Journal of the Mechanics and Physics of Solids
50 (2002) 2237 – 2274
www.elsevier.com/locate/jmps
Mechanics of biomolecules
Gang Bao
Department of Biomedical Engineering, Georgia Institute of Technology and Emory University, Atlanta,
GA 30332, USA
Received 10 January 2002; accepted 5 May 2002
Abstract
Over the last few years molecular biomechanics has emerged as a new ,eld in which theoretical and experimental studies of the mechanics of proteins and nucleic acids have become a
focus, and the importance of mechanical forces and motions to the fundamentals of biology and
biochemistry has begun to be recognized. In particular, single-molecule biomechanics of DNA
extension, bending and twisting; protein domain motion, deformation and unfolding; and the
generation of mechanical forces and motions by biomolecular motors has become a new frontier
in life sciences. There is an increasing need for a more systematic study of the basic issues
involved in molecular biomechanics, and a more active participation of researchers in applied
mechanics. Here we review some of the advances in this ,eld over the last few years, explore
the connection between mechanics and biochemistry, and discuss the concepts, issues, approaches
and challenges, aiming to stimulating a broader interest in developing molecular biomechanics.
? 2002 Elsevier Science Ltd. All rights reserved.
Keywords: A. Chemo-mechanical processes; B. Biological material; B. Constitutive behavior; C. Mechanical
testing; A. Mechano-chemical transduction
1. Introduction
It is well known that mechanical forces can in;uence the growth and form of plants
and animals. Bone, for example, loses density without gravitational force. Cells that
form the wall of a blood vessel behave abnormally without shear ;ow. Recent studies
have con,rmed that mechanical forces can a<ect cell growth, di<erentiation, locomotion, adhesion, signal transduction, and gene expression. It is not clear, however, how
biological tissues and cells sense mechanical forces or deformations and convert them
into biological responses. Nor is it well understood how chemical energy is converted
E-mail address: [email protected] (G. Bao).
0022-5096/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 2 2 - 5 0 9 6 ( 0 2 ) 0 0 0 3 5 - 2
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G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
into directed motions and mechanical forces in cells. In general, there is a lack of
understanding of how mechanics and biochemistry are coupled together at the cellular
and molecular levels. The key to addressing all these issues is to study the motion and
force-induced conformational changes of biomolecules.
As deformable bodies, biomolecules such as proteins, nucleic acids and carbohydrates
change their 3D con,gurations (conformations) under mechanical force. In general, a
biomolecule can change its conformation with changes in temperature, solvent chemistry or molecular interactions. Here we use ‘deformation’ to refer to conformational
changes due to mechanical or Brownian forces; we treat the unfolding and relative
motion of globular domains of a protein as special cases of deformation. How, then,
does a biomolecule deform under force? Why do we care? How do such deformations
a<ect biochemical processes in living cells? How do molecular motors generate force
and motion? In the following sections we review some of the progress made over the
last decade or so in revealing the deformation and dynamics of biomolecules, explore
the connection between mechanics and biochemistry, and discuss the concepts, issues,
approaches and challenges, aiming to stimulating a broader interest. Emphasis is placed
on the basic discoveries and new concepts in molecular biomechanics; more technical
details may be found in the references.
2. The biological relevance of mechanical forces
Like all other living systems on earth, the human body is subject to gravitational
force. Further, from embryonic development to maturation, various organs and tissues
in a human body are subjected to shear ;ow or other mechanical forces or deformations.
These forces are spread amongst cells, causing cellular deformation, altering cell–cell
and cell–ECM (extracellular matrix) interactions, and inducing changes to cell behavior
and functions.
As the basic unit of life, cells are complex biological systems. To perform their
specialized functions, cells must express genetic information; synthesize, modify, sort,
store and transport biomolecules; convert di<erent forms of energy; transduce signals;
maintain internal structures; and respond to external environments. All of these processes may involve one or more mechanical aspects. For example, in cell migration,
contractile forces are generated within the cell which ‘pull’ the cell body forward (Fig.
1a). In cell division, the two copies of each chromosome in the mother cell must be
pulled apart in order for the daughter cells to be born (Fig. 1b). In protein secretion,
protein molecules are packaged in vesicles and transported to the cell membrane by
molecular motor running along ,laments in cells.
Conversely, mechanical forces and deformations can induce biological responses in
cells. There is ample evidence to suggest that many normal and diseased conditions of
cells are dependent upon or regulated by their mechanical environment. The e<ects of
applied forces (or deformations) depend on the type of cells and how the forces are
applied to, transmitted into, and distributed within cells. Before having more detailed
discussions on the molecular issues, it may be helpful to review the structural basis of
a typical animal cell. As shown schematically in Fig. 2a, in a rather simplistic view, a
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2239
Contraction
Stress Fiber
(a)
Dividing Cell
chromosome
centrosome
cytoplasm
nucleus
microtubule
(b)
Fig. 1. (a) During cell migration, contractile forces are generated within the cell by stress ,bers, bundles
of the actomyosin ,lament, which ‘pull’ the cell body forward. (b) During cell division, two copies of a
chromosome in the mother cell are pulled apart by tensile forces from microtubules in order for the daughter
cells to be born.
cell is a ‘cage’ (cytoskeleton) wrapped by the plasma membrane and trapping inside
it a nucleus surrounded by the ‘soup-like’ cytoplasm. The cytoskeleton is a system
of protein ,laments—mostly microtubules, actin ,laments and intermediate ,laments—
that gives the cell its shape and the capability for directed movement (Fig. 2b). In a
physiological environment many types of cells are attached to the extracellular matrix
(ECM), a complex network of polysaccharides and proteins secreted by cells, which
serves as a structural element in tissues. Proteins in the ECM, including collagen,
,bronectin, vitronectin and laminin, also play a regulatory role of cellular function
through binding to various receptor proteins, found on the cell surface (Fig. 2c).
For example, some of these receptors are members of the integrin family of transmembrane proteins. They are composed of two units, and (Fig. 2c) and are expressed on the membrane of a wide variety of cells. Attached to the cell cytoskeleton,
integrins are critical to the mechanical stability of cell adhesion to the ECM and to
other cells. They also serve as biochemical signaling molecules in normal and diseased
states of cells, and are involved in regulating cytoskeletal organization. Recent experiments indicate that mechanical forces can in;uence the association of integrin with
the cytoskeleton to form “focal adhesion complexes” (Chicurel et al., 1998a). Further,
integrins are likely a major force transmitter, since they provide the mechanical linkage
between the cell cytoskeleton and ECM. As such, cells may ‘sense’ mechanical forces
or deformations through both integrin-mediated cell–ECM interactions (Chicurel et al.,
1998b) and the subsequent force balance within the cytoskeleton. This balance may
play a crucial role in regulating the shape, spreading, crawling, and polarity of cells.
The question is again how mechanical force balance is being recognized by cells and
transduced into biological responses.
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G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
Golgi
apparatus
cell
membrane
nucleus
mitochondria
microtubules
actin filaments
(a)
Microtubules, 25-nm diameter
Actin filaments, 8-nm diameter
Intermediate filaments, 13-nm diameter
(b)
actin filament
collagen
(c)
Fig. 2. (a) With an average size of 10 –30 m, an animal cell consists of a cytoskeleton wrapped by the
plasma membrane, and a nucleus surrounded by the cytoplasm containing mitochondria, Golgi apparatus, and
other organelles. (b) The cell cytoskeleton consists of microtubules, actin ,laments, intermediate ,laments
and other binding proteins. (c) Adherent cells attached to the extracellular matrix (ECM) through the focal
adhesion complex consisting of integrin and other binding molecules.
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2241
Despite extensive research e<orts over the last few years, the exact molecular mechanisms responsible for mechanochemical transduction in living cells remain unknown.
However, a number of candidates have been proposed. For example, it is possible that
ion transport through the cell membrane can be altered by mechanical forces, especially
tension in the cell membrane (Hu and Sachs, 1997), thus changing the biochemical
processes in cells. It is also possible that components of the cell cytoskeleton, such as
actin ,laments and microtubules, deform under mechanical forces (Arai et al., 1999),
inducing conformational changes of other proteins attached to them (including motor
molecules), and altering their functions. Further, ECM molecules such as ,bronectin
may deform under force, changing their interactions with cell surface receptors including integrins. It has been found that when the ECM is disrupted, ,bronectin can
contract to a fraction of its original length (Ohashi et al., 1999). This may serve as
a mechanosensitive control of ligand recognition (Krammer et al., 1999). It is also
likely that integrins change their conformations under force, altering their binding to
the ligands and thus changing the downstream biochemical processes. In general, a
ligand is any molecule that binds to a speci,c site on a protein or other molecules,
and a receptor (intracellular or on a cell surface) refers to a protein that binds to a
speci,c extracellular signaling molecule (ligand) and initiates a response in the cell.
Existing experimental results suggest that the binding speci,city and aMnity between
a receptor and a ligand can be changed by mechanical forces.
It has become clear over the last few years that the deformation of nucleic acids and
protein molecules under mechanical force may be the key to understanding mechanochemical transduction in cells (Bao, 2000). In a broad sense, it may also play an
essential role in understanding the connection between mechanics and biochemistry,
which is a critical aspect in mechanics in general and biomechanics in particular. Deformation of biomolecules under force and mechanical forces or motions generated by
motor proteins are just examples of this fundamental connection. In simple words, mechanics is the study of force, motion and deformation, while biochemistry involves the
conformation, binding=reaction and transport of biomolecules. There is ample evidence
suggesting the connections between the two; however, fundamental understanding of
this connection needs to be further established (Zhu et al., 2000).
