chain translocation in a biological context

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CHAIN TRANSLOCATION IN A BIOLOGICAL CONTEXT

INJECTING VIRAL GENOMES INTO HOST CELLS

Roya Zandi Mandar Inamdar

David Reguera Rob Phillips

Joe Rudnick

[CHUCK KNOBLER]

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ANIMAL CELL ENTRY

*Virus gets in by binding to receptor in cell membrane*Whole viral particle enters cell*Virus forms and exits via budding

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*Cells are each surrounded by a rigid (cellulose) wall, which must be “broken” (e.g., by abrasion) in order for viral particles to enter *Consequently, a large number of viral particles enter the cell

simultaneously, where they are disassembled and replicated *New virions leave cell through existing shared-wall channels

Plasmodesmata (shared-wall channels)

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BACTERIAL CELL INFECTION BY VIRUS * Virus binds to receptor and ejects genome *Viral particle stays outside cell! Only its genome enters *Virion leaves via lysis of cell

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104 kBT 10 pN

01.00.5 0

0 0 0.5 1.0

U -(dU/dx)=f

x/L x/L

Bacteriophage Its dsDNA genome, 17000 nm long, is highly stressed in its capsid (30 nm radius), due to: Electrostatic Repulsion DNA is packed at crystalline density and is highly crowded Bending Energy Persistence length, 50 nm, implies DNA is strongly bent

30 nm

Can calculate energy (U) of DNA as a function of length (L-x) inside

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Inte

rnal

For

ce, p

N

Percentage of genome ejected

Osmotic Force:

There is a an opposing force, resisting entry of the chain into the cell, equal to the work per unit length that must be done against the osmotic pressure () in the cell

0 30 60

50

20

0fosmotic

This internal force drives the genome out along its length. But, it falls sharply as ejection proceeds, and…

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EXPERIMENT: COUNTERBALANCE EJECTION FORCE BY ESTABLISHING

AN EXTERNAL OSMOTIC PRESSURE

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

feject = fresist

Capsid permeable to H2O

and to ions, but not to PEG

Measure DNA concentration by 260-nm absorption-- but must distinguish DNA ejected from that remaining in capsid

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Experimental Design

Phages (sedimenting material)

Ejected/digested DNA + PEG

v

v

Spin down phage by centrifugation

PEG8000

PhagesAnd nuclease (not shown explicitly)

Ejected/digested DNA nucleotides

Add receptor

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Evilevitch, Lavelle, Raspaud, Knobler and Gelbart Proc. Nat. Acad. Sci. (USA) 100, 9292 (2003).

UV absorbance of DNA ejected from phage as a function of PEG8000 concentration.

PEG

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Extent of Ejected DNA vs Osmotic Pressure in Solution

020406080100

0 4 8 121620Osmotic pressure of the external solution, atm

Percent of DNA ejected from the phage capsids, %

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3-4 atms

ONLY PART OF GENOME IS DELIVERED TO HOST CELL?!

EFFECT OF GENOME LENGTH ON EJECTION FORCE

P. Grayson M. Inamdar P. Purohit R. Phillips

A. Evilevitch C. M. Knobler W. M. Gelbart

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Stiff chain of length L is “threaded” into a solution of particles that can bind to it at sites separated by distance s; chain diffusion constant is Drod

€

↔s

€ L€

→

TRANSLOCATION (DIFFUSION) INVOLVING PARTICLE BINDING, AND…RATCHETING

€

→

€

←

Binding particles interact with sites on chain via LJ potential (=s, in this case)

mimic of bacterial cytoplasm

viral capsid

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€

τ diffusion =L2

Drod

τ ideal ratchet =L

s

s2

Drod

=Ls

Drod

(=s

Lτ diffusion )

Can then introduce a "ratcheting velocity", v :

τ ratchet =L

vratchet

=L

(Drod /kBT) f=

Ls

Drod

, and hence

fratchet =kBT

s

SUPPPOSE BINDING PARTICLES STICK IRREVERSIBLY AT EACH ENTERING SITE…

[G. Oster et al.]

14€

Q : What is the force exerted by reversibly

binding (ε) particles (volume fraction φ)?

A : (P. - G. de Gennes) The 1D Langmuir pressure!

freversible = Π1D Langmuir =kBT

sln[1+ φeε / kBT ]

ε >>kBT ⏐ → ⏐ ⏐ ε

s>>

kBT

s= f ratchet

BUT, OTHERWISE…

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Rigid rod (with black monomers) of length L moves distance x into cell

(radius Rs) containing N binding particles

Brownian Molecular Dynamics (MBD)

MORE GENERAL TREATMENT OF TRANSLOCATION…

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f(kBT/s)

x(s)

The filled squares show the force calculated directly in the MBD

simulation, for 2Rs=24, L=16, N=100, /kBT=5, and Drod=Do/16;

the open circles show the same for Drod 60 times smaller.

