Tensegrity Structures - medinfo

•To eplain how biological tissues form and function, it is
important to know that there are different type of regulatory
signals:
•chemical
•mechanical
integrated inside the cell
•Many of the molecules that mediate signal transduction
and stimulus-response are physically bound to insoluble
scaffoldings within the cytoskeleton and the nucleus
•In this type of “solid-state” regulatory system,
mechanically-induced structural arrangements could
provide a mechanism for cellular biochemistry
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Cytoskeleton
Cytoskeleton: complex network of filamentous proteins
extending throughout the cytoplasm
Localization: cytoplasm of Eukaryotic cells
Functions:
Cell shape
Anchoring of organelles and cellular structure
Organelle movement
Tensile strength
Cell polarity
Chromosome movement
Motility (crawling, chemokinesis, chemotaxis,
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endo- and exo-citosis)
Components:
Intermediate filaments
Microtubules
Actin microfilaments
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Intermediate Filaments
•Intermediate filaments are important
components of the cell's cytoskeletal
system.
•They may stabilize organelles, like
the nucleus, or they may be involved
in specialized junctions.
•They are distinguished from "thin
filaments" by their size (8-10 nm) and
the fact that thin filaments are
obviously motile.
•Recent evidence indicates that Intermediate Filaments may
also have dynamic properties.
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Microtubules
• They are important in
changing cell shape and
moving organelles.
• They generate forces by
polymerizing and
depolymerizing the protein
tubulin.
•The spindle that moves chromosomes to the mitotic poles at
anaphase is made of microtubules.
•In the developing nervous system, microtubules help growing
neurons search for the appropriate contact cells.
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• Microtubules also generate the small-scale movements
of cilia and flagella (cilia remove wastes from the
lung and sweep eggs from the ovary into the
oviducts.Flagellated cells maintain the flow of water
through the bodies of sponges.)
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Actin microfilaments
•Cytoskeletal actin filaments are
typically nucleated at the plasma
membrane, rather than the
centrosome, which, instead, acts as
an organizational centre for
microtubules.
• When located directly underneath
the plasma membrane,
actin filaments comprise part of the cell cortex, which
regulates the form and motion of the surface of the cell,
playing a key role in whether or not a cell develops
projections such as filopodia or microvilli.
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• A number of external factors generally control actin
filament nucleation and enable the cytoskeleton filaments to
change their characteristics rapidly upon signalling.
• A group of special proteins, two of which are actin-related
proteins (ARPs) that are very similar to actin, are chiefly
responsible for nucleation catalysis.
• This ARP complex is most prominent in locations where
actin filament growth needs to be a brisk process, and is
associated with various signalling molecules and
components of the plasma membrane.
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Cytoskeleton Mathematical Model
Cytoskeleton modeled using two different approaches:
the Maxwell viscoelastic
model
the Tensegrity model
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Maxwell Model
Some assumptions:
• Cytoskeleton is modelled as a sphere.
• Cytoskeleton is deformed by the action of a interaction
force f.
• Deformation induces a change in the radius r.
• The shere contains a rigid structure composed by:
- an elastic spring k2
In parallel with:
- the parallel of two viscous elements
k1
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Maxwell Model
• This model is characterized by visco-elastic properties.
• This mathematical model has been already used in
biomedical field in order to model the behavior of white
blood cells.
• In this model, it is supposed that cytoskeleton is deformed
by the action of force f.
• The cytoskeleton is simulated as a sphere containing a rigid
structure composed by an elastic spring in series with the
parallel of two viscous elements:
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• The spring of elastic constant K2
supplies the force necessary to
maintain the shape, while the elastic
elements K1 and µ1 are associated to
the cellular cytoplasm.
• The deformation of the cytoskeleton due to the action of the
force f induce a change in the radius r of the sphere
represented cytoskeleton following the equation:
K1 ( f − T )
k1 k 2
dr
df / dt
=−
+
−r
dt
µ ( K1 + K 2 ) ( K 1 + K 2 )
µ ( K1 + K 2 )
• k1, k2 and µ1=viscoelastic parameters.
• f =force;
•r = radius.
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Tensegrity Structures
• Tensegrity structure (tensional integrity): the interaction
of a set of discontinuous isolated compression elements
(struts) with a set of continuous tension elements (cables) in
the aim to provide a stable volume and shape in the space.
• The tension elements carry “prestress” (initial stress)
conferring load-supporting capability to the entire structure.
• The role of the compression elements is to provide prestress
in the tension elements.
Together, they form a self-equilibrating stable
mechanical system
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• A characteristic of the tensegrity: in order to express a
resistance to distortion of shape it requires a prestress in its
members even before the external load is applied.
• Examples of tensegrity structures:
spider web: the prestress is provided by discrete
attachments to surrounding objects, such as tree
branches, and is balanced by tension in web
threads.
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• …Examples of tensegrity structures:
foam and leave: the prestress is provided by the
pressure of the inflating fluid (the compression
element in lieu of rigid struts and is carried by lattice
tension elements (liquid films in foams)
gas-liquid foam
plant leave
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• In the absence of the prestress: the intrinsic
resistance to shape distortion is lacking in these
structures because their internal degrees of freedom
of motion are not fully constrained.