Theoretically, the deformation of DNA or protein molecules under applied forces F
may be analyzed by determining the Gibbs free energy of a system
G=U −
F ‘ − TS;
(1)
where U is internal energy, S is entropy, T is temperature, F are generalized forces
including pressure, and ‘ represent generalized displacements (Reichl, 1980). The
system considered here includes not only protein, DNA or RNA molecules under study,
but also the surrounding solvent (water molecules, ions, and other small molecules
such as sugars), and the internal energy U includes chemical potentials. Under applied
forces, the deformation of the biomolecules can be determined by the equilibrium
condition dG=d xi = 0, i.e., the conformational changes of the molecules are such that
the Gibbs free energy attains its global minimum with respect to the variations in
position xi of all the atoms.
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G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
The Gibbs free energy G in Eq. (1) may include the contributions from intermolecular and surface forces such as covalent and Coulomb interactions, van der Waals
forces, electrostatic forces, hydration and hydrophobic forces, and steric and Brownian
forces, in addition to mechanical forces (Israelachvili, 1992). These forces represent
physical and chemical interactions among atoms and molecules in the system. Thus, the
interplay among internal energy, mechanical work and entropy as indicated in Eq. (1)
underlies the fundamental connection between mechanics and biochemistry. Of particular importance is the entropy S of the system which is determined by the total number
of admissible conformations (or con,gurations) of the biomolecules, along with their
interactions with the solvent (such as the formation of ‘caged’ water molecules). The
second law of thermodynamics dictates that an isolated system with constant energy and
volume attains its maximum value of entropy at equilibrium. For example, a polymer
chain such as free DNA in solution has a random coil conformation which maximizes
the entropy. Elongation of a random-coiled biomolecule under tension reduces the total
number of conformational states it can assume, thus decreasing its entropy. When a
DNA molecule is completely straightened, it can only have one conformational state (a
straight line)—a state that requires an applied force. The resistance to elongation and
the tendency to resume the random-coiled conformation in order to maximize entropy
leads to the entropic elasticity of a polymer chain.
3. Conformational dynamics of nucleic acids
Nucleic acids including deoxyribonucleic acids (DNA) and ribonucleic acids (RNA)
are essential to life. A double-stranded helical DNA consists of two long chains held
together by hydrogen bonding within the complementary Watson–Crick base pairs A–T
(adenine–thymine) and G–C (cytosine–guanine). This base pairing gives rise to informational redundancy and allows for chemical ,delity in replication. The base pairing
can be used to assemble new strands of DNA, allowing for information transmission
to o<spring, and can be used to assemble strands of RNA, letting the DNA control
which proteins are manufactured at what times (Calladine and Drew, 1997). These
functions are realized as well as controlled by gene regulatory proteins and other DNA
associating proteins through their interactions with DNA, which often result in DNA
twisting, stretching, and bending. An RNA molecule, on the other hand, typically exists as a single strand rather than a double-stranded helix, though it can also form
double-stranded helices with the complementary DNA. It contains the base uracil (U)
instead of thymine (T). The elasticity of DNA and RNA has very important biological implications. For example, the bending and twisting rigidities of DNA a<ect
how it wraps around histones to form chromosomes, supercoils during replication,
bends upon interactions with proteins, and packs into the con,ned space within a virus
(Strauss and Maher, 1994; Cluzel et al., 1996; Geiselmann, 1997; Strick et al., 1998).
In what follows, attention will be focused on the deformation of DNA due to the extensive studies it has received; much less work has been done on the deformation of
RNA.
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2243
One helical turn = 3.4 nm
Base
3'
C
G
T
C
T
G
5'
A
G
5'
Hydrogen
bond
A
A
C
G
T
C
A
G
T
2 nm
G
C
C
3'
Sugar-phosphate
backbone
(a)
10
2
Measurement
III
Force, f (pN)
10
IV
f = 65 pN
f = 12 pN
1
WLC model
b
10 0
10 -1
II
f = 0.08 pN
FJC model
DNA
I
10-2
θi
f
f
x
0.0
0.5
1.0
1.5
2.0
Relative Extension, x/L
(b)
Fig. 3. (a) A double-helical DNA chain is about 2 nm in diameter and has ten base pairs (bp) per helical
turn. With a short length, it is rodlike, but over long contours it is ;exible. (b) DNA under stretching
has at least four deformation regimes: (I) entropic-elasticity dominant, (II) worm-like chain, (III) contour
elongation and (IV) B-DNA to S-DNA phase transition. The comparison between experiment and models
(FJC and WLC) are given.
3.1. DNA under tension
As shown schematically in Fig. 3a, a DNA molecule is typically in the form of a
right-handed double helix (the B form, or B-DNA) with about 10 base pairs (bp) per
helical turn (3:4 nm pitch). Over short lengths a double-helical DNA chain is rodlike,
but over long contours the chain gradually curves and bends in arbitrary directions,
2244
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
forming a random coil. Upon stretching by an applied force f the coiled DNA elongates, as illustrated by the experimental result shown in Fig. 3b (Strick et al., 2000b;
Baumann et al., 2000). There are at least four di<erent regimes of DNA deformation under stretching with, respectively, (i) f ¡ 0:08 pN, (ii) 0:08 6 f ¡ 12 pN, (iii)
12 6 f ¡ 65 pN and (iv) f ¿ 65 pN. With low applied forces (f ¡ 0:08 pN, see Fig.
3b), the deformability of DNA molecules is largely controlled by the entropic elasticity
(Bustamante et al., 1994, 2000), i.e., resistance to deformation to maximize entropy.
The simplest model of entropic elasticity of DNA or any polymer chain is the Gaussian
chain model that treats the polymer as a chain of statistically independent segments
whose orientations are uncorrelated in the absence of an external force. It can be
demonstrated (de Gennes, 1979) that the resistance to deformation of such a polymer
chain under stretch is entirely due to the changes in entropy. Under small forces, the
Gaussian chain model leads to a linear relationship between force f and the mean
extension x of the chain
fA
3 x
(2)
=
kB T
2 L
where kB is the Boltzmann constant and T the absolute temperature (1 kB T =4:3 pN nm
◦
at 37 C); A and L are, respectively, the persistence length (the distance over which
two segments of the chain remain directionally correlated) and contour length of the
DNA molecule (the length of DNA when it is fully extended). The persistence length
A is an intrinsic parameter of a polymer chain; it is a measure of resistance to shape
changes of the chain due to thermal ;uctuations (Howard, 2001). For double-stranded
DNA, A ≈ 50 nm (Taylor and Hagerman, 1990).
Another model of entropic elasticity of DNA is the freely joined chain (FJC) model
(Doi and Edwards, 1986), which also treats the polymer as a chain of rotationally
uncorrelated segments but imposes inextensibility. It assumes that the chain consists
of N segments of length b = 2A (the Kuhn length, b = L=N ) and has a con,guration
determined by the ensemble of the segments with di<erent orientations represented
by angles i , as shown in the insert of Fig. 3b. Under applied force f, the potential
energy of the chain with any admissible con,guration is
FJC
= −fb
N
cos i :
(3)
i=1
The partition function of the chain is given by
=
e−FJC =kB T =
=
efb
−
dQefb cos =kB T
cos i =kB T
=
N
N
;
efb cos i =kB T
(4a)
i=1
(4b)
where dQ=2 sin d is the number of segments lying in between and +d and is a proportional constant (Cantor and Schimmel, 1980). In deriving Eq. (4b), one uses
the fact that the angle i of each segment is arbitrary and summing over all possible
con,gurations should cover all possible angles at each segment. It can be shown that
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2245
the Helmholtz free energy of the chain can be given by H = U − ST = −kB T ln (e.g., Howard, 2001) from which the mean extension x of the chain can be calculated
to give
fb
4kB T
9H
9
fb
kB T
= L coth
:
(5)
ln
x=−
sinh
−
= kB TN
fb
kB T
kB T
fb
9f
9f
It is readily seen that with small forces f 6 kB T=b, Eq. (5) recovers the Gaussian chain
model given in Eq. (2); when f¡ kB T=A (b = 2A) the FJC model is fairly accurate
compared with the experimental measurements as shown in Fig. 3b. However, at large
forces the FJC model is inaccurate due to the omission of the thin-rod elasticity of
DNA under bending.
With moderate applied loads, 0:08 6 f¡ 12 pN, the bending rigidity of DNA becomes important, and the worm-like chain (WLC) model represents more accurately the
DNA elasticity based on the Lagrangian function (Fixman and Kovac, 1973; Marko
and Siggia, 1995a)
kB T L 2
E=
A ds − fx
(6)
2 0
where is the curvature, and the force f and the extension x are assumed to be
along the same direction (see insert of Fig. 3b). Clearly, the ,rst term in (3) is the
elastic bending energy and the second term is the work done by the applied force.
The partition function and the free energy H of the DNA as a ;exible polymer
chain can be calculated rather elegantly using an analogy with the quantum mechanics
problem of a dipole in an electrical ,eld, as shown by Marko and Siggia (1995a).
Although the exact force-extension relationship of the WLC model is complicated, it
can be approximated accurately by an interpolation formula (Marko and Siggia, 1995a)
x
fA
1
1
= +
− :
kB T
L 4(1 − x=L)2
4
(7)
As revealed by Smith et al. (1992, 1996), for a single -phage DNA molecule with
an applied force f ¡ 12 pN, the force versus extension relationship can be represented
accurately by the WLC model under uniaxial stretch conditions, with A = 53:4 nm and
L = 32:8 m, as illustrated in Fig. 3. It follows from Eq. (7) that when extension x is
small, the DNA molecule behaves like a linear spring (as in the FJC model); however,
the behavior is nonlinear as x becomes large. As illustrated in Fig. 3c, a force of
2–3 pN is able to stretch the DNA to 90% of its contour length, with the force rising
sharply when x approaches L.
