control of cellular information processes (Ingber, 2003b). The biological ground for
these applications has already been laid (Ingber 2003a; 2003b). It is the task of
bioengineers to carry on this work further.
I thank Drs. D. E. Ingber, N. Wang, M. F. Coughlin, and J. J. Fredberg for their
collaboration and support in the course of my research of cellular mechanics. Special
thanks go to Drs. Ingber and Wang for critically reviewing this chapter.
This work was supported by National Heart, Lung, and Blood Institute Grant
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7 Cells, gels, and mechanics
Gerald H. Pollack
The cell is known to be a gel. If so, then a logical approach to the understanding
of cell function may be through an understanding of gel function. Great strides have been
made recently in understanding the principles of gel dynamics. It has become clear that a
central mechanism in biology is the polymer-gel phase-transition ā“ a major structural change
prompted by a subtle change of environment. Phase-transitions are capable of doing work, and
such mechanisms could be responsible for much of the work of the cell. Here, we consider this
approach. We set up a polymer-gel-based foundation for cell function, and explore the extent
to which this foundation explains how the cell generates various types of mechanical motion.
The cell is a network of biopolymers, including proteins, nucleic acids, and sugars,
whose interaction with solvent (water) confers a gel-like consistency. This revelation
is anything but new. Even before the classic book by Frey-Wyssling a half-century
ago (Frey-Wyssling, 1953), the cytoplasmā™s gel-like consistency had been perfectly
evident to any who ventured to crack open a raw egg. The āgel-solā transition as
a central biological mechanism is increasingly debated (Jones, 1999; Berry, et al.,
2000), as are other consequences of the cytoplasmā™s gel-like consistency (Janmey,
et al., 2001; Hochachka, 1999). Such phenomena are well studied by engineers,
surface scientists, and polymer scientists, but the fruits of their understanding have
made little headway into the biological arena.
Perhaps it is for this reason that virtually all cell biological mechanisms build on
the notion of an aqueous solution ā“ or, more speciļ¬cally, on free diffusion of solutes
in aqueous solution. One merely needs to peruse representative textbooks to note the
many diffusional steps required in proceeding from stimulus to action. These steps
invariably include: ions diffusing into and out of membrane channels; ions diffusing
into and out of membrane pumps; ions diffusing through the cytoplasm; proteins
diffusing toward other proteins; and substrates diffusing toward enzymes, among
others. A cascade of diffusional steps underlies virtually every intracellular process,
notwithstanding the cytoplasmā™s character as a gel, where diffusion can be slow enough
to be biologically irrelevant. This odd dichotomy between theory and evidence has
grown unchecked, in large part because modern cell biology has been pioneered
130 G.H. Pollack
by those with limited familiarity with gel function. The gel-like consistency of the
cytoplasm has been largely ignored. What havoc has such misunderstanding wrought?
Problems with the aqueous-solution-based paradigm
Consider the consequences of assuming that the cytoplasm is an aqueous solution. To
keep this āsolutionā and its solutes constrained, this liquid-like milieu is surrounded
by a membrane, which is presumed impervious to most solutes. But solutes need to
pass into and out of the cell ā“ to nourish the cell, to effect communication between
cells, to exude waste products, etc. In order for solutes to pass into and out of the cell,
the membrane requires openable pores. Well over 100 solute-speciļ¬c channels have
been identiļ¬ed, with new ones emerging regularly.
The same goes for membrane pumps: Because ion concentrations inside and outside
the cell are rarely in electrochemical equilibrium, the observed concentration gradients
are thought to be maintained by active pumping mediated by speciļ¬c entities lodged
within the membrane. The text by Alberts et al. (1994) provides a detailed review of
this foundational paradigm, along with the manner in which this paradigm accounts
for many basics of cell function. In essence, solute partitioning between the inside
and the outside of the cell is assumed to be a product of an impermeable membrane,
membrane pumps, and membrane channels.
How can this foundational paradigm be evaluated?
If partitioning requires a continuous, impermeant barrier, then violating the barrier
should collapse the gradients. Metabolic processes should grind to a halt, enzymes
and fuel should dissipate as they diffuse out of the cytoplasm, and the cell should be
quickly brought to the edge of death.
Does this really happen?
To disrupt the membrane experimentally, scientists have concocted an array of
implements not unlike lances, swords, and guns:
r Microelectrodes. These are plunged into cells in order to measure electrical po-
tentials between inside and outside or to pass substances into the cell cytoplasm.
The microelectrode tip may seem diminutive by conventional standards, but to
the 10-Āµm cell, invasion by a 1-Āµm probe is roughly akin to the reader being
invaded by a fence post.
r Electroporation is a widely used method of effecting material transfer into a cell.
By shotgunning the cell with a barrage of high-voltage pulses, the membrane
becomes riddled with oriļ¬ces large enough to pass genes, proteins, and other
macromolecules ā“ and certainly large enough to pass ions and small molecules
r The patch-clamp method involves the plucking of a 1-Āµm patch of membrane
from the cell for electrophysiological investigation; the cell membrane is grossly
Although such insults may sometimes inļ¬‚ict fatal injuries they are not, in fact,
necessarily consequential. Consider the microelectrode plunge and subsequent with-
drawal. The anticipated surge of ions, proteins, and metabolites might be thwarted if
the hole could be kept plugged by the microelectrode shank ā“ but this is not always
Cells, gels, and mechanics 131
the case. Micropipettes used to microinject calcium-sensitive dyes at multiple sites
along muscle cells require repeated withdrawals and penetrations, each withdrawal
leaving multiple micron-sized injuries. Yet, normal function is observed for up to sev-
eral days (Taylor et al., 1975). The results of patch removal are similar. Here again, the
hole in question is more than a million times the cross-section of the potassium ion.
Yet, following removal of the 1-Āµm patch, the 10 Āµm isolated heart cell is commonly
found to live on and continue beating (L. Tung and G. Vassort, personal communi-
Similarly innocuous is the insult of electroporation. Entry of large molecules into
the cell is demonstrable even when molecules are introduced into the bath up to
many hours after the end of the electrical barrage that creates the holes (Xie et al.,
1990; Klenchin et al., 1991; Prausnitz et al., 1994; Schwister and Deuticke, 1985;
Serpersu et al., 1985). Hence, the pores must remain open for such long periods
without resealing. Nor do structural studies in muscle and nerve cells reveal any
evidence of resealing following membrane disruption (Cameron, 1988; Krause et al.,
1984). Thus, notwithstanding long-term membrane oriļ¬ces of macromolecular size ā“
with attendant leakage of critical-for-life molecules anticipated ā“ the cell does not
Now consider the common alga Caulerpa, a single cell whose length can grow to
several meters. This giant cell contains stem, roots, and leaves in one cellular unit
undivided by any internal walls or membranes (Jacobs, 1994). Although battered by
pounding waves and gnawed on by hungry ļ¬sh, such breaches of integrity do not
impair survival. In fact, deliberately cut sections of stem or leaf will grow back into
entire cells. Severing of the membrane is devoid of serious consequence.
Yet another example of major insult lies within the domain of experimental genetics,
where cells are routinely sectioned in order to monitor the fates of the respective
fragments. When cultured epithelial cells are sectioned by a sharp micropipette, the
nonnucleated fraction survives for one to two days, while the nucleated, centrosome-
containing fraction survives indeļ¬nitely and can go on to produce progeny (Maniotis
and Schliwa, 1991). Sectioned muscle and nerve cells similarly survive (Yawo and
Kuno, 1985; Casademont et al., 1988; Krause et al., 1984), despite the absence of
membrane resealing (Cameron, 1988; Krause et al., 1984).
