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Dynamics of DNA Ejection from Bacteriophage

Mandar M. Inamdar,* William M. Gelbart,yand Rob Phillips*z

*Division of Engineering and Applied Science,zKavli Nanoscience Institute, California Institute of Technology, Pasadena, California;

andyDepartment of Chemistry and Biochemistry, University of California, Los Angeles, California

ABSTRACT

translocation of a macromolecular chain along its length through a membrane. The simplest mechanism for this motion is

diffusion, but in the case of phage ejection a significant driving force derives from the high degree of stress to which the DNA is

subjected in the viral capsid. The translocation is further sped up by the ratcheting and entropic forces associated with proteins

that bind to the viral DNA in the host cell cytoplasm. We formulate a generalized diffusion equation that includes these various

pushing and pulling effects and make estimates of the corresponding speedups in the overall translocation process. Stress in

the capsid is the dominant factor throughout early ejection, with the pull due to binding particles taking over at later stages.

Confinement effects are also investigated, in the case where the phage injects its DNA into a volume comparable to the capsid

size. Our results suggest a series of in vitro experiments involving the ejection of DNA into vesicles filled with varying amounts

of binding proteins from phage whose state of stress is controlled by ambient salt conditions or by tuning genome length.

The ejection of DNA from a bacterial virus (i.e., phage) into its host cell is a biologically important example of the

INTRODUCTION

A crucial first step in the life cycle of most bacterial viruses

(i.e., phage) involves binding of the virion to a receptor

protein in the host cell membrane followed by injection of

the phage DNA. The viral genome is typically ;10 mm long,

and its translocation from outside to inside the host cell is

accomplished over times that vary from seconds to minutes.

The wide range of mechanisms responsible for injection of

phage genomes has recently been systematically reviewed

(1–3), including many references to the last few decades of

relevant literature. In this article, we formulate a general

theory of chain translocation that takes into account many of

the physical phenomena involved in actual phage life cycles.

These phenomena include: diffusion of the DNA chain along

its length; driving forces due to stress on the DNA inside the

viral capsid; resisting forces associated with osmotic pres-

sure in the host cell; cell confinement effects that constrain

the injected chain; and ratcheting and pulling forces asso-

ciated with DNA-binding proteins in the host cell cytoplasm.

Considerable effort has been focused on the energetics of

packaging and ejecting DNA in phage. In particular, the-

oretical work (4–11) has shown that the dominant source of

stress on the DNA in the capsid results from strong repulsive

interactions between neighboring portions of double helix

that are confined at average interaxial spacings as small as

2.5 nm. Another major contribution comes from the bending

stress that arises from the capsid radius being smaller than

the DNA persistence length. The force needed to package the

genome against this resistance is provided by a virally en-

coded motor protein that pushes in the DNA along its length.

Recent laser tweezer measurements (12) have confirmed that

this force increases progressively as packaging proceeds, i.e.,

as the chain becomes more crowded and bent, reaching

values as large as 50 pN upon completion. Conversely, the

force ejecting the DNA upon binding of the phage to its

membrane receptor has been shown (13,14) to decrease

monotonically from tens of picoNewtons to zero as crowding

and bending stress are progressively relieved. In this article,

we consider the dynamics of phage ejection and attempt to

distinguish the relative importance of these large, varying,

internal forces and the binding particles in the external so-

lution (bacterial cytoplasm).

It is useful at the outset to consider the simple diffusion

limit of the translocation process. More explicitly, consider

the case in which a chain is threaded through a hole in a

membrane dividing one solution from another. If the chain is

free, i.e., in the absence of pushing or pulling forces and of

binding particles, it will simply diffuse along its length, ex-

periencing a friction associated with its passage through the

membrane and the viscosity of the solution. The time

required for its translocation from, say, the left to the right

will be L2/2D ¼ td, where L is the length of the chain and D

is its effective translational diffusion coefficient.

Suppose now that particles are added to the right-hand

solution, which binds irreversibly to the chain at regularly

spaced sites as soon as they diffuse into the solution. Then, if

s is the spacing between these binding sites, the diffusion of

the chain will be ratcheted each time another length s has

entered the solution (15–17), corresponding to the fact that

the chain cannot move backward through the hole at a site

where a particle is bound. Accordingly, the time it takes

for the entire chain to appear on the right is simply given by

s2/2D—the time required for diffusion between a pair of

neighboring binding sites—times the total number of sites,

L/s. It follows that the overall translocation time in the

presence of perfect ratcheting is reduced by a factor of s/L

over that for free diffusion. When the binding of particles is

Submitted July 14, 2005, and accepted for publication March 15, 2006.