With applied forces 12 6 f ¡ 65 pN, the contour length of DNA increases. Because
of the inextensible assumption in the WLC model, Eq. (7) implies that as x equals L
the force f becomes in,nitely large which is, of course, not realistic. Assuming that
the extension of the DNA backbone depends linearly on the applied force, Eq. (6) can
be modi,ed to give (Odijk, 1995)
2
kB T L
1 L
s
E=
A2 ds −
K
− 1 ds − fx;
(8)
2 0
2 0
s0
2246
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
where K is the stretch modulus and s0 is a reference length (e.g., 0:34 nm=bp). In fact,
Ts=s0 =(s−s0 )=s0 is the local strain. If one approximates DNA as a thin cylindrical rod
of homogeneous isotropic linear elastic material with Young’s modulus Y and radius
R ≈ 1 nm (Voet and Voet, 1995) then
K = R2 Y
A=
R4 ED
YI
=
:
kB T
4kB T
(9)
For -phage DNA in a solution containing 150 nM Na+ ; Y ≈ 300 MPa and K ≈
1000 pN (Bustamante et al., 2000). Solving Eq. (8) for end-to-end distances longer
than the contour length yields (Odijk, 1995)
1 kB T
x
f
=1−
+
(10)
K
L
2 fA
which is valid for applied forces f between 12 and 65 pN.
Finally, when the applied load f reaches about 65 pN, a double-stranded DNA
may undergo a phase transition from B-form DNA (0:34 nm per bp) to S-form DNA
(0:58 nm per bp), with a dramatic increase in length similar to metal yielding, as shown
in Fig. 3b. When f is further increased, the DNA may be overstretched, resulting in
unwinding and unstacking of the molecule (Smith et al., 1992), or even the breakage
of covalent bonds in the DNA backbone. It has been found that the force-extension
behavior of -phage DNA is not very sensitive to di<erent pulling rates, although the
transition from a double helical DNA into two single strands occurs at a higher force
when the pulling rate is larger (Clausen-Schaumann et al., 2000).
3.2. DNA in hydrodynamic 6ow
To understand the interactions between DNA molecules and the surrounding ;uid
during hydrodynamic ;ow, controlled ;ow ,elds were applied to DNA to quantify
relaxation time (Perkins et al., 1994), hydrodynamic drag (Perkins et al., 1995; Larson
et al., 1997), and conformational dynamics (LeDuc et al., 1999; Smith and Chu, 1998;
Smith et al., 1999a). The relaxation time of DNA, de,ned as the time needed for
Brownian forces to globally rearrange its con,guration (Smith and Chu, 1998), was
obtained by attaching a bead to one end of a DNA molecule, exposing it to a uniform
;ow, and recording the time necessary for a fully extended molecule to recoil after the
;ow was stopped. A similar experimental set-up was used to evaluate the dependence
of DNA extension on its contour length L, ;ow velocity v, and the solution viscosity,
& (Perkins et al., 1995). The conformational dynamics of DNA molecules in a shear
;ow (plane Couette ;ow) was studied experimentally (LeDuc et al., 1999; Smith et al.,
1999a). A plane Couette ;ow with shear rate '˙ can be decomposed into a rotational ;ow
with vorticity ! = −'=2
˙ and an elongational ;ow with strain rate ˙ = '=2.
˙ Thus, DNA
molecules in a plane Couette ;ow exhibit both extension and tumbling, as observed
by LeDuc et al. (1999) and Smith et al. (1999a). Unlike the tumbling motion of rigid
ellipsoidal particles in a shear ;ow, which has been analyzed by Je<ery (1922), the
dynamic behavior of DNA molecules in a shear ;ow is more complex due to their
deformations.
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2247
3.3. DNA under torsion
The topology of, and the degree of supercoiling in, DNA plays an important role in
gene replication, transcription, and recombination. The supercoiling of a DNA molecule
can be described by two parameters, twist Tw, the number of helical turns along the
molecule, and writhe Wr, a measure of the coiling of the DNA axis about itself (Voet
and Voet, 1995). For example, a relaxed, linear DNA molecule has a twist Tw equal
to the number of base pairs divided by 10.4 (the number of base pairs per turn for
B-form DNA) and a writhe Wr of zero. The sum of Tw and Wr, Lk = Tw + Wr,
known as the linking number (or link), does not change under continuous deformations
when the two ends of a DNA molecule are torsionally constrained, and is therefore a
topological quantity (White, 1969; Fuller, 1971). The degree of supercoiling (excess
linking number) is characterized by ) = TLk=Lk 0 , i.e., the change in linking number
TLk = Lk − Lk 0 normalized by the linking number Lk 0 of torsionally relaxed DNA. A
DNA molecule with a positive value of ) is overwound, while a DNA with a negative
value of ) is unwound. Most living organism are thought to have partially unwound
DNA with a value of ) ∼ −0:06 (Strick et al., 1998).
DNA supercoiling and the long-range transmission of torsional strain may play
a crucial role in gene expression (Travers and Muskhelishvili, 1998). For example,
it has been found that at the initiation of transcription, the formation of the RNA
polymerase-promoter complex unwinds the DNA, resulting in negative supercoils
(Ansari et al., 1992; Su and McClure, 1994). The rate of gene expression has been
found to be directly dependent on the degree of supercoiling (Condee and Summers,
1992): increased negative supercoiling leads to higher gene expression rates and decreased supercoiling results in the converse.
As shown schematically in Fig. 4a, a relaxed circular or linear DNA can become
supercoiled upon the action of topoisomerases. During the elongation phase of transcription, there is positive supercoiling in front of the transcription ensemble and negative
supercoiling behind the ensemble (Wu et al., 1988). Experimental evidence suggests
that this transcription-induced negative supercoiling can exert a regulatory e<ect over
a distance of 4000 bases or more. It has been shown that when linear (relaxed) DNA
templates are introduced into cultured cells, no transcripts can be detected, whereas
supercoiled circular templates are found to be active (Dunaway and Ostrander, 1993).
Similar to transcription, the initiation of replication also requires negative supercoiling
of the DNA template (Walter and Newport, 2000)—a lack of unwinding prevents the
DNA polymerase from binding.
To better understand the mechanics issues involved in the supercoiling of DNA,
quantitative studies of single linear DNA molecules under twist have been carried out
in which DNA molecules are attached to a treated glass coverslip at one end and
bound to a small magnetic bead at the other end, as shown in Fig. 4b. By applying a magnetic ,eld to the bead, the DNA molecule can be mechanically overwound
and underwound as well as stretched. Two types of experiments were conducted:
force versus extension studies at constant supercoiling, and extension versus supercoiling at constant force (Strick et al., 1996; Allemand et al., 1998). These experiments
have revealed three di<erent regimes of supercoiled -phage DNA in the presence of
2248
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
(a)
Magnets
N
S
Magnetic bead
l
DNA
Glass surface
Relative Extension, l/L
(b)
1.0
8 pN
0.8
1.2 pN
0.6
0.4
0.2 pN
0.2
0.0
-0.1
-0.05
0.0
0.05
0.1
Excess Linking Number, η
(c)
Fig. 4. (a) A relaxed circular DNA is supercoiled. (b) The e<ect of tension on the supercoiling of DNA
is studied with controlled stretch and twist using a magnetic device. (c) Extension as a function of ) of
supercoiled DNA. With a low tensile force (e.g., f =0:2 pN), DNA contraction is the same under overwound
() ¿ 0) and underwound () ¡ 0). With an intermediate force (e.g., f = 1:2 pN), DNA contracts only with
overwounding. With a high force regime (e.g., f = 8 pN), DNA does not contract under both underwound
and overwound.
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2249
tension and with moderate ionic conditions. As illustrated in Fig. 4c, at a low tensile
force (f ¡ 0:5 pN), overwound and underwound DNA molecules behave essentially
the same. Like a phone cord, the DNA molecule shortens ∼ 0:08 m per turn, as
observed experimentally (Boles et al., 1990; Bednar et al., 1994). At an intermediate force regime (0:5 ¡ f ¡ 3 pN), DNA behaves very di<erently in overwinding and
underwinding. The overwound DNA continues to contract, while the extension of an
underwound DNA does not change with ). In the high force regime (f ¿ 3 pN), the
behavior of DNA becomes independent of torsion for both under- and overwound
molecules and assumes a force versus extension curve of a torsion-free DNA.
Models for stretching supercoiled DNA have been developed over the last few years
(Marko and Siggia, 1994, 1995b; Marko, 1997; Moroz and Nelson, 1997; Vologodskii,
and Marko, 1997). For example, by extending the WLC model to include torsional
energy, Moroz and Nelson (1997) was able to successfully model DNA undergoing
small degrees of twist, −0:01 ¡ ) ¡ 0:03, using the Lagrangian function
kB T
E=
2
L
0
[A2 + C(, − !0 )2 ] ds − fx − 2kB T-Lk ;
(11)
where C is the torsional persistence length, , is the twist strain (twist angle per unit
length), !0 =1:85 nm−1 , and - is a nondimensional torque corresponding to the required
twist. Solving Eq. (11) gives (Moroz and Nelson, 1997)
x
1
=1−
2
L
−1=2
- 2
A
Af
1
f − kB TD!02 )
−
+
;
−
+
kB T
2
32
'
L
(12)
where - = !0 )=[1=C + 1=(4A Af=kB T )]; = Af=kB T − -2 , D and ' are, respectively, a
characteristic length and force to be determined. Fitting this formula to the experimental
data yields values of A = 49 nm; C = 120 nm and D = 50 nm. This value of C is
somewhat larger than the experimental result (C = 75 nm) (Crothers et al., 1992).