Finally, and perhaps not surprisingly in light of all that has been said, ordinary
cells in the body are continually in a āmembrane-woundedā state. Cells that suffer
mechanical abrasion in particular ā“ such as skin cells, gut endothelial cells, and muscle
cells ā“ are especially prone to membrane wounds, as conļ¬rmed by passive entry into
the cell of large tracers that ordinarily fail to enter. Yet such cells appear structurally
and functionally normal (McNeil and Ito, 1990; McNeil and Steinhardt, 1997). The
fraction of wounded cells in different tissues is variable. In cardiac muscle cells it is
ā¼20 percent, but the fraction rises to 60 percent in the presence of certain kinds of
performance-enhancing drugs (Clarke et al., 1995). Thus, tears in the cell membrane
occur commonly and frequently even in normal, functioning tissue, possibly due to
Evidently, punching holes in the membrane does not wreak havoc with the cell,
even though the holes may be monumental in size relative to an ion. It appears we
are stuck on the horns of a dilemma. If a continuous barrier envelops the cell and
132 G.H. Pollack
is consequential for function, one needs to explain why breaching the barrier is not
more consequential than the evidence seems to indicate. On the other hand, if we
entertain the possibility that the barrier may be noncontinuous, so that creating yet
another opening makes little difference, we then challenge the dogma on which all
mechanisms of cell biological function rest, for the continuous barrier concept has
Is there an escape?
If the cytoplasm is not an aqueous solution after all, then the need for a continuous
barrier (with pumps and channels) becomes less acute. If the cytoplasm is a gel,
for example, the membrane could be far less consequential. This argument does not
imply that the membrane is absent ā“ only that its continuity may not be essential for
function. Such an approach could go a long way toward explaining the membrane-
breach anomalies described above, for gels can be sliced with relative impunity.
Major insults might or might not be tolerable by the gel-like cell, depending on the
nature of the insult and the degree of cytoplasmic damage inļ¬‚icted. Death is not
obligatory. A continuous barrier is not required for gel integrity, just as we have seen
that a continuous barrier is not required for cell integrity. A critical feature of the
cytoplasm, then, may be its gel-like consistency.
Cells as gels
Gels are built around a scaffold of long-chain polymers, often cross-linked to one
another and invested with solvent. The cytoplasm is much the same. Cellular polymers
such as proteins, polysaccharides, and nucleic acids are long-chained molecules,
frequently cross-linked to one another to form a matrix. The matrix holds the solvent
(water) ā“ which is retained even when the cell is demembranated. āSkinnedā muscle
cells, for example, retain water in the same way as gels. Very much, then, the cytoplasm
resembles an ordinary gel ā“ as textbooks assert.
How the gel/cell matrix holds water is a matter of some importance (Rand et al.,
2000), and there are at least two mechanistic possibilities. The ļ¬rst is osmotic: charged
surfaces attract counter-ions, which draw in water. In the second mechanism, water-
molecule dipoles adsorb onto charged surfaces, and subsequently onto one another
to form multilayers. The ļ¬rst mechanism is unlikely to be the prevailing one because:
(1) gels placed in a water bath of sufļ¬cient size should eventually be depleted of the
counter-ions on which water retention depends, yet, the hydrated gel state is retained;
and (2) cytoplasm placed under high-speed centrifugation loses ions well before it
loses water (Ling and Walton, 1976).
The second hypothesis, that charged surfaces attract water dipoles in multilayers, is
an old one (Ling, 1965). The thesis is that water can build layer upon layer (Fig. 7-1).
This view had been controversial at one time, but it has been given support by several
groundbreaking observations. The ļ¬rst is the now-classical observation by Pashley
and Kitchener that polished quartz surfaces placed in a humid atmosphere will adsorb
ļ¬lms of water up to 600 molecular layers thick (Pashley and Kitchener, 1979); this
implies adsorption of a substantial number of layers, one upon another. The second set
of observations are those of Israelachvili and colleagues, who measured the force re-
quired to displace solvents sandwiched between closely spaced parallel mica surfaces
Cells, gels, and mechanics 133
Fig. 7-1. Organization of water molecules adjacent to charged surface.
(Horn and Israelachvili, 1981; Israelachvili and McGuiggan, 1988; Israelachvili and
WennerstrĀØ m, 1996). The overall behavior was largely classical, following DLVO the-
ory. However, superimposed on the anticipated monotonic response was a series of
regularly spaced peaks and valleys (Fig. 7-2). The spacing between peaks was always
equal to the molecular diameter of the sandwiched ļ¬‚uid. Thus, the force oscillations
appeared to arise from a layering of molecules between the surfaces.
Although the Israelachvili experiments conļ¬rm molecular layering near charged
surfaces, they do not prove that the molecules are linked to one another in the manner
implied in Fig. 7-1. However, more recent experiments using carbon-nanotube-tipped
AFM probes approaching ļ¬‚exible monolayer surfaces in water show similar layering
(Jarvis et al., 2000), implying that the ordering does not arise merely from pack-
ing constraints; and, the Pashley/Kitchener experiment implies that many layers are
Fig. 7-2. Effect of separation on force between closely spaced mica plates. Only the oscillatory part
of the response is shown. After Horn and Israelachvili, 1981.
134 G.H. Pollack
Fig. 7-3. Structured water dipoles effectively āglueā charged surfaces to one another.
possible. Hence, the kind of layering diagrammed in Fig. 7-1 is collectively implied
by these experiments.
When two charged polymeric surfaces lie in proximity of one another, the inter-
facial water layers can bond the surfaces much like glue (Fig. 7-3). This is revealed
in common experience. Separating two glass slides stacked face-to-face is no prob-
lem; when the slides are wet, however, separation is formidable: sandwiched water
molecules cling tenaciously to the glass surfaces and to one another, preventing sep-
aration. A similar principle holds in sand. A foot will ordinarily sink deeply into dry
sand at the beach, leaving a large imprint, but in wet sand, the imprint is shallow.
Water clings to the sand particles, bonding them together with enough strength to
support oneā™s full weight.
The picture that emerges, then, is that of a cytoplasmic matrix very much resembling
a gel matrix. Water molecules are retained in both cases because of their afļ¬nity for the
charged (hydrophilic) surfaces and their afļ¬nity for one another. The polymer matrix
and adsorbed water largely make up the gel. This explains why demembranated cells
retain their integrity.
Embodied in this gel-like construct are many features that have relevance for cell
function. One important one is ion partitioning. The prevailing explanation for the ion
gradients found between extracellular and intracellular compartments lies in a balance
between passive ļ¬‚ow through channels and active transport by pumps. Thus, the low
sodium concentration inside the cell relative to outside is presumed to arise from the
activity of sodium pumps, which transport sodium ions against their concentration
gradient from the cytoplasm across the cell membrane. The gel construct invites an
alternative explanation. It looks toward differences of solubility between extracellular
bulk water and intracellular layered, or āstructuredā water, as well as differences of
afļ¬nity of various ions for the cellā™s charged polymeric surfaces (Ling, 1992). Na+
has a larger hydrated diameter than K+ , and is therefore more profoundly excluded
from the cytoplasm than K+ ; and, because the hydration layers require more energy to
remove from Na+ than from K+ , the latter has higher afļ¬nity for the cellā™s negatively
charged polymeric surfaces. Hence, the cytoplasm has considerably more potassium
Cells, gels, and mechanics 135
than sodium. A fuller treatment of this fundamental biological feature is given in the
recent book by the author (Pollack, 2001).
