1) Polymer looping, the process of a polymer chain forming a loop, is an important biological process but is affected by macromolecular crowding and external tension. 2) Simulations show that macromolecular crowding increases looping probability and slows looping kinetics by decreasing polymer mobility, consistent with experiments. Crowder size also influences looping, with smaller crowders slowing looping more. 3) Even small external tensions dramatically increase looping times, especially for longer chains, in agreement with experiments. Tension cooperatively straightens the chain, making looping more difficult.

1. Polymer looping: Effects of macromolecular
crowding and external tension
Jaeoh Shin
Institute for Physics & Astronomy, University of Potsdam
December 3rd, 2014
References: J. Shin, A. G. Cherstvy, and R. Metzler, Soft Matter (2015).;
J. Shin and W. Sung, J. Chem. Phys. 126, 045101 (2012).
.
MPI-PKS, Dresden
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2. Overview
Introduction
Polymer looping
Looping in the presence of macromolecular crowding
Macromolecular crowding
Model
Results-looping with crowding
Summary
Looping in the presence of tension
DNA loop formation in the presence of tension
Model
Results-looping under tension
Summary
More things
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3. Polymer looping: A thermal noise induced process
It looks simple, but has rich physics - polymer dynamics, stochastic
processes
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4. Polymer looping: it is important (for biology)
It is a fundamental process for many biological functions: RNA
folding, gene regulation, facilitated diffusion, etc.
(R. Phillips et al., “Physical Biology of the Cell” (2012).)
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5. Transcription factor mediated DNA loop
enhancing interactions between transcription factors bound at
distance sites
bringing transcription factors close to RNA polymerase
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6. Quantitative experimental data on looping kinetics are
available
Kinetics of conformational fluctuations in DNA hairpin-loops
in crowded fluids, NJP, 11, 113010 (2013).
Femtonewton Entropic Forces Can Control the Formation of
Protein-Mediated DNA Loops, PRL, 104, 048301 (2010).
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7. Part I : Polymer looping in the presence of macromolecular
crowding
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8. Macromolecular crowding: Cell is very crowded
Volume fraction of macromolecules in
cytoplasm ∼ 30%
orange: actin filament, blue: ribosome
(cryo-EM image of cytoplasm, EMBO reports, 5, 23 (2004))
Mean spacing between the
macromolecules ∼ size of
macromolecules themselves
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9. Molecular crowding can make molecules in cells behave in radically
different ways than in dilute solution1
biochemical reactions
protein folding kinetics/ stability
chromosome structure, ...
Bacterial cytoplasm2
1
S. B. Zimmerman and A. P. Minton, Annu. Rev. Biophys. Biomol. Struct.
(1993) ; H. X. Zhou et al., Annu. Rev. Biophys., (2008)
2
SR McGuffee, AH Elcock, PLos Comp. Biol. 6, e10000694 (2010)
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10. On a large scale
(Seoul subway at rush hour)
Crowding slows down movement
Make us more compact
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11. DNA hairpin loop formation with macromolecules
(O. Stiehl et al., New J. Phys. 11, 113010 (2013))
(FCS: fluorophore, quencher)
(sucrose, PEG, dextran).
Looping kinetics becomes slower & Looping probability increases
with molecular crowding
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12. Simulation Model
Polymer chain
Bead spring model with FENE & LJ potential
UFENE(r) = −
k
2
r2
max ln 1 −
r2
r2
max
. (1)
ULJ(r) = 4 [(σ/r)12
− (σ/r)6
+
1
4
]Θ(21/6
− r). (2)
σ: bead size (∼ Kuhn length of the chain)
Two ends have attraction - mimic the hydrogen bonding of
complementary base pairs
“explicit” Crowding molecules
hard sphere of diameter dcr
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13. Langevin equation
m¨ri = − (UFENE + ULJ) − ξvi + FR
i , (3)
with m ∼ d3
cr and ξ ∼ dcr
Simulate in a periodic box
φ=Ncr ×Vcr /Vbox
.
!!This code is applicable to other systems.!!
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14. Polymer conformation in the presence of MMC
Chain length n=32, dcr =1σ, φ=0.1
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15. (Click to play movie)
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16. Distribution of End-to-end distance
F(r) = −kBT ln(P(r)) : free energy
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17. Looping probability & Polymer size
As φ increases, Pl increases and the chain gets more compact.
This behavior is consistent with the experimental result
(O. Stiehl et al., New J. Phys. 11, 113010 (2013))
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18. Crowder size effect
a. confinement(caging) effect; b. depletion attraction
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19. Time evolution of the end-to-end distance
0
5
10
0 1000 2000 3000
End-to-enddistance,r/σ
Time, t
r=req
r=rc
r=rf
φ=0.1
0
5
10
400 500 600 700
r/σ
t
Tl
Tcl Top
Tul
Looping time: time interval between expanded conformation
r = req and looped conformation (r = rf ).
Unlooping time: vice versa.
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20. Looping and Unlooping times
100
150
200
250
300
0 0.1 0.2 0.3
Loopingtime,Tl
Crowder volume fraction, φ
(a)
dcr=0.75σ
1σ
2σ
4σ
Looping times can increase or
decrease depending on the size
of crowder dcr
50
100
150
200
0 0.1 0.2 0.3
Unloopingtime,Tul
Crowder volume fraction, φ
(c)
dcr=0.75σ
1σ
2σ
4σ
In contrast, unlooping time
increases with all the size of
crowder
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21. Comparison with experiment
τk = Tl ∗ Tul /(Tl + Tul )
τk increases about 2–3 times, consistent with the measured
data.
Real crowding molecules are much more complex than
hard-sphere!