Address reprint requests to R. Phillips, E-mail: phillips@pboc.caltech.edu.

? 2006 by the Biophysical Society

0006-3495/06/07/411/10 $2.00

doi: 10.1529/biophysj.105.070532

Biophysical Journal Volume 91 July 2006 411–420411

Page 2

reversible—they do not remain bound indefinitely, thereby

allowing some sites to diffuse backward through the hole—

the translocation time is increased by a factor of (1 1 2K)

compared to perfect ratcheting, where K is the ratio of off-

and on-rates for particle binding (15,16). Finally, note that

the ideal ratcheting time of Ls/2D corresponds to a velocity

of 2D/s and hence, by the Stokes-Einstein relation, to a force

of 2 kBT/s pulling the chain into the particle-containing

solution (18).

When the particle binding is reversible, however, it turns

outthattherecanbeadifferentmechanismfromtheratcheting

dynamics, one that can significantly shorten the translocation

time below Ls/2D ¼ tidealratchet. This effect requires that the

diffusive motion of the chain is slow enough and is due to the

fact that the entropy of reversibly bound particles increases

when there is more chain for them to explore. As a result, the

entropy is an increasing function of chain length available in

the right-hand solution. Indeed, in the limit of fully equili-

brated binding, the system is equivalent to a one-dimensional

Langmuir adsorption problem (18,19) (P. G. de Gennes,

privatecommunication,2002;seealsoReversibleForcefrom

the Binding Proteins, this article) . More explicitly, the one-

dimensional Langmuir pressure can be written in the form

P1D¼ (kBT/s) ln f11 exp ((e 1 m)/kBT)g, where e . 0is the

energy lowering of the adsorbing particles upon binding and

m is their chemical potential in solution. Note that in the limit

of large binding energy ((e 1 m)/kBT ? 1), this pressure

reducessimplyto(e1m)/s,which—becausepressureisforce

in a one-dimensional system—can be directly interpreted as

the force pulling on the chain due to the reversible binding of

particles. Note further, in the large binding energy limit, that

thisforce isnecessarilylarge comparedtotheideal ratcheting

force, 2 kBT/s (18).

Ambjornsson and Metzler (19) have recently clarified the

various timescales that determine the different regimes of

chain translocation in the presence of chaperones, i.e., bind-

ing particles. The first, t0, is the time needed for the chain to

diffuse a distance of order s, the separation between binding

sites. The second and third are toccand tunocc, the char-

acteristic times that a binding site remains occupied and un-

occupied, respectively. The values toccand tunoccare related

by the equilibrium relation,

tocc

tunocc

¼ exp

e1m

kBT

??

:

(1)

Finally, tunocccan be approximated by the typical time it

takes for a particle to diffuse a distance of order Rð;c?1=3

between binding free particles,

0

Þ

tunocc¼R2

2D0’

1

D0c2=3

0

;

(2)

where D0is the diffusion coefficient of the particles, and c0is

their number density. One can then distinguish between three

different regimes:

1. Diffusive regime: t0? tunocc, tocc. Here, the binding

particles are irrelevant to the chain translocation because

the chain diffuses its full length in a time too short for the

particles to bind.

2. Irreversible binding regime: tunocc? t0? tocc. Here,

particles bind essentially irreversibly on a timescale short

compared to the time it takes for the chain to diffuse a

distance between binding sites. We shall refer to this as

the ratcheting regime.

3. Reversible binding regime: tunocc, tocc, ? t0. Here,

diffusion of the chain along its length is slow compared

to the time required for an on/off equilibrium of the bind-

ing particles to be achieved. We shall refer to this as the

Langmuir regime.

It is also important to clarify some relevant length scales

involved in the problem. Specifically, we distinguish be-

tween two extremes of how the separation, s, between bind-

ing sites compares with the range, d, of the attractive

interaction between binding particle and the chain. Pure and

perfect ratcheting will arise when tunocc? t0? tocc, in-

dependent of the relative values of d and s. Imperfect

ratcheting will arise when tunocc, tocc, ? t0, but d ? s. The

translocation time for the imperfect ratchet is higher than the

perfect ratchet by a factor of (1 1 2K). Finally, when tunocc,

tocc, ? t0and d ? s, we will have a Langmuir force acting

on the chain. Note that, if the binding free energy between

DNAandthebindingproteinsisverylarge,thenK?1.Also,

when the range of attraction d is comparable to the spacing

between the binding sites s, the reversible binding of the

proteins will result in a Langmuir force on the DNA chain. In

the rest of the article we use K ? 1 and d ? s. A schematic of

the role of these various effects is shown in Fig. 1.