• In the presence of a prestress: the structural
elements move relative to one another until they
attain a configuration which provides equilibrium
between external shear forces and those carried by
the structural elements.
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• A key feature of any tensegrity structure is the
interconnectedness of its elements(the manner in
which structural elements are mutually connected
and the degree of relative motion between
interconnecting elements at their junctions).
• It is likely that the cytoskeleton together with the
extracellular matrix form a tensegrity structure
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• Model structures built of sticks and elastic strings
according to the rules of tensegrity architecture.
• It qualitatively mimics many of the phenomena that
have been observed in living cells including the
effects of substrate adhesion on cell shape, cell
polarity and cytoskeleton remodelling.
• These tensegrity structures also exhibit a nearly
linear dependence between the stiffness of the entire
structure and the applied stress over a wide range of
stresses.
• This peculiar “linear stiffening” response is a
fundamental property of living cells as well as
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tissue.
Tensegrity Model
• For simplicity we focus on a
tensegrity structure
containing 6 rigid struts
interconnected by 24
linearly elastic cables.
• This structure is a simple
tensegrity but it embodies
the same essential features
observed in structures with
different arrangements and
numbers of structural elements.
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• The cables are viewed as elastic elements which support
only tension forces.
• The struts are viewed as rigid bars under compression. They
are slender and support no lateral load.
• At the initial state compression forces in the struts balance
tension forces in the cables ( the initial tension is referred to
as a prestress).
• Within the cytoskeleton:
- microfilaments and intermediate filaments may play the
role of cables;
- microtubules may play the role of struts.
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• The origin O of a rectangular
Cartesian coordinate system
Oxyz is placed at the center of
the structure.
• The Ox-axis is parallel to the
struts CC and C’C’.
• Oy-axis is parallel to the struts
AA and A’A’;
• Oz-axis is parallel to the struts BB and B’B’.
•It is assumed that the struts are of unit length.
• At the initial state the length of each cable segment is:
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• The distance between parallel struts is:
• The structure is stretched in the direction of the Ox-axis by
force of magnitude T (stretching force).
• This stretch causes:
- changes in the distances between the pairs of parallel
struts from s0 to sx for the struts AA and A’A’, sy for
the struts BB and B’B’, and sz for the struts CC and
C’C’
- changes in the cable lengths from l0 to
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• Changes in the distances between parallel struts are referred
as extensions:
• Relationships between distances sx, sy, and sz and cable
lengths l0, l1, and l2 are:
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Two cases of interconnectedness are
considered
Pinned Structure
Looped Structure
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Pinned Structure: frictionless
pin joints at their junctions,
the tension force in
each cable segment depends on
its length;
forces acting at each
end of a strut or a cable reduce
to a single force (tension for
cables and compression for
struts) and no couples.
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At the molecular level the Pinned Structure
could correspond to the case where different
cytoskeleton filaments are cross-linked or
physically bound to one another through
intermolecular binding interactions (
microtubules and intermediate filaments
through kinesin)
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• The work of stretching force T during an incremental axial
extension of the structure dsx is equal to the work of tensile
forces in the cables F0, F1, and F2 during corresponding
changes of cable lengths dl0, dl1, and dl2:
• By substituting into this equation the following equations
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• In the model it is assumed that the cables are
linearly elastic (Hookean) and carry only tensile
forces
force vs length relationship:
• k = cable stiffness
• lR = resting (unstressed) length of the cable segment
0< lR<=l0
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Results of Tensegrity Model with
Pinned Structure
The quantitative analysis of the architectural basis of
cell shape stability begins :
• by varying prestress in the pinned six!strut
tensegrity structure;
• by numerically varying the initial cable strain ? at a
given cable stiffness k:
• Results shown were obtained for k=1
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• The stretching force T increased
nonlinearly with increasing axial
extension ? sx:
• Structural stiffness E
(E=T/ ? sx) increased with increasing
stretching force T
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• The lateral extension ? sy
increased whereas the lateral
extension ? sz first increased, then
decreased with increasing axial
extension ? sx
• Cable strains
were much smaller than
the fractional change of the
structure length in the
uniaxial direction
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Looped structure: cables run
through frictionless loops at the
junctions and and thus they can
slide relative to the struts,
the tension force in the
cables depends on the overall
cable length not the length of an
individual segment
loops joints transmit
only tension and compression
forces to cables and struts,
respectively, no couples
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At the molecular level structure could
correspond to the case when those
filaments are not cross-linked and can
slide relative to each other ( intermediate
filaments across actin stress fibres)
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• Forces in each cable are equal throughout
deformation:
• Forces depend on the overall cable length
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• Force vs length relationship:
• K denote cable stiffness
• LR is the overall resting length of the cable:
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Results of Tensegrity Model with
Looped Structure
• Results shown here were obtained with the same
values of initial cable strain ? and the axial distance
sx as in the pinned case.