There have been extensive theoretical studies on the supercoiling of DNA (Schlick,
1995; Moroz and Nelson, 1997). The elastic rod model treats the DNA molecule as
a long, thin and initially straight elastic isotropic rod with a circular cross section.
Although a ,rst-order approximation, this model works remarkably well in predicting the supercoiling of DNA under torsion, including the onset of supercoiling,
the abrupt nonplanar buckling, and the DNA con,gurations as determined by ) (Yang
et al., 1998). However, as pointed out by Marko and Siggia (1994), thermal ;uctuations and entropic e<ects, which have been taken into account in the modi,ed elastic
thin-rod model (e.g., Moroz and Nelson, 1997), play important roles in stabilizing the
interwound superhelical DNA.
It has been well established that DNA replication, condensation and transcription rely
on the conformational matches between DNA and the associated proteins (Erie et al.,
1994). However, deformation of DNA can alter this conformational match, hence the
structural rigidity of DNA as well as related protein enzymes comes into play (Bustamante et al., 2000). Single-molecule biomechanics studies of DNA over the last few
years have revealed many intriguing features of the elasticity of DNA under stretching,
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G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
twisting, and in hydrodynamic ;ow. The constitutive modeling e<orts, such as the
development of the worm-like chain model, have helped capture the essense of the
DNA deformation under force. However, many important issues remain to be addressed.
For example, we still do not have a clear picture of how mechanical forces are applied
to DNA in cells, be they through DNA–protein interactions or due to hydrodynamic
;ow. We still lack a good understanding of how mechanical forces on DNA contribute
to the regulation and control of gene replication and transcription in cells. For example,
if a DNA molecule deforms under force, how does such deformation a<ect the onset
and rate of gene transcription? What is the distribution of torsional deformation along
DNA? What is the force required to open the transcription complex as determined by
the tension and torsion in DNA? Answering these questions will not only advance our
understanding of DNA mechanics, but also help understand the functions of protein
enzymes involved in gene replication and transcription.
4. Motion and deformation of proteins
Proteins are cellular machines and constitute about 60% of a cell’s dry weight. They
perform most of the cellular functions such as transducing signals, providing structural
support, transporting biomolecules, regulating biological responses, and catalyzing biochemical reactions for metabolism. Proteins are formed by an assortment of 20 di<erent
amino acids arranged in a speci,c sequence encoded in the corresponding gene. Typically consisting of hundreds to thousands of amino acids, proteins can have complex
structures that are maintained by hydrophobic and electrostatic interactions, and hydrogen bonding. There are four di<erent levels of structural organization in proteins. The
primary structure of a protein is its amino acid sequence. As shown in Fig. 5, the helices and sheets formed due to regular hydrogen-bond interactions are protein’s secondary structure. Protein domains consisting of -helices and -sheets compacted into
folded globular units are called the tertiary structure. Protein quaternary structure refers
to multiple domains linked by polypeptides (loops) or secondary structures. A native,
folded protein is a solid-like polymeric material since its domains are closely compacted. Speci,cally, many proteins or protein domains have an approximately rounded
shape, refer to as globular proteins.
An essential feature of proteins is that the functions of a protein are determined
by its 3D conformation, i.e., the spatial arrangement of the atoms in its folded structure. As illustrated in Fig. 5d, the recognition and binding of two proteins is largely
dictated by the geometric and chemical complementarity of the interacting surfaces,
ensuring binding speci,city in a ‘lock-and-key’ fashion. Such biomolecular recognition
underlies almost all cellular processes. In other words, although nonspeci,c binding between proteins may occur, it is the speci,c, noncovalent binding of two proteins in the
intra- and extra-cellular environment that enables cells to survive and function. However, as described in more detail below, protein molecules are deformable, thus their
conformations can be altered by mechanical forces. Such alterations can a<ect protein–
protein and protein–DNA recognition, binding and unbinding, leading to changes in
downstream biochemical processes that control cellular behavior and function. Protein
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2251
Fig. 5. The structure of a protein includes: (a) helices, (b) sheets, (c) domains. (d) The complicated
3D structure of proteins dictates their recognition and binding in a ‘lock-and-key’ fashion.
deformation, therefore, is an important concept in molecular biomechanics (Bao, 2000;
Zhu et al., 2000).
Proteins are not static rigid structures. Rather, they are dynamic and undergo constant
motions and structural changes in cells under normal physiological conditions. These
Y movements of domains as well as small-scale
changes include large-scale (∼ 5–50 A)
Y random movements of secondary structures or domains, or ‘breathing’. The
(∼ 0:5 A)
time scales for protein motion and deformation in biological processes may span many
orders of magnitude, ranging from one femtosecond (10−15 s) to a few seconds. For
example, hinge motion of a protein may take only one nanosecond, whereas local
denaturing may take a few seconds (McCammon and Harvey, 1987). To illustrate,
consider a globular protein immobilized on a surface through a lever-arm (e.g., an
-helix), as shown schematically in Fig. 6a. Under an applied force F the small motion
of the protein in the vertical direction can be analyzed based on a mass-spring-dashpot
system shown in Fig. 6b. The corresponding governing equation for displacement y is
myZ + )ẏ + ky = F;
(13a)
where m is the mass and ) the viscous drag coeMcient of the protein, and k is the
elastic constant of the lever arm. Neglecting the inertial e<ect and assuming that the
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G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
Fig. 6. The motion of a protein under applied force F. (a) A globular protein immobilized on a surface
through an -helix. (b) The mass-spring-dashpot system as a model for protein motion.
protein is stationary at t = 0, we have
F
y(t) = (1 − e−(k=)) ):
)
(13b)
If the globular protein has a mass of 100 kDa (1 Dalton = 1:66 × 10−24 g) and a drag
coeMcient of 60 pN s=m (Howard, 2001), and the lever arm has k = 6 pN=nm, the
relaxation time of the protein motion is - = )=k = 10 ns. Note that the above analysis
is valid only if FkB T=a, where a is a characteristic length of the protein.
Energetically, proteins are not very stable in that small changes in temperature
or pH of the solution can convert the proteins from a biologically active (native)
state to a biologically inactive (denatured) state (Voet and Voet, 1995). Speci,cally,
although the changes in enthalpy and entropy between these two states can both
be large (∼100 kcal=mol), the corresponding free energy change is only about 5 –
15 kcal=mol, which is just a few times the energy of a single hydrogen bond in water,
1 kcal=mol (6:92 pN nm per bond). This may be a fundamental reason why a small
force acting on a protein could generate a large conformational change.
The motion and deformation of protein molecules can have various modes under
di<erent forces. These modes include domain motion (Subbiah, 1996), domain deformation and unfolding (Erickson, 1997; Rief et al., 1997; Kellermayer et al., 1997;
Tskhovrebova et al., 1997; Oberhauser et al., 1998), and denaturing of secondary structures (Idiris et al., 2000), as shown schematically in Fig. 7. In general, in domain hinge
motion (Fig. 7a), the individual domains have very limited deformation; the motion
largely consists of rotations of domains around a ;exible hinge (e.g., loops, -helices or
-sheets that join the domains together), or relative (in-plane and out-of-plane) ‘shear’
motions between domains. Mechanistically, these three modes of protein motion resemble the three fracture modes (Mode I, II, III). It may also have mixed mode motions,
such as a combination of rotation about a hinge and twist about an axis of the domain.
Such motions are typically driven by Brownian forces or moments, which may involve
forces of 0.1–1 pN. When force becomes large, e.g. 1–100 pN, the domains may begin
to deform (Fig. 7b). For protein molecules such as titin and tenascin, domain unfolding
occurs when the applied force is ∼ 100 pN (Tskhovrebova et al., 1997; Oberhauser
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2253
Fig. 7. Modes of protein deformation: (a) domain hinge motion, (b) domain deformation and unfolding,
(c) unfolding of secondary structures.
et al., 1998). The denaturing (i.e. melting) of -helices and -sheets (Fig. 7c) may
require an even larger force. Although very rough, the above order-of-magnitude estimates for the force scales may provide some guidance to the analysis of protein
deformation.
Protein molecules are viscoelastic in nature, similar to all polymeric materials, especially those in aqueous environment. As such, their deformations are rate-dependent. In
what follows, we discuss some basic features of protein hinge motion, domain deformation and unfolding, and the implications of protein deformation to receptor–ligand
binding. Since the study of protein deformation is still in its infancy, many basic issues
remain to be addressed. For example, how does the deformation of a protein relates
to its 3D structure? What are the deformation mechanisms? How protein deformations
are coupled to protein function? What is the dynamic behavior of proteins? In addition
to single-molecule studies of protein deformation and dynamics, it is also necessary
to study the deformation of a cluster of two or more proteins binding together, and
protein assemblies.
4.1. Protein hinge motion
Polypeptide chains of more than ∼200 amino acids usually fold into two or more
globular clusters known as domains. These structurally independent domains have an
average size of ∼2:5 nm and possess unique 3D geometries. Most protein domains
have speci,c functions such as the binding of small molecules (ligands) and other
protein domains. The ligand-binding sites in multidomain proteins are often located in
the clefts between domains, making the interactions with di<erent ligands ;exible.
