v

   f  m    v   F
s
 t

Angular Momentum Equation:
q

 q  I
   ( I )  f  tˆ  Q
s
t
Inextensibility & Unshearability Constraint:
v
   v    tˆ
s
Compatibility Condition:


   
s
t
6
Linear Material (Constitutive) Law
Material 
properties
[Pos/B209,
Lillian et al.]
1. Stiffness
tensor
2. Intrinsic
curvature
q( s)  B( s) ( s)   0 ( s) 
Restoring
moment
Curvature of
deformed state
Note: 3. Chirality (right-handedness of the molecule) can also be captured
in the rod-constitutive law [Goyal et al., J. Comp. Phys., 2005].
7
Linear Material (Constitutive) Law
•
The material properties are sequence-dependent and hence are
non-uniform along the rod-length.
•
The stiffness tensor includes two-axes bending and torsional
stiffness.
•
Bending stiffness is effectively isotropic on long length-scales due
to high intrinsic twist of the molecule [Maddocks and co-workers].
Question:
How does the non-uniform stiffness affect DNA looping?
8
Strain Energy of DNA Loop
L
0
0
U    ( s)   0 ( s)  B( s) ( s)   0 ( s) ds
T
0
Non-uniform
stiffness
To analyze the influence of non-uniform stiffness on looping, we set intrinsic
curvature to zero in the rod model.
9
Non-uniform Stiffness
Insight:
k1
k2
Stiffness
Stretch
Energy
k1
F/ k1
F2/ 2k1
k2
F/ k2
F2/ 2k2
F
Strain and Strain Energy tends to concentrate in soft regions
(Both distribute in the inverse proportion of stiffness)
Problem Set-up:
Pure Torsion
Stiff
Soft
Stiff
Computed Result:
Untwist localized in soft zone
10
Two Computed Lac-R DNA Loops
(under-twisted and over-twisted)
Twist surplus(+)/
deficit(-)
deg./bp
34kT
38kT
(uniform
stiffness)
(uniform
stiffness)
19kT
20kT
(non-uniform
stiffness)
(non-uniform
stiffness)
LacR
Protein
Under-twisted DNA loop
LacR
Protein
Over-twisted DNA loop
Strain energy U of DNA loop shown in kT
k = Boltzmann constant and T = absolute room temperature in Kelvin
11
Figure Description
The figure shows a simulation example pertaining to LacR-DNA loops
where the stiffness is lowered by an order of magnitude at a specified
location (see next slide). The results are contrasted with those predicted
by the rod with uniform stiffness. The color scale shows the distribution
of twist surplus (+) or deficit (-) over the nominal twist of 34.6° per basepair step.
Observations
•
Twist/ untwist and bending localizes in the softer region.
•
Strain energy of the loop is lowered with non-uniform stiffness.
12
Description of the Rod with Non-Uniform Stiffness:
Length L = 77 base-pairs ≈ 26 nm
0.2 L
0.2 L
0.6 L
An order of magnitude softer
than the rest of the domain
L
•
1
B( s)ds) is same as that of the uniform rod.
Average stiffness (=
L 0
•
Average Bending stiffness = 50nm  kT [Hagerman, 1988].
•
Average Torsional stiffness = 75nm  kT [Strick et al., 1996].
13
Possible Sources of Non-uniform Stiffness
• Sequence-dependence:
2 H-bonds in A-T base-pairs vs 3 H-bonds in G-C base-pairs. A-T rich
regions are expected to be softer.
• Base-pair flipping (kink-ability):
Base-pair flipping unconstrains the two strands of DNA and might lower the
stiffness by more than an order of magnitude. (The net stiffness of two
independent strands is the sum of their individual stiffness. For example, the
bending stiffness of individual strand is 0.75 nm-kT [Smith et al., 1996]. The
total bending stiffness of the two unconstrained strands would be 1.5 nm-kT
(imagine two bending springs in parallel), which is << 50 nm-KT stiffness of
double-stranded DNA.)
• Melting:
Melting also unconstrains the two strands of DNA. Local melting may occur
at RNAP binding site.
14
Conclusions/ Insights from Rod Model Simulations
•
Non-uniform stiffness reduces energetic cost of looping.
•
Non-uniform stiffness alters loop topology by localizing deformations
(twist and bending) in soft regions.
Additional Thoughts
•
Softer regions of DNA might be more prone to melting/ kinking due to strain
energy concentration.
Please also visit:
1.
2.
1981-Pos/B209: Computational rod theory predicts experimental characteristics of DNA
looping by the Lac repressor, Todd D. Lillian, Sachin Goyal, Noel C. Perkins, Jens-Christian
Meiners, Jason D. Kahn.
9:30 am, Wed, Mar 7, 2574-Plat, Modeling the Entropic Cost of DNA Looping, David P.
Wilson, Todd D. Lillian, Bachelors, Sachin Goyal, Noel C. Perkins, Alexei Tkachenko, Jens C.
Meiners.
15
Online Reference
Acknowledgements:
(NSF, ONR, LLNL)
Website contents:
(handout (PPT), publications)
Special thanks to:
(Andricioaei et al,
Tkachenko et al.)
Questions/ comments
e.mail to:
http://www.whoi.edu/sites/sgoyal
(Go paperless, go blue!)
Sachin Goyal
[email protected]
Todd Lillian
[email protected]
David Wilson
[email protected]
Edgar Meyhofer
[email protected]
Jens-Christian Meiners
[email protected]
Noel Perkins
[email protected]
16