Before proceeding further, it is instructive to make some

numerical estimates. Within this simple translocation model

all timescales are naturally referenced to that for pure

FIGURE 1

diffusion in the process of phage DNA ejection. The DNA cross-section is

not shown to scale: its diameter is 2–3 nm, as compared with a capsid radius

that is 10 times larger. The spring denotes schematically the stored energy

density resulting in a force F acting along the length L?x of chain remaining

in the capsid. The small spheres denote particles giving rise to an external

(cytoplasmic) osmotic pressure Posmotic, and the green particles labeled i

and i 1 1 are successive binding particles. (The schematic and the model

were inspired by Fig. 10.10 in (17).)

Schematicshowingthe various physical effectsthat assist bare

412Inamdar et al.

Biophysical Journal 91(2) 411–420

Page 3

translational diffusion of a chain along its length, and hence

to the diffusion coefficient D introduced earlier. In reality,

however, the DNA ejection process is enormously more

complicated, since the chain moving through the tail of the

phage is feeling not only the friction associated with the few

hydration layers surrounding it but also the viscous effects

arising from interaction with the inner surface of the tail just

nanometers away. Furthermore, this chain portion is con-

nected to the lengths of chain inside the capsid and outside in

the cell cytoplasm. The chain remaining inside the capsid

moves by reptating through neighboring portions of still-

packaged chain and/or by overall rotation of the packaged

chain. All of these latter motions involve viscous dissipation

that is insufficiently well-characterized to enable realistic

estimates of diffusion timescales, even though one can dis-

tinguish between the different dependences on chain length

for each of these dynamical processes (9,20). If the dominant

source of dissipation is due to the friction/attraction between

the DNA and the phage tail, the diffusion coefficient will be

independent of the amount of DNA ejected (21). On the

other hand, the diffusion coefficient D, in general, may

depend on the amount of DNA ejected. To keep the matters

simple,weassumethatitispossibletodefineaneffectivedif-

fusion coefficient D, and define the unit of time, td¼ L2/2D.

This way we can make predictions of how the ejection time-

scaleswiththegenomelength,forexample,withoutknowing

what the actual value of D is, although in the end it may be

found that this picture of diffusion is too simple and a length-

dependent diffusion coefficient will have to be involved.

A strong upper bound for D can be obtained by con-

sidering the part of the dissipation arising as the chain moves

through the tail portion of the virus. Taking into account only

the friction between the DNA and the fluid in the tail we

have, for example (20,22), z ¼ 2p lh/ln(D/d). Here z is the

friction coefficient, l is the length of the tail, h is the viscosity

of water, D is the inner diameter of the tail, and d is the di-

ameter of the double-stranded DNA. Taking l ¼100 nm, h¼

10?9pN – s/nm2, D ¼ 4 nm (23), and d ¼ 2 nm, we find

z ¼ 9 3 10?7pN – s/nm and hence a diffusion coefficient

(D ¼ kBT/z) of 5 3 106nm2/s. For a typical phage genome

length (L) of 10 mm, this in turn leads to a diffusional

translocation time (td¼ L2/2D) of ;10 s, not unlike ejection

times measured for phage l (24) (P. Grayson, private

communication, 2005). Recall, however, that this estimate is

based on a value for D that is a strong upper bound, because

of all the viscous dissipation contributions that were

neglected, suggesting that the actual unassisted diffusional

time is likely several orders-of-magnitude larger than this

10-s estimate. Indeed, the outcome of the work presented

below is that the translocation time is shortened beyond tdby

severalorders ofmagnitudebya combinationofeffectsdom-

inated by pressure in the capsid and binding particles in the

external solution. This simple estimate provides us with an

interesting insight into the dissipation mechanisms involved,

and suggests two possibilities:

1. The friction of water (and hence, dissipation) is much

larger at such short length scales.

2. As mentioned above, there are several other dissipation

mechanisms, which are not taken into account.