Many protein molecules undergo hinge motion in cells, which has many biological implications (Subbiah, 1996; see also http://bioinfo.mbb.yale.edu/MolMovDB/). In
2254
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
Fig. 8. (a) The structure of E. coli periplasmic dipeptide binding protein. (b) A model for rotational hinge
motion of a protein with two domains linked by an elastic hinge.
these proteins, one or more domains rotate about a hinge due to Brownian force or
molecular interactions such as ligand binding. For example, the motor molecule myosin
‘moves’ on an actin ,lament by generating hinge motion of its head upon ATP binding
and hydrolysis. While some motor molecules rely on hinge motion to convert chemical
energy into mechanical force, other proteins may utilize hinge motion to transduce
signals, regulate interactions, and facilitate enzymatic activities. As an example, the
structure of E. coli periplasmic dipeptide binding protein is displayed in Fig. 8a. It has
two domains linked by a hinge consisting of two -sheets; its hinge motion has been
well documented (http://bioinfo.mbb.yale.edu/MolMovDB/).
Biologically, both the magnitude and rate of the rotational hinge motion are important. Some insight can be gained by performing a Langevin dynamics analysis of
protein hinge motion in which proteins having two domains linked by a hinge is considered (Fig. 8b). The angle of rotation of a domain about the center of the hinge
is assumed to obey the stochastic di<erential equation (McCammon et al., 1976)
˙ + k(t) = R(t)
Z + )r (t)
I0 (t)
or
Z +
I0 (t)
0
t
˙ ds + k(t) = R(t);
(t − s)(s)
(14a)
(14b)
where I0 is the moment of inertial of the domain, )r is the rotational viscous drag
coeMcient, (t) is a dissipation function, and k the elastic constant of the hinge. The
Brownian torque R(t) in Eq. (14) is a random function of time and represents the
collective e<ect of the random collisions of the surrounding molecules in the solvent
with the protein. The ensemble average of R(t) is zero. Similar to almost all the studies
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2255
of stochastic processes, Boltzmann’s ergodic hypothesis is adopted (Co<ey, 1985) and
the time average of a parameter a[ is assumed to be identical to its ensemble average
a.
In most of Langevin dynamics studies of Brownian motion it is assumed that the
values of the Brownian torque at di<erent times are completely uncorrelated, i.e., R(t)
is a ‘white noise’ random process with the autocorrelation function
R(t)R(t + s) = 2kB T)r 2(s);
(15)
where the noise strength 2kB T)r comes from the equipartition theory of classical thermodynamics (van Kamper, 1981; Risken, 1984). In this case Eq. (14a) is employed.
Although this ‘white noise’ random process model is usually adequate, it may not
always be accurate if the solvent is very viscous. In fact, in living cells the viscosity of cytoplasm can be three orders of magnitude higher compared with water
(Crick and Hughs, 1950). As an alternative, colored noise may be considered and
the autocorrelation function of R(t) may obey an exponential-decay function with
time
R(t)R(t + s) = kB T)r exp(−|s|);
(16)
where 1= is the correlation time of the random process, which is an intrinsic property
of the solvent. Consequently, Eq. (14b) should be used in this case according to the
;uctuation–dissipation relation (Uhlenbeck and Ornstein, 1930; Wang and Uhlenbeck,
1945; Fox, 1978). It is possible to analyze protein hinge motion by solving Eqs. (14)
– (16), or to simulate the protein hinge motion using molecular dynamics calculations.
However, it remains to be very challenging to experimentally observe or quantify
protein hinge motion due to its characteristic time scale (1–100 ns).
4.2. Protein domain deformation and unfolding
Protein domains can sustain deformation or even unfolding under normal physiological forces. It has been revealed that, under stretching with AFM or optical tweezers,
the immunoglobulin domains of muscle molecule titin may deform and unfold, with a
characteristic sawtooth pattern (Rief et al., 1997; Kellermayer et al., 1997), as shown
schematically in Fig. 9. Clearly, a force of ∼ 100 pN is required for domain unfolding
of titin. The FN-III domains of the ECM protein tenascin unfold upon stretch in a
similar fashion (Oberhauser et al., 1998). Single-molecule biomechanics studies, both
experimental and theoretical, have been performed on the forced unfolding of ECM
molecule ,bronectin, which regulates many cellular functions through its binding to
integrin (Krammer et al., 1999; Ohashi et al., 1999; Craig et al., 2001). Experimental
evidence suggests that, during cell adhesion and locomotion, the FN-III domains of ,bronectin may unfold and refold due to the tensile and contractile forces the cells apply
to the ECM (Ohashi et al., 1999). More important, this unfolding and refolding may
have signi,cant implications to cell signaling, since the resulting structural changes
of the RGD loop seems to regulate the binding between ,bronectin and integrin, as
revealed by molecular dynamics simulations (Craig et al., 2001; Vogel et al., 2001).
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G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
Force (pN)
Unfolding of one protein domain
100
0
50
100
150
200
Extension (nm)
Fig. 9. A typical force versus extension curve for the deformation and unfolding of proteins with multiple
domains.
It is possible that cells utilize force-induced domain deformation and unfolding in
proteins to alter molecular recognition, thus coupling mechanical force with biochemical
reactions.
Although proteins are structurally far more complex than nucleic acids, their deformations share certain common features with that of DNA. For example, analysis
of the deformation and unfolding of individual domains in titin has revealed that the
force-extension relationship of titin can be described by the WLC model (Eq. (7)),
with A = 0:4 nm, and L = 0 + n (n = 0; 1; 2; : : :) where 0 = 58 nm; ranges from 28
to 29 nm. A similar expression was found to ,t very well the measured force-extension
curve for tenascin (Oberhauser et al., 1998). It is intriguing that the WLC model seems
to work well for domain deformation of protein molecules containing multiple, individually folded domains with -sandwich structures.
Protein domain deformation can be a result of protein–protein and protein–DNA
interactions, since it typically only requires forces of 1–10 pN. But what is the extent of
protein unfolding in cells under physiological conditions? It is likely that only proteins
associated with the cell cytoskeleton and ECM may sustain mechanical forces large
enough to cause domain unfolding, which requires ∼ 100 pN. The contractile machinery
of the actin ,lament system in a cell can generate forces of 10 –100 nN (Galbraith and
Sheetz, 1997), resulting in signi,cant deformation of the ECM. Further, cells under
stretch and shear due to injury, motion, growth and blood ;ow may deform, leading
to large forces in the cytoskeleton. It will be very valuable if direct observations of
domain unfolding of proteins in cells can be made.
The unfolding of biomolecules may also provide insight into the toughness of natural materials such as abalone shell. It has been revealed through AFM studies that
molecules in the ∼ 30 nm thick organic adhesive layer in the abalone shell exhibit sawtooth force-extension curves similar to that of titin and tenascin (Smith et al., 1999b).
The successive unfolding of individual modules in a long molecule seems to be more
fracture resistant, leading to both high strength and toughness. Therefore, the understanding of the deformation and unfolding of protein molecules may lead to novel
engineering design concepts of nanostructured materials and composites.
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2257
4.3. Receptor–ligand binding
Perhaps one of the most important aspects of protein deformation under force is
its e<ect on receptor–ligand binding, an essential process in cells (Lau<enburger and
Linderman, 1993), because speci,c molecular recognition and interactions rely on conformational matches and charge complementarity between the receptor and the ligand.
In general, protein–protein and protein–DNA interactions are determined by many factors, including electrostatic double-layer force, van der Waals force, and ‘steric’ repulsion forces, hydrogen-bonding, and hydrophobic contacts. Most of these factors operate
within short ranges (Israelachvili, 1992). Therefore, the 3D geometry local to the binding pocket of the receptor and ligand contributes signi,cantly to the characteristics
of their binding. Good conformational matches usually lead to strong and long-lasting
bound states, whereas poor conformational matches do the converse. The underlying
reason is that receptor–ligand binding is realized largely through noncovalent bonds
such as hydrogen bonds, which are rather weak individually but, in suMcient number,
can be strong collectively. However, hydrogen bonding operates only within narrow geometric ranges; thus, to form a large number of hydrogen bonds, a good conformational
match between the receptor and ligand at the binding pocket is necessary.
Both domain hinge motion and domain deformation of proteins can a<ect receptor–
ligand binding. As mentioned above, many proteins have the binding pocket for a
ligand formed between the domains and located near the hinge. Therefore, depending
on the magnitude and rate of domain hinge motion, receptor–ligand binding may be
altered or even blocked. Speci,cally, if the rate of hinge motion is much faster than
the time required for certain ligands to di<use into the binding pocket, it is likely that
binding of these ligands is prohibited. This may serve as a ‘selection’ mechanism for
speci,c receptor–ligand binding.
When mechanical forces are applied to a receptor (and ligand), the receptor may
deform, thereby altering the conformational match between the receptor and ligand, as
illustrated schematically in Fig. 10. Most likely it does not take much deformation to
signi,cantly reduce the binding strength because, for hydrogen bonds, increasing the
Y can reduce binding strength (in
distance between the hydrogen-binding atoms by 1 A
ligand
receptor
Without Deformation
ligand
hydrogen
bonds
f
f
receptor
With Deformation
Fig. 10. Receptor–ligand binding can be a<ected by protein deformation. (a) A good conformational match
between the receptor and ligand leads to strong binding and reaction. (b) When the receptor deforms under
force, the binding aMnity decreases due to poor conformational match.
2258
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
terms of the free energy) from 1 kcal=mole (∼ 7 pN nm) to a much lower value. This
suggests that small deformations of proteins could have large in;uences on receptor–
ligand binding. It is important to point out that the speci,c e<ect of mechanical forces
on receptor–ligand binding may depend on the mode and the magnitude of the deformation of a receptor or a ligand (or both), which are in turn determined by the structure
of the molecules, the solvent surrounding, and loading history (including rate). It is
possible that deformation only alters the kinetic rates (Evans and Ritchie, 1997; Merkel
et al., 1999). However, in certain cases, only deformation can expose the binding site,
thus switching between the ‘on’ and ‘o<’ states of the receptor. It is also possible that
deformation can change the speci,city of a receptor/ligand pair, i.e. binding to ligand
B instead of ligand A upon deformation. The latter two e<ects are both well known
for molecules such as integrins that can be activated biochemically. It is likely that
force-induced protein deformation may yield similar results.
