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
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
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: email@example.com.
? 2006 by the Biophysical Society
Biophysical JournalVolume 91 July 2006411–420 411
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
When the particle binding is reversible, however, it turns
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,
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
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,
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,
where D0is the diffusion coefficient of the particles, and c0is
their number density. One can then distinguish between three
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
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
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
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
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