The outline of the article is as follows. In the next section

we include the effect of capsid pressure by formulating a

Fokker-Planck description of translocation driven by a com-

bination of diffusion and spatially varying force, i.e., a force

pushing the chain from one side that depends on the length of

chain remaining on that side (corresponding to the portion

still in the capsid and hence experiencing stress due to

crowding and bending). We evaluate the mean-first-passage-

time (MFPT) for translocation of an arbitrary length and

thereby calculate the length ejected as a function of time,

using estimates of the spatially varying ejection force from

recent theories of phage-packaging energetics. We find that

the translocation times are 2–3 orders-of-magnitude faster

than the diffusional time. We also treat the case of ejection

into a volume comparable to the capsid size (mimicking, say,

studies in which phage are made to eject into small vesicles

that have been reconstituted with receptor protein (25,26))

and find the dependence of ejection time on the relative sizes

of the phage capsid and the vesicle. In DNA Ejection in the

Presence of DNA Binding Proteins, we treat the further

speedup in translocation due to ideal ratcheting and the

Langmuir force arising from the reversible particle binding,

respectively. We find that the simple ratcheting effect is

small compared to that arising from the entropic force of

reversible particle binding. The effect of reversible particle

binding decreases the translocation time by another order of

magnitude beyond that due to capsid pressure effects. Fi-

nally, these particle binding effects are shown to be sufficient

to work against resistance forces due to external (i.e.,

cytoplasmic) pressure. In the final section (Discussion and

Conclusion), we conclude with a discussion of related work

by others, of additional contributions to ejection dynamics

that will be studied in future theoretical work (in particular,

the effect of RNA polymerase acting on the ejected DNA),

and of experiments planned to test the various predictions

made in this work.

KINETICS OF EJECTION DRIVEN BY

PACKAGING FORCE

As discussed in the Introduction, we focus here on a chain

that has been confined in a viral capsid and is ejected from it

through a hollow tail just big enough to accommodate its

diameter. To elucidate the essentials of this ejection process,

we describe the translocation of the chain as a diffusion-in-

a-field problem (21,27,28). In this case, involving the trans-

location of a linear polymer along its length, the diffusion

coordinate is a scalar, i.e., the length of chain x that has been

ejected from the tail of the virus. The external field is de-

scribed by the potential energy U(x) that gives rise to the

DNA Ejection from Bacteriophage 413

Biophysical Journal 91(2) 411–420

Page 4

force F(x) ¼ ?dU(x)/dx, pushing on the chain when a length

x of it has been ejected. This force is due to the remaining

chain length L?x being confined inside the capsid and

thereby subjected to strong self-repulsion (Urep) and bending

(Ubend). The corresponding potential U(x) ¼ Urep(L?x) 1

Ubend(L?x) is the free energy calculated in recent theories

of DNA packaging in viral capsids (7,8,11). This energy is

seen to decrease dramatically as ejection proceeds (i.e., as x

increases), and so does the magnitude of its slope that con-

stitutes the driving force for ejection.

The one-dimensional dynamics of a diffusing particle in

the presence of an external field is a classic problem in sto-

chastic processes (29), and, as argued above, can be tailored

to treat the translocation of phage DNA under the action of

an ejection force F(x) ¼ ?dU(x)/dx. Accordingly, the

probability p(x, t) of finding a length x ejected at time t is

given by the Fokker-Planck equation

?

As a part of this stochastic description of the translocation-

under-a-force process, it is natural to define a mean-first-

passage-time (MFPT), t(x), which gives the average time it

takes for a length x to be ejected in the presence of the

external field U(x), namely (30),

Zx

It is useful to consider several limits of this general equa-

tion, the first corresponding to the familiar case of no exter-

nal field. From U [ 0, the integrals in Eq. 4 reduce trivially

to x2/2D, giving the expected diffusion time, t(x) ¼ x2/2D.

For the case of constant force, i.e., U ¼ ?Fx 1 constant,

the integrals in Eq. 4 can also be evaluated analytically,

giving (16)

@pðx;tÞ

@t

¼@

@x

D@pðx;tÞ

@x

1D

kBT

@UðxÞ

@x

pðx;tÞ

?

:

(3)

tðxÞ ¼1

D

0

dx1exp ?Uðx1Þ

kBT

? ?Zx

x1

dx2exp

Uðx2Þ

kBT

??