Similarly, the gel construct provides an explanation for the cell potential. The cell is
ļ¬lled with negatively charged polymers. These polymers attract cations. The number
of cations that can enter the cell (gel) is restricted by the cationsā™ low solubility in
structured water. Those cations that do enter compete with water dipoles for the cellā™s
ļ¬xed anionic charges. Hence, the negative charge in the cell is not fully balanced by
cations. The residual charge amounts to approximately 0.3 mol/kg (Wiggins, 1990).
With net negative charge, the cytoplasm will have a net negative potential. Indeed,
depending on conditions, membrane-free cells can show potentials as large as 50 mV
(Collins and Edwards, 1971). And gels made of negatively charged polymers show
comparable or larger negative potentials, while gels built of positively charged poly-
mers show equivalent positive potentials (Fig. 7-4). Membranes, pumps, and channels
evidently play no role.
Hence, the gel paradigm can go quite far in explaining the cellā™s most fundamen-
tal attributes ā“ distribution of ions and the presence of a cell potential. These are
equilibrium processes; they require no energy for maintenance.
The cell is evidently not a static structure, but a machine designed to carry out a mul-
titude of tasks. Such tasks are currently described by a broad variety of mechanisms,
apparently lacking any single identiļ¬able underlying theme ā“ at least to this author.
For virtually every process there appears to be another mechanism.
Whether a common underlying theme might govern the cellā™s many operational
tasks is a question worth asking. After all, the cell began as a simple gel and evolved
from there. As it specialized, gel structure and processes gained in intricacy. Given
such lineage, the potential for a simple, common, underlying, gel-based theme should
not necessarily be remote. Finding a common underlying theme has been a long-
term quest in other ļ¬elds. In physics, for example, protĀ“ gĀ“ s of Einstein continue the
search for a unifying force. That nature works in a parsimonious manner, employing
variations of a few simple principles to carry out multitudinous actions, is an attractive
notion, one I do not believe has yet been seriously pursued in the realm of cell function,
although simplicity is a guiding principle in engineering.
If the cell is a gel, then a logical approach to the question of a common under-
lying principle of cell function is to ask whether a common underlying principle
governs gel function. Gels do āfunction.ā They undergo transition from one state to
another. The process is known as a polymer-gel phase-transition ā“ much like the tran-
sition from ice to water ā“ a small change of environment causing a huge change in
Such change can generate work. Just as ice formation has sufļ¬cient power to
fracture hardened concrete, gel expansion or contraction is capable of many types of
work, ranging from solute/solvent separation to force generation (Fig. 7-5). Common
examples of useful phase-transitions are the time-release capsule (in which a gel-sol
transition releases bioactive drugs), the disposable diaper (where a condensed gel
undergoes enormous hydration and expansion to capture the āloadā), and various
136 G.H. Pollack
Fig. 7-4. KCl-ļ¬lled microelectrodes stuck into gel strips at slow constant velocity, and then with-
drawn. (a) Typical anionic gel, polyacrylamide/polypotassiumacrylate, shows negative potential.
(b) Typical cationic gel, polyacrylamide/polydialyldimethylammonium chloride, shows positive
potential. Courtesy of R. GĀØ lch.
artiļ¬cial muscles. Such behaviors are attractive in that a large change of structure can
be induced by a subtle change of environment (Fig. 7-6).
Like synthetic gels, the natural gel of the cell may have the capacity to undergo
similarly useful transitions. The question is whether they do. This question is perhaps
more aptly stated a bit differently, for the cell is not a homogeneous gel but a collection
of gel-like organelles, each of which is assigned a speciļ¬c task. The more relevant
question, then, is whether any or all such organelles carry out their functions by
The short answer is yes: it appears that this is the case. Pursuing so extensive a
theme in a meaningful way in the short space of a review article is challenging; for
Cells, gels, and mechanics 137
Fig. 7-5. Typical stimuli and responses of artiļ¬cial polymer hydrogels. After Hoffman, 1991.
Temperature, solvent composition, pH, ions
electric field, UV, light, specific molecules, or
Fig. 7-6. Phase-transitions are triggered by subtle shifts of environment. After Tanaka et al.,
138 G.H. Pollack
Fig. 7-7. Calcium and other divalent cations can bridge the gap between negatively charged sites,
resulting in zipper-like condensation.
a fuller development I refer the reader to Pollack (2001). In this venue I focus on a
single aspect: the relevance of phase-transitions in the production of motion.
Gels and motion
The classes of motion produced by phase-transitions fall largely into two categories,
isotropic and linear. In isotropic gels, polymers are randomly arranged and sometimes
cross-linked. Water is held largely by its afļ¬nity to polymers (or proteins, in the case
of the cell). The gel is thus well hydrated ā“ and may in the extreme contain as much as
99.97 percent water (Osada and Gong, 1993). In the transitioned state, the dominant
polymer-water afļ¬nity gives way to a higher polymer-polymer afļ¬nity, condensing
the gel into a compact mass and expelling solvent. Thus, water moves, and polymer
Linear polymers also undergo transition ā“ from extended to shortened states. The
extended state is stable because it maximizes the number of polymer-water contacts
and therefore minimizes the systemā™s energy. Water builds layer upon layer. In the
shortened state the afļ¬nity of polymer for itself exceeds the afļ¬nity of polymer for
water, and the polymer folds. It may fold entirely, or it may fold regionally, along
a fraction of its length. As it folds, polymer and water both move. And, if a load is
placed at the end of the shortening ļ¬lament, the load can move as well.
Phase-transitions are inevitably cooperative: once triggered, they go to completion.
The reason lies in the transitionā™s razor-edge behavior. Once the polymer-polymer
afļ¬nity (or the polymer-water afļ¬nity) begins to prevail, its prevalence increases;
hence the transition goes to completion. An example is illustrated in Fig. 7-7. In
this example, the divalent ion, calcium, cross-links the polymer strands. Its presence
thereby shifts the predominant afļ¬nity from polymer-water to polymer-polymer. Once
a portion of the strand is bridged, ļ¬‚anking segments of the polymer are brought closer
together, increasing the proclivity for additional calcium bridging. Thus, local action
enhances the proclivity for action in a neighboring segment, ensuring that the reaction
proceeds to completion. In this way, transitions propagate toward completion.
Evidently, the polymer-gel phase-transition can produce different classes of motion.
If the cell were to exploit this principle, it could have a simple way of producing a broad
array of motions, depending on the nature and arrangement of constituent polymers.
Cells, gels, and mechanics 139
Fig. 7-8. Textbook view of secretion. Chemicals are packed in vesi-
cles, which work their way to the cell surface, poised for discharge.
In all cases, a small shift of some environmental variable such as pH or chemical
content, for example, could give rise to a cooperative, all-or-none response, which
could produce massive mechanical action.
As representative examples of such action, two fundamental cellular processes are
considered ā“ secretion and contraction. The ļ¬rst involves the propulsive motion of
small molecules, the second the motion of protein ļ¬laments. (Additional details on
these and other mechanisms can be found in Pollack (2001)).
Secretion is the mechanism by which the cell exports molecules. The molecules are
packed into small spherical vesicles, which lie just within the cell boundary, awaiting
export (Fig. 7-8). According to prevailing views, the vesicle is a kind of āsoupā
surrounded by a membrane ā“ a miniature of the prevailing view of the cell itself. For
discharge, the vesicle docks with the cell membrane; cell and vesicle membranes fuse,
opening the interior of the vesicle to the extracellular space and allowing the vesicleā™s
contents to escape by diffusion. Although attractive in its apparent simplicity, this
mechanism does not easily reconcile with several lines of evidence.