While protein deformation can e<ect receptor–ligand binding, the converse is also
true, i.e. ligand binding can induce protein conformational change (Alberts et al., 1994).
Such conformational changes can serve as a means to transduce biochemical signals,
or to facilitate an enzymic reaction. For example, when a ligand is bound to the receptor, the rotational motion of the domains about the hinge can be altered signi,cantly.
In fact, during the receptor–ligand binding processes, both the receptor and the ligand may change their conformations in order to have the ‘best’ ,t, i.e. to realize the
most energetically favorable state. Such ‘induced ,t’ may be driven by entropic e<ects
(Bonggrand, 1999), hydrophobic interactions, and the formation of hydrogen bonds,
e.g. ‘caged’ water molecules trapped in the binding pocket. Here again, mechanical
properties of a protein such as its rigidity distribution come into play, since the conformational changes are ultimately determined by the structure of the protein.
5. The mechanics of molecular motors
Motor molecules, a special class of proteins, play an essential role in many cellular
processes, including muscle contraction, cell movement, cell division, vesicle transport,
signal transduction, and DNA replication, condensation and transcription (Alberts et al.,
1994). Speci,c motor molecules include the kinesin and dynein superfamily (Hirokawa,
1998; Okada and Hirokawa, 1999), the myosin superfamily (Mermall et al., 1998),
and numerous proteins involved in the interactions with DNA. These motor molecules
directly convert chemical energy into mechanical work via conformational changes
induced by ATP hydrolysis. As more detailed 3D structures of motor molecules (e.g.
Cramer et al., 2000) and images of their movements (e.g. Walker et al., 2000) become
available, the structure-function relationship of molecular motors has begun to emerge.
It is highly likely that molecular motors utilize protein conformational changes to store
energy, generate motion, and control/regulate motor function (Vale and Milligan, 2000;
Kikkawa et al. 2001). However, for most of the motor proteins, their mechanics, i.e.
how they move, how their conformational changes are related to ATP hydrolysis, how
the force generation is related to their structural rigidity, is still largely unknown.
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2259
Fig. 11. Kinesin and myosin motors. (a) Upon ATP hydrolysis, myosin molecules generate sliding motion
between antin and myosin thick ,lament through the swing of the lever arm. (b) Kinesin molecules ‘walk’
along microtubules by alternating the binding and unbinding of its two heavy chains to the microtubule.
5.1. Kinesin and myosin motors
To date the most extensively studied motor molecules are muscle myosin and conventional kinesin. Myosin II (skeletal and smooth muscle myosin) is a two-headed
linear motor that generates sliding between actin and myosin ,laments; only one head
attaches to actin at a given time. As shown in Fig. 11a, each myosin head consists of
a globular motor domain and a long lever-arm domain (Rayment et al., 1993). ATP
hydrolysis results in the swing (rotational hinge motion) of the lever arm, converting
chemical energy into mechanical motion. Although early measurements of myosin mechanics revealed an average step size of 11 nm and a force of ∼ 4 pN (Finer et al.,
1994), it was suggested that the step size is only ∼ 4 nm for skeletal muscle myosin
and ∼ 7 nm for smooth muscle myosin (Molloy et al., 1995; Howard, 1997). The
4 –11 nm step size due to the ‘working stroke’ of the myosin head and the 36 nm
separation between two consecutive binding sites of myosin on actin ,lament implies
that multiple myosin molecules must work together to move continuously along actin,
which is indeed the case in muscle contraction and cell movement. It is still debatable, however, as to how many steps per each hydrolyzed ATP, how large is the force
generated by myosin, and how conformational changes of myosin are related to ATP
hydrolysis and actin binding.
Kinesin is also a two-headed linear motor, transporting vesicles along microtubules.
As illustrated in Fig. 11b, each head consists of a heavy chain with an ATP binding
2260
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
domain and a light chain on which a cargo is attached. The two light chains form a
coiled-coil neck domain, as can be seen from Fig. 11b. Unlike myosin, a single kinesin
molecule can move along a microtubule for several micrometers before dissociating,
with a step size of 8 nm (Schnitzer and Block, 1997), the distance between two subunits
of tubulin. With very low applied mechanical load, kinesin advances one step per each
ATP hydrolysis; however, it remains to be seen if this one-to-one correspondence also
holds true at high load. It has been uncovered that the force required to stop a single
kinesin molecule is about 5 –7 pN (Meyhofer and Howard, 1995; Kojima et al., 1997),
and the rate of kinesin movement decreases linearly with applied load (Svoboda and
Block, 1994).
The 3D structure of kinesin indicates that the two putative tubulin-binding regions
are separated by ∼ 5 nm, implying that there must be conformational changes in kinesin
in order to separate the two heads by 8 nm. Indeed, the work by Rice et al. (1999)
suggests that the neck linker of kinesin undergoes a large conformational change upon
ATP binding, assisting the attachment of one head to microtubule. The other head
then moves 16 nm forward driven by ATP hydrolysis and the re-zippering of the neck
linker.
Despite extensive studies on kinesin, it is not well understood how the 8 nm mechanical step is generated, and how it is related to the conformational changes of the
neck linker, and ATP hydrolysis. The hand-over-hand model hypothesizes that the two
kinesin heads are symmetric and, with each step of kinesin, the coiled-coil neck domain
◦
rotates 180 . However, as revealed by recent experimental evidence, such rotation does
not occur during kinesin movement (Hua et al., 2002). As an alternative model, it was
suggested that the two kinesin heads are not symmetric in that one is always ahead
of the other during each successive cycle, i.e., the two heads move in an inchworm
fashion, consisting with the experimental observations. A better understanding of the
mechanochemical coupling in conventional kinesin, which involves its structural features, conformational changes, binding to microtubule, ATP hydrolysis, and Brownian
motion, will undoubtedly lead to great advances of the study of the kinesin superfamily, as well as the myosin superfamily, since the two motor families share certain
similarities.
5.2. ATP synthase
ATP synthase is a special motor enzyme, which can either pump protons across
an insulating membrane against the electrochemical gradient using ATP hydrolysis or
manufacture ATP from ADP and phosphate using the energy derived from a transmembrane protonmotive gradient (Boyer, 1993). The general structure of ATP synthase is
shown schematically in Fig. 12 in which the and 2 subunits are omitted for clarity.
It consists of a transmembrane component, F0 , comprising the proton channel, and a
soluble component, F1 -ATPase, containing the catalytic sites. It has been observed that,
◦
when ATP is added, the F1 -ATPase rotates 120 step-wise, with one ATP molecule
hydrolyzed per step (Noji et al., 1997). As perhaps the world’s smallest rotary engine,
ATP synthase carries out both its synthetic and hydrolytic cycles through conformational changes of its domains in the F1 component (Fig. 12a). For example, it has
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2261
Fig. 12. (a) Schematic of the structure of ATP synthase (the and 2 subunits are not shown). It has a
transmembrane portion, F0 , and a soluble component, F1 , which contains catalytic sites located at the interfaces. (b) Hydrolysis of ATP induces hinge motion of the domain, which in turn drives the rotation
of F1 .
been suggested that in pumping protons, the rotation of the '-subunit in the F1 component, which contains catalytic sites located at the interfaces, is due to the hinge motion
of the domain driven by ATP hydrolysis (Fig. 12b). In analyzing energy transduction
in ATP synthase, it is critical to understand how the protein enzyme converts the free
energy of nucleotide binding into elastic energy. Here, protein deformation serves as
the key mechanism for mechanochemical coupling, generating mechanical torque from
the free energy of ATP binding, or producing ATP using torques generated by the
protonmotive force (Elston et al., 1998; Wang and Oster, 1998).
5.3. Motors involved in DNA–protein interactions
In order to conduct gene transcription, replication and DNA condensation, cells rely
on many motor proteins, including RNA polymerase, DNA polymerase, and topoisomerases, to name only a few. E. coli RNA polymerase, for example, carries out the
synthesis of an RNA copy of the template DNA by progressing along it at speeds
more than 10 nucleotides per second. T7 DNA polymerase, on the other hand, catalyses DNA replication at rates of more than 100 bases per second. It has been uncovered
over the last few years that forces applied to enzymes can a<ect DNA replication and
transcription in cells. To reveal the e<ect of mechanical force, single-molecule studies of RNA and DNA polymerases have been conducted using optical tweezers, as
shown schematically in Fig. 13. Speci,cally, a polymerase is immobilized on a surface
2262
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
Fig. 13. A single-molecular experiment of DNA transcription against applied force. A tensile force is applied
to an immobilized RNA polymerase through the transcribing DNA which is attached to an optically trapped
bead. The inset shows a typical velocity versus force curve for transcription.
and a tensile force is applied to it through the transcribing DNA, which is attached
to a bead trapped by the optical tweezers. Transcription of DNA is then proceeded
against the applied force. It was found that at saturating nucleotide triphosphate (NTP)
concentrations, E. coli RNA polymerases stall at applied forces of 14 –25 pN (Yin et
al., 1995; Wang et al., 1998), and the transcription velocity decreases with applied
force (Wang et al., 1998, see the insert of Fig. 13). The RNA polymerases have di<erent intrinsic transcription rates and di<erent propensities to pause and stop (Davenport
et al., 2000). It was also found that the velocity of T7 DNA polymerase along their
single stranded DNA templates is sensitive to tension in the templates (Wuite et al.,
2000), suggesting that deformation of DNA under tensile forces may a<ect DNA–
polymerase interactions. Further, torsion in DNA molecules appears to in;uence the
relaxation of DNA supercoils by topoisomerase molecules (Strick et al., 1996, 2000a),
indicating again that force in DNA can alter DNA–protein interactions during gene
expression and transmission. Although the exact roles of mechanical force in these
interactions remain elusive, it is possible that deformation of DNA may change the
energy barriers the motor molecules need to overcome or change the conformation
(and therefore functionality) of the motor proteins.