: (4)

tConstantForceðxÞ ¼x2

D

exp½?bFx?1bFx ? 1

ðbFxÞ2

:

(5)

Here we have written b for 1/kBT, and taken F ¼ ?dU(x)/

dx . 0 to denote the constant force driving translocation of

the chain to the right (see Fig. 1). In DNA Ejection in the

Presence of DNA Binding Proteins, we will apply Eq. 5

locally, over each segment of length s associated with a

binding site, to calculate the ideal ratcheting corrections to

force-driven translocation. Note that simple and ratcheted

diffusion are overwhelmed by force-driven translocation

when bFL ? 1 and bFs ? 1, respectively.

In the most general instance of spatially varying external

field U(x), as in the case of capsid-pressure-driven translo-

cation, the integrals in Eq. 4 must be evaluated numerically.

In this way we calculate t(x) from Eq. 4 for the U(x)

determined from a recent treatment (7,11) of the packaging

energetics in phage capsids. This provides a one-to-one cor-

respondence between each successive time t(x) and the frac-

tion of chain ejected x(t)/L at that instant.

In Purohit et al. (7,11), the shape of the l-phage capsid is

approximated as spherical, and the DNA inside the capsid is

assumed to be organized in a hexagonally packed inverse-

spool. The potential U(x) is expressed as a combination of

the bending energy and the repulsive interaction between the

DNA strands, and is given by

UðxÞ ¼ UrepðL ? xÞ1UbendðL ? xÞ

¼

12pkbTj

ffiffiffi

The values F0and c are experimentally determined con-

stants (31) describing the interaction between neighboring

DNA strands, j is the persistence length of DNA, d is the

interstrand spacing, Routand Rinare the radius of the capsid

and the inner radius of the DNA spool, respectively, and N(r)

is the number of hoops of DNA at a distance r from the spool

axis. We are interested in finding the internal force on the

phage genome as a function of genome length inside the

capsid. We do so using Eq. 6 and simple geometrical con-

straints on the phage genome inside the capsid. The number

of loops N(r) in Eq. 6 is given by z(r)/d, where

zðrÞ ¼ ðR2

r from the central axis of the DNA spool. The actual volume

available for the DNA—V(Rin, Rout)—can be related to the

genome length L?x in the capsid, and the interstrand spacing

d, giving an expression for Rinin terms of d, Rout, and L?x.

This relation can be substituted for Rinin Eq. 6, which then

can be minimized with respect to d to give the equilibrium

interstrand spacing as a function of the genome length L?x

inside the capsid. In this way we determine the total packing

energy as a function of genome length inside the capsid

(L?x) or as a function of the DNA length ejected x, i.e., U(x).

Using this result and Eq. 4, we can evaluate the MFPT, t(x),

for the DNA ejection in l as a function of the length ejected.

The corresponding fraction ejected, x(t)/L, is shown as a

function of time in Fig. 2, with the label ‘‘no confinement’’;

note that time here is measured in units of L2/D.

The value of D can be estimated on the basis of this simple

model by the following procedure. The experiment by

Novick and Baldeschwieler (24) showed that in a buffer

containing 10 mM of Mg12it took ;50 s for phage l to

completely eject its genome. The values for F0and c in

buffers containing Mg21have been measured (31). Since the

values measured for 5 mM and 25 mM Mg21were not

significantly different, we assume that the forces at 10 mM

will be identical, i.e., F0¼ 12,000 pN/nm2and c ¼ 0.3 nm.

Using these values in Eq. 4 and numerically evaluating it for

x ¼ L ¼ 48,500 3 0.34 nm, we find the total time for l to

eject its genome of 48.5 kbp is t ? (105nm2/D) s. Then, since

this value is experimentally estimated to be ;50 s (24), we

infer that D ? 103nm2/s. This is approximately three orders-

of-magnitude smaller than the D estimated in Introduction,

consistent with all the sources of dissipation that were left

ffiffiffi

3

p

F0ðL ? xÞðc21cdÞexpð?d=cÞ

p

d

Rin

r

3

ZRout

NðrÞ

dr:

(6)

out? r2Þ1=2is the height of the capsid at distance

414 Inamdar et al.

Biophysical Journal 91(2) 411–420

Page 5

out of that estimate. Note that once the parameter D has been

fixed, there are no other free parameters in the model and

hence all further deductions from the model are predictive.

As will be shown below, the model developed thus far

predicts the dynamics of in vitro ejection of phage DNA into

lipid bilayer vesicles.