The ļ¬rst is that the vesicle is by no means a clear broth. It is a dense matrix of tangled
polymers, invested with the molecules to be secreted. Getting those molecules to dif-
fuse through this entwining thicket and leave the cell is as implausible as envisioning
a school of ļ¬sh escaping from an impossibly tangled net.
A second concern is the response to solvents. Demembranated vesicle matrices
can be expanded and recondensed again and again by exposure to various solutions,
but these solutions are not the ones expected from classical theory. When condensed
matrices from mast cells or goblet cells (whose matrices hydrate to produce mucus) are
exposed to low osmolarity solutions ā“ even distilled water ā“ they remain condensed
even though the osmotic draw for water ought to be enormous (Fernandez et al.,
1991; Verdugo et al., 1992). Something keeps the network condensed, and it appears
to be multivalent cations, in some cases calcium and in other cases the molecule
to be secreted, which is commonly a multivalent cation. These multivalent cations
cross-link the negatively charged matrix and keep it condensed, even in the face of
solutions of extremely low osmolarity (Fig. 7-9).
A third issue is that discharge does not appear to be a passive event. It is of-
ten accompanied by dramatic vesicle expansion. Isolated mucin-producing secretory
140 G.H. Pollack
Fig. 7-9. Phase-transition model of secretion. The phase-transition is triggered as extracellular Na+
replaces the multivalent cation holding the anionic network condensed.
vesicles, for example, undergo a 600-fold volume expansion within 40 ms (Verdugo
et al., 1992). Vesicles of nematocysts (aquatic stinging cells) are capable of linear
expansion rates of 2,000 Āµm/ms (Holstein and Tardent, 1984). Such phenomenal
expansion rates imply something beyond mere passive diffusion of solutes and water.
Given these features, it is no surprise that investigators have begun looking for
mechanistic clues within the realm of the phase-transition, where expansion can be
large and rapid. A feature of secretory discharge consistent with this mechanism is that
discharge happens or doesnā™t happen depending on a critical shift of environment ā“
the very hallmark of the polymer-gel phase-transition. Goblet-cell and mast-cell
matrices condense or expand abruptly as the solvent ratio (either glycerol/water or
acetone/water) is edged just past a threshold or the temperature edges past a threshold,
the transition thresholds in both cases lying within a window as narrow as 1 percent
of the critical value (Verdugo et al., 1992). Hence, the phase-transitionā™s signature
criterion is satisļ¬ed. The abrupt expansion and hydration would allow the relevant
molecules to escape into the extracellular ļ¬‚uid.
Such a system might work as follows. When the condensed matrix is exposed to
the extracellular space, sodium displaces the divalent cross-linker. No longer cross-
linked, the polymer can satisfy its intense thirst for hydration, imbibing water and
expanding explosively, in a manner described as a jack-in-the-box (Verdugo et al.,
1992). Meanwhile, the messenger molecules are discharged. Diffusion may play some
role in release, but the principal role is played by convective forces, for multivalent
ions are relatively insoluble in the layered water surrounding the charged polymers
(Vogler, 1998), and will therefore be forcefully ejected. Hence, discharge into the
extracellular space occurs by explosive convection.
As is now well known, muscle sarcomeres, or contractile units, contain three ļ¬lament
types (Fig. 7-10): thick, thin, and connecting ā“ the latter interconnecting the ends of
Cells, gels, and mechanics 141
Fig. 7-10. Muscle sarcomere contains three ļ¬lament types, bounded by Z-lines.
the thick ļ¬lament with respective Z-lines and behaving as a molecular spring. All three
ļ¬laments are polymers: thin ļ¬laments consist largely of repeats of monomeric actin;
thick ļ¬laments are built around multiple repeats of myosin; and connecting ļ¬laments
are built of titin (also known as connectin), a huge protein containing repeating
immunoglobulin-like (Ig) and other domains. Together with water, which is held with
extreme tenacity by these proteins (Ling and Walton, 1976), this array of polymers
forms a gel-like lattice.
Until the mid-1950s, muscle contraction was held to occur by a mechanism not
much different from the phase-transition mechanism to be considered. All major
research groups subscribed to this view. With the discovery of interdigitating ļ¬la-
ments in the mid-1950s, it was tempting to dump this notion and suppose instead
that contraction arose out of pure ļ¬lament sliding. This supposition led Sir Andrew
Huxley and Hugh Huxley to examine independently whether ļ¬laments remained at
constant length during contraction. Back-to-back papers in Nature, using the optical
microscope (Huxley and Niedergerke, 1954; Huxley and Hanson, 1954) appeared
to conļ¬rm this supposition. The constant-ļ¬lament-length paradigm took hold, and
has held remarkably ļ¬rm ever since ā“ notwithstanding more than thirty subsequent
reports of thick ļ¬lament or A-band shortening (Pollack, 1983; 1990) ā“ a remarkable
disparity of theory and evidence. The motivated reader is invited to check the cited
papers and make an independent judgment.
With the emerging notion of sliding ļ¬laments, the central issue became the nature
of the driving force; the model that came to the forefront was the so-called swinging
cross-bridge mechanism (Huxley, 1957). In this model, ļ¬lament translation is driven
by oar-like elements protruding like bristles of a brush from the thick ļ¬laments,
attaching transiently to thin ļ¬laments, swinging, and propelling the thin ļ¬laments to
slide along the thick. This mechanism explains many known features of contraction,
and has therefore become broadly accepted (Spudich, 1994; Huxley, 1996; Block,
1996; Howard, 1997; Cooke, 1997).
On the other hand, contradictory evidence abounds. In addition to the conļ¬‚icting ev-
idence on the constancy of ļ¬lament length, which contradicts the pure sliding model,
a serious problem is the absence of compelling evidence for cross-bridge swing-
ing (Thomas, 1987). Electron-spin resonance, X-ray diffraction, and ļ¬‚uorescence-
polarization methods have produced largely negative results, as has high-resolution
electron microscopy (Katayama, 1998). The most positive of these results has been
an angle change of 3ā—¦ measured on a myosin light chain (Irving et al., 1995) ā“ far
142 G.H. Pollack
short of the anticipated 45ā—¦ . Other concerns run the gamut from instability (Irving
et al., 1995), to mechanics (Fernandez et al., 1991), structure (Schutt and Lindberg,
1993; 1998), and chemistry (Oplatka, 1996; 1997). A glance at these reviews conveys
a picture different from the one in textbooks.
An alternative approach considers the possibility that the driving mechanism does
not lie in cross-bridge rotation, but in a paradigm in which all three ļ¬laments shorten.
If contiguous ļ¬laments shorten synchronously, the event is global, and may qualify
as a phase-transition. We consider the three ļ¬laments one at a time.
First consider the connecting ļ¬lament. Shortening of the connecting ļ¬lament re-
turns the extended, unactivated sarcomere to its unstrained length. Conversely, applied
stress lengthens the connecting ļ¬lament. Shortening may involve a sequential fold-
ing of domains along the molecule, whereas stretch includes domain unfoldings ā“
the measured length change being stepwise (Rief et al., 1997; Tskhovrebova et al.,
1997). Similarly in the intact sarcomere, passive length changes also occur in steps
(Blyakhman et al., 1999), implying that each discrete event is synchronized in parallel
over many ļ¬laments.