Solid curve is computed from the full, coupled, equations for chain diffusion in the presence of binding particles; dashed curve is obtained from assumption of fast equilibration of particle binding.

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€

A(x,n)Langmuir

= −nε − kBT log(6x /s)!

n!(6x /s − n)!− kBT(N − n)log

V

(N − n)v

∂ρ(x,n, t)

∂t=

∂

∂xDrod (

1

kBT

∂A(x,n)

∂xρ +

∂ρ

∂x)

+∂

∂nDn (

1

kBT

∂A(x,n)

∂nρ +

∂ρ

∂n), Dn ≅ σ

N

VDo

∂ρ (x, t)

∂t=

∂

∂xDrod (

f (x)

kBTρ +

∂ρ

∂x)

f (x) = dn∂A(x,n)

∂x∫ exp(−A(x,n) /kBT)

dnexp(−A(x,n) /kBT)∫

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Fast equilibration of particle binding

€

⇒

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f(kBT/s)

x(s)

Dashed curve is obtained from solution to the quasi-equilibrium equation for (x,t); solid curve is computed by solving the full, coupled, diffusion equation for (x,n,t).

The filled squares show the force calculated directly in the MBD

simulation, for 2Rs=24, L=16, N=100, /kBT=5, and Drod=Do/16;

the open circles show the same for Drod 60 times smaller.

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€

∂(x, t)

∂t=

∂

∂xDrod (

dU /dx)

kBTρ (x, t) +

∂ρ (x, t)

∂x)

t(x) =1

Drod

dx10

x

∫ exp(−U(x1)

kBT) dx2 exp(

U(x2)

kBT)

x1

x

∫

tconstant force(x; f ) =x 2

Drod

exp(− fx /kBT) + fx /kBT −1

( fx /kBT)2

t(x)U(x)+ratchet = tconstant force(s; f i =dU

dxi=1

x / s

∑ x= is)

t(x)⇒ x(t)⇒x(t)

Ltot

≡ fraction ejected

TRANSLOCATION, INCLUDING PUSHING AND PULLING FORCES

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104 kBT 10 pN

01.00.5 0

0 0 0.5 1.0

U -(dU/dx)=f

x/L x/L

RECALL THATDNA is packed at crystalline density and is highly crowded, hence involving

a large energy of self-repulsion AND because its persistence length is larger than the capsid size, a significant

bending energy is also involved

ENERGY ‘COST” (U) IS RELIEVED AS EJECTED LENGTH (x) INCREASES

€

U(x) =Urepulsion (L − x) + Ubending (L − x)

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EFFECT OF RATCHET ON U(x)

Langmuir

0.5 x 10-4 1 (L2/D) 4 6Internal force + Langmuir

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EFFECT OF LANGMUIR FORCE ON U(x) -- ADD /s TO fi’s:

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EFFECT OF OSMOTIC FORCE -- SUBTRACT fosmotic FROM fi’s:

€

f ratchet < ( fosmotic:3 atm ≈1pN) < fLangmuir (<< f internalmax )

driving force drops below 1pN when fraction ejected reaches 50%

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€

on ⇔ off , ΔG = ε + kT lnφ > 0

K =[off ]

[on]= e−βΔG =

1

φe−βε =

koff

kon

(<1)

BINDING/UNBINDING (ON/OFF) EQUILIBRIUM

COMPETING TIME SCALES FOR TRANSLOCATION

€

(1

kon

↔)τ off ≈(average particle spacing)2

Do

≈1

(N /V )2 / 3 Do

(1

koff

↔)τ on =1

Kτ off > τ off

Third time scale is τ diff ≡s2

Drod

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diffusion ratcheting pulling

τoff τon

WHERE IS τdiff=s2/Drod ON THE TIME SCALE OF BINDING/UNBINDING?

€

diffusion : τ diff < τ off < τ on

τ trans =L2

Drod

, F = 0

ratcheting : τ off < τ diff < τ on

τ trans =Ls

Drod

≡ τ ratchet , F =kBT

s

pulling : τ off < τ on < τ diff

τ trans =Ls /Drod

ln(1+ φeβΔG )< τ ratchet , F =

kBT

sln(1+ φeβΔG ) ≈

ΔG

s>>

kBT

s

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FUTURE WORK: GENOME EJECTION -- PHAGE

Investigate effects on injection, of:

internal osmotic pressure (all) DNA-binding proteins (e.g., T5) RNA polymerase (e.g., T7)

Build mimics of the bacterial cell, i.e., reconstituted vesicles -- either lipid bilayers or A-B-A block copolymers

Complement with single-cell, in vivo, studies, monitoring -- in real time -- the entry of the viral genome into bacterial cytoplasm [P. Grayson, R. Phillips]

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L. T. Fang, C. M. Knobler, W. M. Gelbart

+ DNA-binding proteins…