6. The connection between mechanics and biochemistry
As illustrated above, mechanics and biochemistry are tightly coupled together in the
studies of DNA elasticity, protein conformational dynamics, receptor–ligand binding,
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2263
and motor molecules. Speci,cally, forces and motions (including deformations) are
involved in conformational dynamics, binding, reaction and transport of biomolecules.
For example, bending, twisting and stretching of DNA are clearly mechanical processes;
however, they are crucial to the expression and transmission of genetic information,
which is an important subject of biochemistry. Force-induced protein conformational
changes can alter many biochemical processes, including reaction kinetics, biomolecular recognition, and protein–DNA interactions. In addition, as cellular machines, motor
molecules convert chemical energy directly into mechanical motion. Mechanical forces
may also a<ect protein traMcking and secretion, ion transport through membranes, and
enzymatic activities in living cells. It is clear that in performing mechanics analysis of
the deformation and dynamics of biomolecules, it is necessary to consider the biochemical aspects such as the sequences of DNA and RNA, the conformations of proteins,
the chemistry of the solvent, polymerization and depolymerization of ,laments, and the
reaction kinetics.
6.1. Mechanochemical transduction
Mechanochemical transduction in cells involves two di<erent aspects. The ,rst is
how mechanical forces and motions are being ‘sensed’ by cells and transduced into
biochemical and biological responses, the second is how chemical reactions generate
mechanical forces or motions through molecular motors and machines, both are still
largely unknown. A possible mechanism for mechanochemical transduction in cells is
the induced conformational changes of proteins, as suggested in Bao (2000) and Zhu
et al. (2000). For example, the receptor-mediated signaling processes in cells can be
regulated by deformation of ECM molecules such as ,bronectin, resulting in the exposure of an RGD (arginine-glycine-aspartate) sequence for binding to integrin (SanchezMateos et al., 1996), whose functions include both adhesion and signal transduction.
As discussed earlier, integrins are likely a major force transmitter, because they provide
the mechanical linkage between ECM and cytoskeleton. As such, these receptors may
also serve as a force sensor. In addition to the ability of switching between resting
and active states of proteins upon ligand occupancy, it is conceivable that cells utilize
more continuous conformational changes in response to various forces to produce more
gradual exposure of the functional domain or a<ect the binding rate for the downstream
signaling molecules, thereby transducing mechanical forces and deformations into biochemical responses. Taking together, it is essential to study protein deformation in
order to understanding mechanochemical transduction in cells.
6.2. Reaction kinetics
Altered kinetics of biochemical reaction by mechanical forces is another major aspect
of mechanochemical coupling (Khan and Sheetz, 1997). As shown in Fig. 14, the
applied forces can lower the energy barrier of molecular unbinding, thus in;uencing
the reaction kinetic rates (Evans, 2001). To illustrate, consider the binding between
two proteins, a receptor and a ligand, under an applied force f in the direction of
pulling them apart. Based on the fracture analysis of solids, Bell (1978) predicted that
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
Interaction Energy
2264
f : mechanical force
Distance
binding
unbinding
f=0
f=0
Fig. 14. The applied mechanical force can lower the energy barrier of molecular unbinding, thus in;uencing
receptor–ligand reaction kinetics.
the applied force f will increase the unbinding rate and suggested that the equilibrium
association constant Keq of receptor–ligand binding is dependent on the applied force
f according to (Bell et al., 1984)
fx
0
;
(17)
Keq (f) = Keq exp −
2kB T
0
where Keq
is the value of Keq at f = 0, and x is a characteristic length. More recently,
Evans and Ritchie (1997) developed a theoretical model for bond failure under force
based on Kramers’ rate theory (Kramers, 1940). This model predicts that the bond
strength varies with loading rate, which was experimentally veri,ed by measuring the
unbinding force of biotin–avidin and biotin–streptavidin pairs (Merkel et al., 1999).
The energy-based approach discussed above is simple and elegant. However, it is
often necessary to quantify the deformation of proteins in order to add the mechanical
work to the binding energy. This is true whether the applied force is in the unbinding
direction or not. For example, even if the applied force on the receptor is perpendicular
to the unbinding direction of ligand, as shown in Fig. 10, the conformational changes
of receptor may still alter the binding strength. In addition, depending on the loading
history, protein deformation can be local as well as global, which a<ects not only the
strength but also the number of contributing hydrogen bonds. For example, when the
loading rate is low, bonds may dissociate nearly spontaneously without inducing global
deformation of the receptor–ligand complexes, thereby only involving interactions local
to the binding pocket. In contrast, high loading rate may result in global deformation,
which may alter the dissociation path by involving interactions with structures away
from the binding pocket. Thus, protein deformation may play an essential role in
determining the rate of bond dissociation; its e<ects on the changes of energy landscape
should be included for a complete description of the relation between force and kinetic
rates.
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2265
6.3. E:ect of solvent
A basic feature of the deformation of proteins and DNA is that they are always in
an aqueous environment in order to function. Solvent chemistry, including the polar
character of water and ionic strength, plays an essential role in the conformational
dynamics of biomolecules. For example, one of the major determinants of protein
folding is to bury the nonpolar side chains in the core of the domain. The 3D structure
of a protein is formed and stabilized largely by establishing hydrogen bonds among its
amino acids, and with water molecules (Creighton, 1993). However, the concentrations
of ions in the solvent can signi,cantly change the structural stability of a protein or
DNA molecule, altering its deformability. For example, the elasticity of -phage DNA
molecules in di<erent bu<er solutions was found to change with the ionic strength I
(Baumann et al., 1997)
1
I=
(18)
ci Zi2 ;
2 i
where ci is the molar concentration of the ith ionic species and Zi is its ionic charge
(Voet and Voet, 1995). Speci,cally, the persistence length A was found to decrease as
I increases (Baumann et al., 1997), obeying a nonlinear Poisson–Boltzmann theory
A = A0 + 0:0324=I;
nm
(19)
where A0 = 50 nm and I is in molar units. The stretch modulus K (Eqs. (8) – (9)),
however, was found to increase with increasing I , contradicting with the classical
elasticity theory in which the bending and stretching rigidities are related by A =
KR2 =4kB T for a circular rod as given by Eq. (9).
6.4. Mechanochemical equivalence?
There is ample experimental evidence to suggest that there might be some equivalence between mechanical and biochemical e<ects as far as cells and biomolecules are
concerned. For example, mechanical forces can induce neural cell growth, so could
biochemical stimuli (Smith et al., 2001). Mechanical forces can denature proteins, so
could ions in solution (and temperature, see Paci and Karplus, 2000). Mechanical
forces can alter receptor–ligand binding, so could other ligands that bind at di<erent
sites (allostery). It is possible that in cells mechanical forces or motions function the
same as biochemical agents, such as ligands, ions, growth factors and hormones. This
might be the fundamental reason why living cells can alter their behavior and functions in response to changes in the mechanical environment. Although the e<ect of
mechanical forces can be understood theoretically from a thermodynamics view point
as indicated by Eq. (1), it is still necessary to elucidate the basic features of the possible mechanochemical equivalence, especially when nonequilibrium thermodynamical
processes are involved. Needless to say, not all biochemical e<ects can be induced mechanically, nor can all mechanical e<ects on cells be reproduced biochemically. This
possible mechanochemical equivalence is again related to the most critical aspect of the
2266
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
connection between mechanics and biochemistry, i.e., the 3D conformations of proteins
determine their functions, and these conformations can be altered by mechanical forces.
7. Major mechanics issues and challenges
The development of molecular biomechanics as an emerging ,eld inevitably involves
new concepts in mechanics, new theoretical and experimental approaches, and new
challenges. Given below is a brief discussion of these aspects.
7.1. New features in mechanics
In addition to the direct mechanochemical coupling discussed in Section 6, entropic
elasticity and Brownian forces play essential roles in the mechanics of proteins and
nucleic acids. As mentioned before, the tendency of maximizing entropy resists deformation that imposes restrictions on the accessible conformational states. Brownian
forces, on the other hand, act in many processes even when the entropy of a system is
constant. Although the entropic contribution to free energy is important for polymeric
materials such as rubber, entropic elasticity may become dominant in the deformation of singlel biomolecules. Indeed, as illustrated by de Gennes (1979), under forces
¡ 0:08 pN, the elasticity of DNA is almost entirely due to changes in entropy, leading
to a linear force-extension relationship. The derivation of the FJC model for DNA
under tension given in Section 3.1 is clearly di<erent from the continuum mechanics
approach in that: (1) statistical distribution of the orientation of the chain segments
is considered and (2) the entropy of the chain under tension is calculated. The use
of free energy in calculating extension, however, is analogous to using strain energy
function to calculate strain. Note that if temperature T =0◦ F, the elastic spring constant
in Eq. (2) (and in FJC and WLC models) becomes zero, indicating that there is no
resistance to deformation. These features may also have engineering implications. Since
a DNA molecule has a speci,c sequence, is very stable and easy to synthesize chemically, it may become an important material for nanotechnology (Quake and Scherer,
2000). Thus, understanding the entropic elasticity of DNA may help design functional
nanodevices.