Next, consider the thick ļ¬lament. Thick ļ¬lament shortening could transmit force
to the ends of the sarcomere through the thin ļ¬laments, thereby contributing to active
sarcomere shortening. Evidence for thick ļ¬lament shortening was mentioned above.
Although rarely discussed in contemporary muscle literature, the observations of thick
ļ¬lament length changes are extensive: they have been carried out in more than ļ¬fteen
laboratories worldwide and have employed electron and light microscopic techniques
on specimens ranging from crustaceans and insects to mammalian heart and skeletal
muscle ā“ even human muscle. Evidence to the contrary is relatively rare (Sosa et al.,
1994). These extensive observations cannot be summarily dismissed merely because
they are not often discussed.
Thick ļ¬lament shortening cannot be the sole mechanism underlying contraction.
If it were, the in vitro motility assay in which thin ļ¬laments translate over individual
myosin molecules planted on a substrate could not work, for it contains no ļ¬laments
that could shorten. On the other hand, ļ¬lament shortening cannot be dismissed as ir-
relevant. Thick ļ¬lament shortening could contribute directly to sarcomere shortening.
It could be mediated by an alpha-helix to random-coil transition along the myosin
rod, which lies within the thick ļ¬lament backbone (Pollack, 1990). The helix-coil
transition is a classical phase-transition well known to biochemists ā“ and also to those
who have put a wool sweater into a hot clothes dryer and watched it shrink.
The thin ļ¬lament may also shorten. There is extensive evidence that some structural
change takes place along the thin ļ¬lament (Dos Remedios and Moens, 1995; Pollack,
1996; KĀØ s et al., 1994). Crystallographic evidence shows that monomers of actin can
pack interchangeably in either of two conļ¬gurations along the ļ¬lament ā“ a ālongā
conļ¬guration and a shorter one (Schutt and Lindberg, 1993). The difference leads
to a ļ¬lament length change of 10ā“15 percent. The change in actin is worth dwelling
on, for although it may be more subtle than the ones cited above, it may be more
universal, as actin ļ¬laments are contained in all eukaryotic cells.
Isolated actin ļ¬laments show prominent undulations. Known as āreptationā be-
cause of its snake-like character, the constituent undulations are broadly observed:
in ļ¬laments suspended in solution (Yanagida et al., 1984); embedded in a gel (KĀØ s a
Cells, gels, and mechanics 143
Fig. 7-11. Reptation model. An actin ļ¬lament snakes its way toward the center of the sarcomere, past
myosin cross-bridges, which may well interconnect adjacent thick ļ¬laments. From Vogler, 1998.
et al., 1994); and gliding on a myosin-coated surface (Kellermayer and Pollack, 1996).
Such undulations had been presumed to be of thermal origin, but that notion is chal-
lenged by the observation that they can be substantially intensiļ¬ed by exposure to
myosin (Yanagida et al., 1984) or ATP (Hatori et al., 1996). These effects imply a
speciļ¬c structural change rather than a thermally induced change.
In fact, structural change in actin is implied by a long history of evidence. Molec-
ular transitions had ļ¬rst been noted in the 1960s and 1970s (Asakura et al., 1963;
Hatano et al., 1967; Oosawa et al., 1972). On exposure to myosin, actin monomers un-
derwent a 10ā—¦ rotation (Yanagida and Oosawa, 1978). Conformational changes have
since been conļ¬rmed not only in probe studies, but also in X-ray diffraction stud-
ies, phosphorescence-anisotropy studies, and ļ¬‚uorescence-energy transfer studies,
the latter showing a myosin-triggered actin-subdomain-spacing change of 17 percent
(Miki and Koyama, 1994).
That such structural change propagates along the ļ¬lament is shown in several ex-
perimental studies. Gelsolin is a protein that binds to one end (the so-called ābarbedā
end) of the actin ļ¬lament, yet the impact of binding is felt along the entire ļ¬lament:
molecular orientations shift by 10ā—¦ , and there is a three-fold decrease of the ļ¬lamentā™s
overall torsional rigidity (Prochniewicz et al., 1996). Thus, structural change induced
by point binding propagates over the entire ļ¬lament. Such propagated action may
account for the propagated waves seen traveling along single actin ļ¬laments ā“ ob-
servable either by cross-correlation of point displacements (deBeer et al., 1998) or
by tracking ļ¬‚uorescence markers distributed along the ļ¬lament (Hatori et al., 1996;
1998). In the latter, waves of shortening can be seen propagating along the ļ¬lament,
much like a caterpillar.
Could such a propagating structural transition drive the thin ļ¬lament to slide along
the thick? A possible vehicle for such action is the inchworm mechanism (Fig. 7-11).
By propagating along the thin ļ¬lament, a shortening transition could propel the thin
ļ¬lament to reptate past the thick ļ¬lament, each propagation cycle advancing the
ļ¬lament incrementally toward the center of the sarcomere.
Perhaps the most critical prediction of such a mechanism is the anticipated quantal
advance of the thin ļ¬lament. With each propagation cycle, the ļ¬lament advances by
a step (Fig. 7-12). The advance begins as an actin monomer unbinds from a myosin
cross-bridge; it ends as the myosin bridge rebinds an actin momomer farther along the
thin ļ¬lament. Hence, the ļ¬lament-translation step size must be an integer multiple of
the actin-repeat spacing (see Fig. 7-12). The translation step could be one, two, . . . or
n times the actin-repeat spacing along the thin ļ¬lament.
144 G.H. Pollack
Fig. 7-12. Reptation model predicts that each advance of the thin ļ¬lament will be an integer multiple
of the actin-monomer repeat along the ļ¬lament.
This signature-like prediction is conļ¬rmed in several types of experiment. The thin
ļ¬lament advances in steps; and, step size is an integer multiple of the actin-monomer
spacing (Figs. 7-13ā“7-16). This is true in the isolated molecular system, where the
single myosin molecule translates along actin (Kitamura et al., 1999); in the intact
sarcomere, where thin ļ¬laments translate past thick ļ¬laments (Blyakhman et al., 1999;
Yakovenko et al., 2002); and in isolated actin and myosin ļ¬laments sliding past one
another (Liu and Pollack, 2004).
In the sarcomere experiments, the striated image of a single myoļ¬bril is projected
onto a photodiode array. The array is scanned repeatedly, producing successive traces
Fig. 7-13. Time course of single sarcomere shortening in single activated myoļ¬brils. From
Yakovenko et al. (2002).
Cells, gels, and mechanics 145
Fig. 7-14. Analysis of steps in records such as those of Fig. 7-13. Steps are integer multiples
of 2.7 nm.
of intensity along the myoļ¬bril axis. Hence, single sarcomeres can be tracked. The
sarcomere-length change is consistently stepwise (Fig. 7-13). Analysis of many steps
showed that their size is an integer multiple of 2.7 nm, the actin-monomer spacing
projected on the ļ¬lament axis (Fig. 7-14).
Similar results are obtained when a single actin ļ¬lament slides past a single thick
ļ¬lament (Liu and Pollack, 2004). Here the trailing end of the actin ļ¬lament is attached
to the tip of a deļ¬‚ectable nanolever (Fauver et al., 1998), normal to its long axis. On
addition of ATP, the actin ļ¬lament slides, bending the nanolever. Displacement of
the actin ļ¬lament is monitored by tracking the position of the nanolever tip, which is
projected onto a photodiode array. Representative traces are shown in Fig. 7-15. They
reveal the stepwise nature of ļ¬lament translation. Step size was measured by applying
an algorithm that determined the least-squares linear ļ¬t to each pause. The vertical
displacement between pauses gave the step. Fig. 7-16 shows a histogram obtained
from a large number of steps. The histogram is similar to that of Fig. 7-14. It shows,
once again, steps that are integer multiples of 2.7 nm, both when the actin ļ¬lament
slides forward during contraction and when the actin ļ¬lament is forcibly pulled in
the sarcomere-lengthening direction. Hence, the results obtained with single isolated
ļ¬laments and single sarcomeres are virtually indistinguishable.