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22. Facilitation or inhibition of looping
10
2
103
10
4
10 100
Loopingtime,Tl
Chain length, nσ
(a)
n
2ν+1
Tl, φ=0
dcr=4σ, φ=0.2
1σ, 0.2
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23. Effects of crowder
Statics : makes the chain more compact conformation
Dynamics : Diffusive motion of polymer segments is slowed
down
Crowder size dependence of the looping times
100
150
200
250
300
1 10
Loopingtime
Size of crowder, dcr/ σ
(b)
2*Rg
φ=0.2
0.3
φ=0
Depending on the size of
crowder dcr , the looping can be
facilitated or inhibited
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24. Looping time from MFPT calculation
Consider the looping process as a one-dimensional barrier crossing
process3
T(dcr , φ) =
req
rf
dr exp(βF(r))
1
Dee(dcr , φ)
L
r
dr exp(−βF(r ))
(4)
F(r) = −kBT ln(P(r)) : free energy
Dee : diffusivity of the relative motion of end monomers
β = 1/(kBT)
3
A. Szabo et al., J. Chem. Phys. 72, 4350 (1980); N. M. Toan et al., J.
Phys. Chem. B, 112, 6094 (2008).
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25. Diffusion of a tracer particle in a crowded medium
Crowding molecules slow down the motion
0.1
1
10
100
1 10 100
—
δ
2
x/σ
2
Time, ∆
2D0∆
φ=0
dcr=4σ, φ=0.1
0.3
dcr=1σ, φ=0.1
0.3
MSD, 4σ, 0.1
Mean squared displacement
0.2
0.3
0.5
1
0 0.1 0.2 0.3D(φ)/D0 φ
dcr=1σ
4 σ
.
.
Diffusivity (∼ 1 /viscosity) decreases more sharply with smaller
crowder
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26. Using F(r) and Dee, we calculate the looping time
100
1000
10000
10 100
Loopingtime
Chain length, n
(a)
Tl, φ=0
dcr=4σ, φ=0.2
dcr=1σ, φ=0.2
Tl, Rouse, 4σ
Tl, Rouse, 1σ MFPT gives correct trend
for different dcr .
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27. Summary of Part I
We characterized dual effects of the molecular crowding on
the looping in terms of an effective solution diffusivity and
excluded volume effects.
Our results are consistent with the experimental data on DNA
hairpin loop formation.
Since the polymer looping is an elementary process for other
biochemical processes, macromolecular crowding also affects
those processes, such as gene transcription.
.
.
Reference: J. Shin, A. G. Cherstvy, and R. Metzler, Soft Matter,
(2015).
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28. Part II: DNA loop formation in the presence of tension
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29. In an experiment
a small tension of 100 fN (=0.1 pN) scale can increase the looping
time about 10 times!! (Y. -F. Chen et al.(2010))
(Physics, Synopsis, February 1 2010)
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30. DNA loop formation in the presence of tension
Yellow: micron-sized bead, black: (ds)DNA, blue: DNA binding
protein (LacI)
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31. Model
DNA is modeled as bead-spring chains with bending energy
tension is applied to end monomers
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32. End-to-end distance distribution
F(r) = −kBT log(P(r))
Due to the high bending energy, P(r) has a peak near r ≈ L (chain
length).
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33. Looping time as function of tension
A minute tension(∼ 100 fN) greatly increases the looping time.
cf.) forces exerted by molecular motor≈ 10pN.
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34. Theoretical model: One-dimensional description of the
looping
Mean first-passage time from req to rf is,
T(req) =
req
rf
dr exp(βF(r))
1
Dee
L
r
dr exp(−βF(r )) (5)
In the presence of tension f , the free energy changes to
F(r) = F0(r) − fr
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35. Relative change of the looping time
MFPT in dimensionless form
DT(f )
L2
= Ψ(
fL
kBT
,
rf
L
,
lp
L
)
(6)
The looping time changes
more dramatically for longer
chain.
(symbols : simulation data, lines :
MFPT calculation) Two results are in
an excellent agreement.
cf.) kBT ≈ 4pN × nm (at room temperature)
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36. Comparison with the experiment
(symbol: experimental
data, line: theoretical
prediction)
.
.
We can explain the experimental data with our minimum model!
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37. Summary of Part II.
Due to the cooperativity of the chain, a minute tension can
dramatically change looping time, especially for longer chain.
Our minimum model can explain astonishing experimental
data.
This study suggests a cells might use tension to control gene
expression.
.
.
Reference: J. Shin and W. Sung, J. Chem. Phys. 126, 045101
(2012).
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38. Future work
Polymer looping has been studied for a few decades, but there are
still many interesting problems, including
polydisperse crowding molecules
internal loop formation
looping in a bath of active particles
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39. Other recent works
Semiflexible polymer loop inside a cavity
(A) (B) (C)
Φ 0
0
10
20
30
lp Σ
50
100
n
0.0
0.2
0.4
Pl
Φ 0
0
10
20
30
lp Σ
50
100
n
6
8
log Tl
The looping probability & looping time shows oscillating behavior
as function of chain length.
J. Shin, A. G. Chersty, R. Metzler, submitted to Macro lett. (2014).
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40. Other recent works
J. Shin, A. G. Chersty, R. Metzler, PRX (2014) : Sensing viruses by mechnical
tension of DNA
J. Shin, A. G. Chersty, R. Metzler, NJP (2014) : Segregation of Ring polymers
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41. Acknowledgment
Prof. W. Sung, Pohang University of Science and Technology
Prof. R. Metlzer, Universit¨at Potsdam
Dr. A.G. Cherstvy, Universit¨at Potsdam
.
Funding: Federal Ministry of Education and Research, Germany
(BMBF Project)
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