The energy involved in Brownian motion scales with kB T which, at room temperature, equals 4:1 pN nm. For large-scale materials and structures, Brownian forces and
motions are negligible. However, the motion and deformation of DNA and proteins
can be driven by Brownian forces. The reason is simple: these biomolecules are of
nanometer size, and their deformations require only piconewton forces. Therefore, the
energy of 1kB T is enough to cause signi,cant conformational ;uctuations. In fact,
Brownian forces due to the ;uctuation of water molecules can assist the breakage of
hydrogen bonds in a protein domain (Lu and Schulten, 2000). It is clear that Brownian motion induced di<usion is essential for the encountering, rotation, binding and
reaction of proteins in the cytoplasm, and for protein–DNA interactions in the cell
nuclei (Kramers, 1940; Hanggi et al., 1990). It is likely that cells utilize ‘biased’ or
‘recti,ed’ Brownian motion to facilitate speci,c cellular processes including binding,
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2267
transport, signaling, and energy conversion (Simon et al., 1992; Astumian, 1997; Fox
and Choi, 2001), i.e., generate order from disorder, although its thermodynamic basis
is not well understood (Feynman et al., 1963). The interplay between Brownian forces,
viscous drag, and chemical energy due to ATP hydrolysis may provide the basis for a
rich class of phenomena in living cells and is an exciting research topic in molecular
biomechanics.
Similar to the studies of engineering materials, it is important to determine the
constitutive behavior of proteins and nucleic acids in order to predict their deformations under force. This involves both the formulation of the constitutive equations, and
the determination of the material constants. Such equations may be in the form of
force-extension, torsion-twist, or stress–strain relations. However, in performing mechanics analysis of the deformation of proteins, the continuum assumption may not be
valid. Further, it may not always be necessary nor desirable to de,ne local stresses and
strains, which usually depend on the speci,c amino acid sequence and the 3D globular structure unique to the protein molecule under study. Therefore, the constitutive
behavior for one protein de,ned in terms of stresses and strains may not be applicable to other proteins. The global deformation, such as overall elongation, unfolding or
twisting, however, may be insensitive to the amino acid sequence and thus easier to
use in making predictions. Most of the biomechanics studies of proteins and nucleic
acids concerns mainly their deformations, or conformational changes under force, not
their mechanical failure strength. Therefore, it may be more convenient to use forces
(or moments) and extensions (or rotations) in the deformation analysis, although in
some cases stresses need be de,ned in order to satisfy stress boundary conditions such
as shear stress and pressure.
7.2. Major challenges
Like any emerging ,eld, the development of molecular biomechanics faces many
challenges. For example, to understand how forces regulate cell function it is necessary
to know how the applied forces are transmitted through cell–ECM and cell–cell contacts
and distributed within the cells. Contractile forces generated by cells during locomotion
(Dembo et al., 1996; Dembo and Wang, 1999) and mitosis (Burton and Taylor, 1997)
have been measured with a deformable-substrate method or microelectromechanical
systems (MEMS) technology (Galbraith and Sheetz, 1997). However, to date it still
lacks good experimental and theoretical methods to quantify how these forces are
distributed among various subcellular structures inside a cell. The reason is that a
signi,cant portion of forces is supported as well as generated by the cell cytoskeleton;
however, cells are active and the cytoskeleton structures are dynamic, with simultaneous
polymerization and depolymerization of the major cytoskeletal components including
microtubules, actin ,laments, and intermediate ,laments. This poses a major challenge
to both modeling (possibly ,nite element) and experimental e<orts.
Structural (static and dynamic) analysis and constitutive modeling of the cell cytoskeleton, major subcellular components such microtubules, actin ,laments and stress
,bers, membrane systems, cell–ECM linkages, and the interactions among them is another challenging task. This includes the viscoelastic behavior and reorganization of
2268
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
cytoskeletal networks, bending and buckling of various ,laments, docking, fusion and
budding of vesicles, the formation and strength of the cell adhesion complexes, and the
strength and structural principles of elastin, collagen, and other large protein assemblies.
Since the diameter of these ,laments is on the order of 10 nm, their deformation (and
constitutive models) may exhibit interesting features not found in large scale structures.
Further, one may use the constitutive equations of these ,laments as building blocks
for the development of a structural model for the cell cytoskeleton. Analyzing these
subcellular structures will not only provide a better understanding of the biological issues involved, but may also lead to new engineering design principles and approaches
for developing advanced materials and composites.
It is very critical and challenging to quantify the deformation of DNA, RNA and
protein molecules under applied mechanical forces and moments. Experimental techniques for force measurement in single-molecule biomechanics studies have been developed over the last decade or so including atomic force microscope (AFM, with a
force range of 0.01–100 nN), optical tweezers (0.01–150 pN), micropipette/microneedle
(0.01–1000 pN), surface force apparatus (10 nN–1 N) and magnetic beads (0.01–
100 pN) (Mehta et al., 1999; Leckband, 2000; Strick et al., 2000b). Quantifying protein
deformation, however, is more diMcult because a typical protein is about 5 –50 nm in
size; its deformation is usually a few nanometers or even just a few angstroms (Alberts et al., 1994; Creighton, 1993), which cannot be observed directly using optical
microscopy. As a potential ‘deformation gauge’, the ;uorescence resonance energy
transfer (FRET) between donor and acceptor dye molecules is extremely sensitive to
their relative distance (¡10 nm); it has been used for studying the conformational
changes during RNA folding (Zhuang et al., 2000), and coexisting conformations of
,bronectin in cell culture (Baneyx et al., 2001). When properly labeled with FRET dye
molecule pairs, proteins undergoing deformations of ¿ 1 nm may be detected. It may
also be possible to study conformational dynamics of proteins such as domain motion using FRET in ;uorescence correlation spectroscopy (FCS). Another challenge is
to generate controlled deformation in a single-molecule experiment. Unlike large-scale
structures, proteins are so small that it is hard to anchor them, grab them and deform them mechanically without causing alterations in their conformation. It may be
necessary to genetically engineer ‘handles’ or ‘tethers’ on proteins to facilitate the
experimental studies. Despite all these diMculties, perhaps systematically classifying
and quantifying the deformation of proteins and DNA is one of the most important
research topics in molecular biomechanics. For example, we need to understand how
di<erent modes of deformation and dynamics of proteins are related to their structural
rigidities (or sequences), quantify the time scales of protein motions and deformations
as determined by the structural features, and explore whether protein function can be
predicted directly from its mechanical properties.
Molecular dynamics (MD) simulations are based upon numerical solutions of the
classical Newtonian equations of motion in which the force exerted on each atom
is given by the negative gradient of the potential energy function with respect to
the position of the atom (McCammon and Harvey, 1987; Wang et al., 2001). These
simulations have become the ‘workhorse’ for exploration of structure dynamics of
biomolecules, protein/DNA interaction, and the e<ect of solvent. Examples of MD
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
2269
simulations include the predictions of protein folding, unfolding, motion, unbinding, and
more recently its elasticity. For example, the forced unfolding of the FN-III domains of
,bronectin can be simulated (Krammer et al., 1999; Vogel et al., 2001). The unbinding
between biotin and streptavidin molecules has also been analyzed using MD simulations
(Grubmuller et al., 1996). However, currently MD simulations can be performed only
for a short time period, typically a few nanoseconds. The results of these simulations
are often qualitative rather than quantitative due to the limitations of the force ,elds,
i.e., the potential functions used to characterize the atomic and molecular interactions
such as covalent and Coulomb interactions, electrostatic and van der Waals forces,
and hydrogen bonding. These force ,elds are usually obtained using direct quantum
mechanics calculations, or based on experimental measurements, of the interaction of
limited pairs of atoms. Since many biomolecular processes in cells occur on a time
scale ranging from a few microsecond to a few second, it is necessary to seek a new
numerical approach to simplify the simulations while still preserve the basic features
of the molecular interactions. It is unclear if a continuum-based approach similar to a
,nite element method based on the minimization of free energy is valid for studying
molecular biomechanics issues.
Theoretical modeling of the deformation and dynamics of proteins and nucleic acids
is a very rich area, and some of the basic approaches in applied mechanics are still
valid. For example, the dynamics of proteins can be modeled at di<erent levels: for
example as a rigid body, a mass-spring-dashpot system, an Euler beam, or a full 3D
structure. This is analogous to the study of large-scale engineering systems such as
the motion of a tall building. It is worth mentioning, however, that the mechanics
of biomolecules is quite di<erent from conventional engineering mechanics in that
mechanics needs to be integrated with thermodynamics, statistical physics, biochemistry
and molecular biology.
8. Concluding remarks
As a basic discipline in engineering and science, mechanics has made tremendous
progress over the last century. There is little doubt that mechanics can play an important role in advancing life sciences and medicine. Cellular and molecular biomechanics, addressing issues in cell locomotion, cell adhesion, cell spreading, cell–ECM
interactions, and the dynamics of cell cytoskeleton, is important to wound healing,
immunology, cell and tissue engineering, and cancer studies. Molecular biomechanics
issues in functional genomics, proteomics and nanotechnology have also begun to attract attention, including those in protein and DNA conformational dynamics, di<usion,
reaction, secretion and transport of biomolecules, and the structure-function relationship
of molecular motors and machines. The rapid accumulation of sequence and structural
information about proteins and nucleic acids, and the fast development of advanced
technologies over the last few decades has provided a basis for systematic and quantitative studies of the mechanics of biomolecules. This presents great challenges and
opportunities for researchers in solid mechanics.
2270
G. Bao / J. Mech. Phys. Solids 50 (2002) 2237 – 2274
Acknowledgements
GB would like to thank Andrew Tsourkas and Yangqing Xu for stimulating discussions. This work was supported in part by the Coulter Foundation, and a Cutting Edge
Research Award from Georgia Institute of Technology.
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