Agreement between these results and the modelā™s prediction lends support to the
proposed thesis. Conventional mechanisms might generate a step advance during each
cross-bridge stroke; and, with a fortuitously sized cross-bridge swinging arc, the step
could have the appropriate size. But it is not at all clear how integer multiples of the
146 G.H. Pollack
Fig. 7-15. Step-wise interaction between single actin and single thick ļ¬lament. Horizontal bar rep-
resents 1 s except in traces 1 and 2, where it equals 0.5 s. The two traces in the box show backward
steps. Arrows indicate positions of some pauses. Top trace shows noise level, and has same scale as
fundamental size might be generated in a simple way, although they are observed
frequently (Figs. 7-14, 7-16). By contrast, the detailed quantitative observations de-
scribed above are direct predictions of the reptation mechanism.
In sum, contraction of the sarcomere could well arise out of contraction of each
of the three ļ¬laments ā“ connecting, thick, and thin. Connecting and thick ļ¬la-
ments appear to shorten by local phase-transitions, each condensation shortening the
Fig. 7-16. Continuous histogram of step-size distribution from data similar to those shown on
Fig. 7-15, with bin width of 1.0 nm and increments of 0.1 nm.
Cells, gels, and mechanics 147
respective ļ¬lament by an incremental step. Because these two ļ¬laments lie in series,
ļ¬lament shortening leads directly to sarcomere shortening. The thin ļ¬lament appears
to undergo a local, propagating transition, each snake-like cycle advancing the thin
ļ¬lament past the thick by an increment. Repeated cycles produce large-scale transla-
tion. (A similar process may occur in the in vitro motility assay, where the myosins
are ļ¬rmly planted on a substrate rather than in the lattice of ļ¬laments; the ļ¬lament
may āsnakeā its way along.)
The incremental steps anticipated from these transitions are observable at various
levels of organization. These levels range from the single ļ¬lament pair (Liu and
Pollack, 2004) and single myoļ¬brillar sarcomere (Blyakhman et al., 1999; Yakovenko
et al., 2002), to bundles of myoļ¬brils (Jacobson et al., 1983), to segments of whole
muscle ļ¬bers (Granzier et al., 1987). Hence, the transitions are global, as the phase-
transition anticipates. It is perhaps not surprising that phase-transitions arise in all
three ļ¬lamentary elements. This endows the system with a versatile array of features
that makes muscle the effective machine that it is. Indeed, muscle is frequently referred
to as the jewel in mother natureā™s crown of achievements.
Two examples of biological motion have been presented, each plausibly driven by
phase-transitions and each producing a different type of motion. Isotropic structures
such as secretory vesicles undergo condensations and expansions, whereas ļ¬lamen-
tary bundles such as actin and myosin produce linear contraction. Linear contraction
can also occur in microtubules, another of natureā™s linear polymers: when cross-linked
into a bundle, microtubules along one edge of the bundle are often observed to shorten
(McIntosh, 1973); this may mediate bending, as occurs for example in a bimetal strip.
Hence, diverse motions are possible.
Given such mechanistic versatility, it would not be surprising if the phase-transition
were a generic mechanism for motion production, extending well beyond the examples
considered here. Phase-transitions are simple and powerful. They can bring about
large-scale motions induced by subtle changes of environment. This results in a kind
of switch-like action with huge ampliļ¬cation. Such features seem attractive enough
to imply that if nature has chosen the phase-transition as a common denominator of
cell motion (and perhaps sundry other processes) it has made a wise choice.
The consent of Ebner and Sons to reprint ļ¬gures from Pollack, Cells, Gels and the
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8 Polymer-based models of cytoskeletal networks
Most plant and animal cells possess a complex structure of ļ¬lamentous proteins
and associated proteins and enzymes for bundling, cross-linking, and active force generation.
This cytoskeleton is largely responsible for cell elasticity and mechanical stability. It can also
play a key role in cell locomotion. Over the last few years, the single-molecule micromechanics
of many of the important constituents of the cytoskeleton have been studied in great detail by
biophysical techniques such as high-resolution microscopy, scanning force microscopy, and
optical tweezers. At the same time, numerous in vitro experiments aimed at understanding
some of the unique mechanical and dynamic properties of solutions and networks of cytoskele-
tal ļ¬laments have been performed. In parallel with these experiments, theoretical models have
emerged that have both served to explain many of the essential material properties of these net-
works, as well as to motivate quantitative experiments to determine, for example concentration
dependence of shear moduli and the effects of cross-links. This chapter is devoted to theoretical
models of the cytoskeleton based on polymer physics at both the level of single protein ļ¬la-
ments and the level of solutions and networks of cross-linked or entangled ļ¬laments. We begin
with a derivation of the static and dynamic properties of single cytoskeletal ļ¬laments. We then
proceed to build up models of solutions and cross-linked gels of cytoskeletal ļ¬laments and we
discuss the comparison of these models with a variety of experiments on in vitro model systems.
Understanding the mechanical properties of cells and even whole tissues continues
to pose signiļ¬cant challenges. Cells experience a variety of external stresses and
forces, and they exert forces on their surroundings ā“ for instance, in cell locomotion.
The mechanical interaction of cells with their surroundings depends on structures
such as cell membranes and complex networks of ļ¬lamentous proteins. Although
these cellular components have been known for many years, important outstanding
problems remain concerning the origins and regulation of cell mechanical properties
(Pollard and Cooper, 1986; Alberts et al., 1994; Boal, 2002). These mechanical factors
determine how a cell maintains and modiļ¬es its shape, how it moves, and even how
cells adhere to one another. Mechanical stimulus of cells can also result in changes
in gene expression.
Cells exhibit rich composite structures ranging from the nanometer to the microm-
eter scale. These structures combine soft membranes and rather rigid ļ¬lamentous
Polymer-based models of cytoskeletal networks 153
proteins or biopolymers, among other components. Most plant and animal cells, in
fact, possess a complex network structure of biopolymers and associated proteins and
enzymes for bundling, cross-linking, and active force generation. This cytoskeleton
is often the principal determinant of cell elasticity and mechanical stability.
Over the last few years, the single-molecule properties of many of the important
building blocks of the cytoskeleton have been studied in great detail by biophysi-
cal techniques such as high-resolution microscopy, scanning force microscopy, and
optical tweezers. At the same time, numerous in vitro experiments have aimed to
understand some of the unique mechanical and dynamic properties of solutions and
networks of cytoskeletal ļ¬laments. In parallel with these experiments, theoretical
models have emerged that have served both to explain many of the essential mate-
rial properties of these networks, as well as to motivate quantitative experiments to
determine the way material properties are regulated by, for example, cross-linking
and bundling proteins. Here, we focus on recent theoretical modeling of cytoskeletal
solutions and networks.