An interesting application of our estimates is to experi-

ments in which viruses eject their DNA into lipid vesicles

(2,24,26,32). Here lipid vesicles are reconstituted with the

receptors recognized by the phage of interest, and then mixed

with a solution of the phage. The phage binds to the receptor

and ejects its DNA into the vesicle. We argue that the amount

of DNA ejected into the vesicle and the corresponding time

dependsontheradiusofthevesicle.Inparticular,ifthevesicle

hasaradiuscomparabletothatoftheviralcapsid,therewillbe

a buildup of pressure inside the vesicle due to the ejected

DNA. Ultimately, the ejection process will come to a halt

when the force on the DNA from the capsid equals the force

from the vesicle side—this can be thought of similarly from

the free energy perspective as a free-energy minimizing con-

figuration.Hence,theejectionwillnot,ingeneral,becomplete.

We can work out the ejection rate for this process as

follows. If x is the length of genome ejected into the vesicle,

we denote the free energies of the DNA inside the viral

capsid and the vesicle by Ucapsid(L?x) and Uvesicle(x), re-

spectively. The total free energy will be given by

UðxÞ ¼ UcapsidðL ? xÞ1UvesicleðxÞ:

As explained before, we already know Ucapsid(L?x) (see

Eq. 6); the expression for Uvesicle(x) can be obtained similarly

by assuming that the vesicle is like a spherical capsid and the

DNA configuration inside is similar to that inside the viral

capsid. Our assumed structure for the DNA in the vesicle is a

highly idealized model, though we note that electron micros-

copy on such vesicles demonstrates that DNA within them

(7)

can adapt to highly ordered configurations (26). In the limit

where the vesicle radius is large compared to that of the

phage capsid we will recover the free injection result (DNA

ejecting from phage into the surrounding solution).

The injection process will stop when the total free energy

reaches a minimum, i.e., the total force on the DNA is zero.

The predicted time for DNA injection is given by Eq. 4. We

have worked out the kinetics of the ejection for bacterio-

phage l (radius ? 29 nm) ejecting its genome into vesicles of

radius 29, 50, and 100 nm. The phage is taken to be sus-

pended in a solution of Mg12ions, and similarly the vesicle,

with concentration that approximately gives the same values

for F0and c, as discussed earlier. This yields a prediction for

the kinetics of injection for different vesicle radius. It can be

seen from Fig. 2 that when the size of the vesicle is com-

parable to the capsid size there is only a partial ejection of the

DNA. When the vesicle size is almost twice the size of the

capsid, nearly the entire genome is ejected, except for the last

part of the DNA, which takes extra time because of the re-

sistance offered to it from the DNA inside the vesicle.

Finally, when the vesicle is more than three times the size of

the capsid, DNA gets completely ejected from the phage

capsid as if there were no vesicle. It is interesting to note that

in the initial stages of ejection, all the curves for various

vesicle sizes fall on one another because there is no resis-

tance to theinjection, but asthe ejection proceeds,eachcurve

reflects a different resistance.

It is also possible that the arguments given above for in

vitro ejection into vesicles could be relevant to thinking

about ejection into the crowded environment of a bacterial

cell (33,34). As a result of the crowding within the host

bacterium, the viral DNA may be subject to confinement

effects like those induced by vesicles.

DNA EJECTION IN THE PRESENCE OF DNA

BINDING PROTEINS

The Escherichia coli cell has as many as 250 types of DNA

binding proteins (35). Some fraction of these proteins likely

binds either specifically or nonspecifically to the phage

genome as it enters the host bacterium. Accordingly, we

consider what happens if the phage DNA is swarmed with

binding proteins upon its entry into the host cell. Depending

on the binding on/off rates, binding site density, and the

strength of binding, we have a corresponding speedup of the

DNA injection into the bacterial cell, relative to the pure

force-driven case. In this section we explore this effect and

see how, in addition to the speedup, it helps the phage inject

its DNA against the osmotic pressure in the host cell.

Throughout the following analysis of particle binding

effects, we assume that the chain is stiff on length scales

(e.g., tens of nanometers for double-stranded DNA genomes)

large compared to the size of the relevant binding particles

(typically a few nanometers). We also assume that the bind-

ing particles are comparable in size to the distance between

FIGURE 2

of radius 29, 50, and 100 nm. The capsid radius of the phage is 29 nm. It can

be seen that the amount of DNA injection increases as the ratio of the vesicle

radius to the capsid radius increases. On the timescale depicted here, there

will be essentially no ejection due to pure diffusion (which takes place

instead at times of order 1, in units of L2/D).

Ejection time for phage-l injecting its genome into vesicles

DNA Ejection from Bacteriophage 415

Biophysical Journal 91(2) 411–420