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• As in pinned structure,
stretching force T increased
nonlinearly with increasing
axial extension ? sx
• Unlike the pinned structure,
structural stiffness E exhibited a
softening effect at higher values
of prestress (stiffness E
decreased after an initial
increase in response to
increasing stretching force 37T)
• As in the pinned case, lateral
extension ? sy decreased
• Unlike pinned structure, lateral
extension ? sz first decreased and
then increased with increasing
axial extension ? sx
• When stretched uniaxially, the
looped structure extends much
more than the cable elongates and
this effect is independent of
prestress
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Conclusions on both Tensegrity
Model stuctures
• In comparison with the pinned structure, the looped
is more compliant at a given prestress.
• The asymmetry in the dependences of lateral
extensions ? sy and ? sz on the axial extension ? sx
implies that the looped structure is not isotropic.
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• Plots show that the predicted stretching force T vs
axial extension ? sx relationships of the structure
were nonlinear, even though the force vs extension
relationship of the cables was linear
mechanical
properties of individual structural elements are not
the sole determinants of mechanical properties of the
integrated structure during its shape distortion.
• Prestress and architectural features of the structure
were found to contribute importantly to its
mechanical properties, a finding consistent with
results obtained in studies with living cells.
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Parallels between the behaviors of
tensegrity structures and
observations in living cell
• Tensegrity structure exhibits structural stiffness E
that increases with increasing level of prestress
Cytoskeleton stiffness measured in living
endothelial cells increases with increased cell
spreading which is, in turn, mediated by the cell
extension and the cytoskeleton reorganization
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• Tensegrity structure exhibits initial stiffness that
increases with increasing prestress
the initial cytoskeleton stiffness in spread
endothelial cells has been shown to be higher than
that in round cells, a finding consistent with the
possibility that the initial cell stiffness is provided
by the prestress in the cytoskeleton.
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• Pinned tensegrity structure exhibits stiffness greater
than that of the looped structure
Cross-linking in isolated actin filament networks
increases their ability to resist shape distortion and
cytoskeleton stiffness increases when ATP is
depleted and actomyosin cross-bridges are fixed
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• Tensegrity structure undergoes larger fractional
changes of length than its cables when it stretches
uniaxially "
Living cells, such as neurites can similarly elongate
when mechanically stressed even though individual
actin filaments and microtubules are not very
extensible.
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• Tensegrity structure exhibits stiffening that is not linear
over the entire range of observed prestresses
; the stiffening response is close to linear for
some prestresses within this range
This specialized form of stiffening behavior is exhibited by
living endothelial cells as well as many biological tissues
.
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Differences between the behaviours
of tensegrity model and those
observed in cells
• The dependence of structure's stiffness on applied
load is nonlinear whereas in cells it appears linear.
• The more tensed the structure, the less stiffening it
exhibits, contrary the behaviour observed in cells
where more distended cells exhibit greater stiffening
than less distended ones.
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• Possible reason of these discrepancies: model assumption
that cables are extensible and linearly elastic and that struts
are rigid.
Infact:
• Measurements on isolated cytoskeleton filaments indicate
that tension-bearing elements (actin filaments) are very little
extensible and that compression-bearing elements
(microtubules) buckle under compression.
Tensegrity Model with buckling compression
elements
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Tensegrity Model with buckling
compression elements
• The structure is the same of the previous model (6
struts interconnected with 24 cable segments).
• Struts not rigid but slender and buckle under
compression.
The struts are buckled and
dashed line segments AAx,
BBx, and CCx
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• At the initial configuration:
- all the struts are of the same length L0:
Young’s modulus and resting crosssectional area for cable
Young’s modulus and resting crosssectional area for strut
Experimental value
- length of cable segments:
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• The structure is stretched in x-direction by applying force T
changes in the lengths of the struts from L0 to LI,
LII, and LIII
changes in the distance between the pairs of
parallel strut form s0 to sx, sy, and sz
changes in the length of cable segments for l0 to
l1, l2, and l3.
• Relationships between LI, LII, LIII , sx, sy, sz , l1, l2, and l3:
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• Tensile forces F1, F2, and F3 are in cables;
• External stretching force T is applied uniaxially (xdirection):
• It is assumed that the cables are linearly elastic and carry
only tensile forces:
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• Compression forces PI, PII, and P III exert on struts:
• The struts are viewed as elastic slender columns which carry
compression forces:
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Results
Stretching force T increases
nonlinearly with increasing
axial extension ? sx
This is consistent with the
behaviour observed in living
cells
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The structural stiffening
K=T/ ? sx increases with
increasing T.
This linearity cannot be
obtained from the previous
model where the struts are
viewed as rigid
This suggests that buckling
of the cytoskeleton
compression-bearing
elements (microtubules) is a
key determinant of cell of
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cell deformability.
Remarks about Tensegrity Structure
The tensegrity structure:
• Can mimic the behaviour observed in living cells
exposed to mechanical stresses.
• The model with rigid bar predicts forces and
deformation with an overestimation;
• The model with slender struts predicts value of
forces and deformations which fall within the same
order of magnitude as those observed cells.
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