One of the principal components of the cytoskeleton, and even one of the most
prevalent proteins in the cell, is actin. This exists in both monomeric or globular
(G-actin) and polymeric or ļ¬lamentary (F-actin) forms. Actin ļ¬laments can form
a network of entangled, branched, and/or cross-linked ļ¬laments known as the actin
cortex, which is frequently found near the periphery of cells. In vivo, this network is
far from passive, with both active motion and (contractile) force generation during
cell locomotion, and with a strong coupling to membrane proteins that appears to play
a key role in the ability of cells to sense and respond to external stresses.
In order to understand these complex structures, quantitative models are needed
for the structure, interactions, and mechanical response of networks such as the actin
cortex. Unlike networks and gels of most synthetic polymers, however, these networks
have been clearly shown to possess properties that cannot be modeled by existing
polymer theories. These properties include rather large shear moduli (compared with
synthetic polymers under similar conditions), strong signatures of nonlinear response
(in which, for example, the shear modulus can increase by a full factor of ten or
more under modest strains of only 10 percent or so) (Janmey et al., 1994), and unique
dynamics. In a very close and active collaboration between theory and experiment over
the past few years, a standard model of sorts for the material properties of semiļ¬‚exible
polymer networks has emerged, which can explain many of the observed properties
of F-actin networks, at least in vitro. Central to these models has been the semiļ¬‚exible
nature of the constituent ļ¬laments, which is both a fundamental property of almost any
ļ¬lamentous protein, as well as a clear departure from conventional polymer physics,
which has focused on ļ¬‚exible or rod-like limits. In contrast, biopolymers such as
F-actin are nearly rigid on the scale of a micrometer, while at the same time showing
signiļ¬cant thermal ļ¬‚uctuations on the cellular scale of a few microns.
This chapter begins with an introduction to models of single-ļ¬lament response
and dynamics, and in fact, spends most of its time on a detailed understanding of
these single-ļ¬lament properties. Because cytoskeletal ļ¬laments are the most impor-
tant structural components in cells, a quantitative understanding of their mechan-
ical response to bending, stretching, and compression is crucial for any model of
the mechanics of networks of these ļ¬laments. We shall see how these fundamental
154 F.C. MacKintosh
Fig. 8-1. Entangled solution of semiļ¬‚exible actin ļ¬laments. (A) In physiological conditions, individ-
ual monomeric actin proteins (G-actin) polymerize to form double-stranded helical ļ¬laments known
as F-actin. These ļ¬laments exhibit a polydisperse length distribution of up to 70 Āµm in length. (B)
A solution of 1.0 mg/ml actin ļ¬laments, approximately 0.03% of which have been labeled with
rhodamine-phalloidin in order to visualize them by ļ¬‚orescence microscopy. The average distance
Ī¾ between chains in this ļ¬gure is approximately 0.3 Āµm. (Reprinted with permission from Mac-
Kintosh F C, KĀØ s J, and Janmey P A, Physical Review Letters, 75 4425 (1995). Copyright 1995 by
the American Physical Society.
properties of the individual ļ¬laments can explain many of the properties of solutions
The biopolymers that make up the cytoskeleton consist of aggregates of large globular
proteins that are bound together rather weakly, as compared with most synthetic,
covalently bonded polymers. Nevertheless, they can be surprisingly strong. The most
rigid of these are microtubules, which are hollow tube-like ļ¬laments that have a
diameter of approximately 20 nm. The most basic aspect determining the mechanical
behavior of cytoskeletal polymers on the cellular scale is their bending rigidity.
Even with this mechanical resistance to bending, however, cytoskeletal ļ¬la-
ments can still exhibit signiļ¬cant thermally induced bending ļ¬‚uctuations because of
Brownian motion in a liquid. Thus such ļ¬laments are said to be semiļ¬‚exible or worm-
like. This is illustrated in Fig. 8-1, showing ļ¬‚uorescently labeled F-actin ļ¬laments
on the micrometer scale. The effect of the Brownian forces on the ļ¬lament leads to
increasingly contorted shapes over larger-length segments. The length at which sig-
niļ¬cant bending ļ¬‚uctuations occur actually provides a simple yet quantitative charac-
terization of the mechanical stiffness of such polymers. This thermal bending length,
or persistence length p , is deļ¬ned in terms of the the angular correlations (for exam-
ple, of the local orientation along the polymer backbone), which decay exponentially
with a characteristic length p . In simple terms, however, this just says that a typical
ļ¬lament in thermal equilibrium in a liquid will appear rather straight over lengths
that are short compared with this persistence length, while it will begin to exhibit
a random, contorted shape only on longer-length scales. The persistence lengths of
a few important biopolymers are given in Table 8-1, along with their approximate
diameter and length.
Polymer-based models of cytoskeletal networks 155
Table 8-1. Persistence lengths and other parameters of various biopolymers
(Howard, 2001; Gittes et al., 1993)
Type Approximate diameter Persistence length Contour length
DNA 2 nm 50 nm ā¼
F-actin 7 nm ā¼
ā¼1ā“5 mm 10s of Āµm
Microtubule 25 nm
The worm-like chain model
Rigid polymers can be thought of as elastic rods, except on a small scale. The me-
chanical description of these is essentially the same as for a macroscopic rod with
quantitative differences in parameters. The important role of thermal ļ¬‚uctuations,
however, introduces a qualitative difference from the macroscopic case. Because the
diameter of a ļ¬lamentous protein is so much smaller than other length scales of inter-
est ā“ and especially the cellular scale ā“ it is often sufļ¬cient to think of a ļ¬lament as
an idealized curve that resists bending. This is the essence of the so-called worm-like
chain model. This can be described by an energy of the form,
Hbend ds (8.1)
where Īŗ is the bending modulus and t is a (unit) tangent vector along the chain. The
variation (derivative) of the tangent is a measure of curvature, which appears here
quadratically because it is assumed that there is no preferred direction of curvature.
Here, the chain position r (s) is described in terms of a coordinate s corresponding to
the length along the chain backbone. Hence, the tangent vector
These quantities are illustrated in Fig. 8-2.
The bending modulus Īŗ has units of energy times length. A natural energy scale
for a rod subject to Brownian ļ¬‚uctuations is kT , where T is the temperature and k
is Boltzmannā™s constant. This is the typical kinetic energy of a molecule or particle.
The persistence length described above is simply given by p = Īŗ/(kT ), because the
ļ¬‚uctuations tend to decrease with stiffness Īŗ and increase with temperature. As noted,
this is the typical length scale over which the polymer forgets its orientation, due to
the constant Brownian forces it experiences in a medium at ļ¬nite temperature.
More precisely, for a homogeneous rod of diameter 2a consisting of a homogeneous
elastic material, the bending modulus should be proportional to the Youngā™s modulus
E. The Youngā™s modulus, or the stiffness of the material, has units of energy per
volume. Thus, on dimensional grounds, we expect that Īŗ ā¼ Ea 4 . In fact (Landau and
Īŗ = Ea 4 .
The prefactor in front of Ea 4 depends on the geometry of the rod (in other words, its
cross-section). The factor Ļa 4 /4 is for a solid rod of radius a. For a hollow tube, such
156 F.C. MacKintosh
Fig. 8-2. A ļ¬lamentous protein can be regarded as an
elastic rod of radius a. Provided the length of the rod
is very long compared with the monomeric dimension a,
and that the rigidity is high (speciļ¬cally, the persistence
length p a), this can be treated as an abstract line or
s curve, characterized by the length s along its backbone.
A unit vector t tangent to the ļ¬lament deļ¬nes the local
orientation of the ļ¬lament. Curvature is present when this
orientation varies with s. For bending in a plane, it is suf-
ļ¬cient to consider the angle Īø (s) that the ļ¬lament makes
with respect to some ļ¬xed axis. The curvature is then
as one might use to model a microtubule, the prefactor would be different, but still
of order a 4 , where a is the (outer) radius. This is often expressed as Īŗ = E I , where
I is the moment of inertia of the cross-section (Howard, 2001).
In general, for bending in 3D, there are two independent directions for deļ¬‚ections
of the rod or polymer transverse to its local axis. It is often instructive, however, to
consider a simpler case of a single transverse degree of freedom, in other words,
motion conļ¬ned to a plane, as illustrated by Fig. 8-2. Here, the integrand in Eq. 8.1
becomes (ā‚Īø/ā‚s)2 , where Īø(s) is simply the local angle that the chain axis makes
at point s, relative to any ļ¬xed axis. Using basic principles of statistical mechan-
ics (Grosberg and Khokhlov, 1994), one can calculate the thermal average angular
correlation between distant points along the chain, for which
|sā’s |/ s
eā’|sā’s |/2 p .
cos[Īø(s) ā’ Īø (s )] cos ( Īø) (8.2)
As noted at the outset, so far this is all for motion conļ¬ned to a plane. In three
dimensions, there is another direction perpendicular to the plane that the ļ¬lament can
move in. This increases the rate of decay of the angular correlations by a factor of
two relative to the result above:
t(s) Ā· t(s ) = eā’|sā’s |/ p , (8.3)
where p is the same persistence length deļ¬ned above. This is a general deļ¬nition
of the persistence length, which also provides a purely geometric measure of the
mechanical stiffness of the rod, provided that it is in equilibrium at temperature
T . In principle, this means that one can measure the stiffness of a biopolymer by
simply examining its bending ļ¬‚uctuations in a microscope. In practice, however, it is
usually better to measure the amplitudes of a number of different bending modes (that
is, different wavelengths) in order to ensure that thermal equilibrium is established
(Gittes et al., 1993).
Force-extension of single chains
In order to understand how a network of ļ¬laments responds to mechanical loading, we
need to understand at least two things: the way a single ļ¬lament responds to stress; and
the way in which the individual ļ¬laments are connected or otherwise interact with each
Polymer-based models of cytoskeletal networks 157
other. We address the single-ļ¬lament properties here, and reserve the characterization
of the way ļ¬laments interact for later.
A single ļ¬lament can respond to forces in at least two ways. It can respond to
both transverse and longitudinal forces by either bending or stretching/compressing.
On length scales shorter than the persistence length, bending can be described in
mechanical terms, as for elastic rods. By contrast, stretching and compression can
involve both a purely elastic or mechanical response (again, as in the stretching,
compression, or even buckling of macroscopic elastic rods), as well as an entropic
response. The latter comes from the thermal ļ¬‚uctuations of the ļ¬lament. Perhaps
surprisingly, as will be shown, the longitudinal response can be dominated by entropy
even on length scales small compared with the persistence length. Thus, it is incorrect
to think of a ļ¬lament as truly rod-like, even on length scales short compared with p .
The longitudinal single-ļ¬lament response is often described in terms of a so-called
force-extension relationship. Here, the force required to extend the ļ¬lament is mea-
sured or calculated in terms of the degree of extension along a line. At any ļ¬nite
temperature, there is a resistance to such extension due to the presence of thermal
ļ¬‚uctuations that make the polymer deviate from a straight conformation. This has been
the basis of mechanical studies, for example, of long DNA (Bustamante et al., 1994).
In the limit of large persistence length, this can be calculated as follows (MacKintosh
et al., 1995).
We consider a ļ¬lament segment of length that is short compared with the persis-
tence length p . It is then nearly straight, with small transverse ļ¬‚uctuations. We let the
x-axis deļ¬ne the average orientation of the chain segment, and let u and v represent
the two independent transverse degrees of freedom. These can then be thought of as
functions of x and time t in general. For simplicity, we shall mostly consider just one
of these coordinates, u(x, t). The bending energy is then
Īŗ ā‚ 2u
= = Īŗq 4 u q ,
Hbend dx (8.4)
2 4 q
where we have represented u(x) by a Fourier series
u(x, t) = u q sin(q x). (8.5)
As illustrated in Fig. 8-3, the local orientation of the ļ¬lament is given by the slope
ā‚u/ā‚ x, while the local curvature is given by the second derivative ā‚ 2 u/ā‚ x 2 . Such a
description is appropriate for the case of a nearly straight ļ¬lament with ļ¬xed boundary
conditions u = 0 at the ends, x = 0, . For this case, the wave vectors q = nĻ/ ,
where n = 1, 2, 3,. . . .
We assume that the chain has no compliance in its contour length, in other words,
that the total arc length ds is unchanged by the ļ¬‚uctuations. As illustrated in Fig. 8-3,
for a nearly straight ļ¬lament, the arc length ds of a short segment is approximately
given by (d x)2 + (du)2 = d x 1 + |ā‚u/ā‚ x|2 . The contraction of the chain relative
to its full contour length in the presence of thermal ļ¬‚uctuations in u is then
1 + |ā‚u/ā‚ x|2 ā’ 1 d x |ā‚u/ā‚ x|2 .
= dx (8.6)
158 F.C. MacKintosh
u ( x, t ) du
Fig. 8-3. From one ļ¬xed end, a ļ¬lament tends to wander in a way that can be characterized by u(x),
the transverse displacement from an initial straight line (dashed). If the arc length of the ļ¬lament is
unchanged, then the transverse thermal ļ¬‚uctuations result in a contraction of the end-to-end distance,
which is denoted by . In fact, this contraction is actually distributed about a thermal average value
. The mean-square (longitudinal) ļ¬‚uctuations about this average are denoted by Ī“ 2 , while
the mean-square lateral ļ¬‚uctuations (that is, with respect to the dashed line) are denoted by u 2 .
The integration here is actually over the projected length of the chain. But, to leading
(quadratic) order in the transverse displacements, we make no distinction between
projected and contour lengths here, and above in Hbend .
Thus, the contraction
= q 2uq .
Conjugate to this variable is the tension Ļ„ in the chain. Thus, we consider the effective
ā‚ 2u ā‚u
H= dx Īŗ +Ļ„ = (Īŗq 4 + Ļ„ q 2 )u q .
2 4 q
Under a constant tension Ļ„ , therefore, the equilibrium amplitudes u q must satisfy
|u q |2 = , (8.9)
(Īŗq 4 + Ļ„ q 2 )
and the contraction
= kT . (8.10)
(Īŗq 2 + Ļ„ )
There are, of course, two transverse degrees of freedom, and so this last answer
incorporates a factor of two appropriate for a chain ļ¬‚uctuating in 3D.
Semiļ¬‚exible ļ¬laments exhibit a strong suppression of bending ļ¬‚uctuations for
modes of wavelength less than the persistence length p . More precisely, as we see
from Eq. 8.9 the mean-square amplitude of shorter wavelength modes are increasingly
suppressed as the fourth power of the wavelength. This has important consequences
for many of the thermal properties of such ļ¬laments. In particular, it means that the
longest unconstrained wavelengths tend to be dominant in most cases. This allows us,
for instance, to anticipate the scaling form of the end-to-end contraction between
points separated by arc length in the absence of an applied tension. We note that it is a
length and it must vary inversely with stiffness Īŗ and must increase with temperature.
Thus, as the dominant mode of ļ¬‚uctuations is that of the maximum wavelength, , we
0ā¼ / p . More precisely, for Ļ„ = 0,
expect the contraction to be of the form
kT 2 2