Session 14

� Stability of a structure based on a balanced synergy

of continuous tension and discrete compression

components

� Richard Buckminster Fuller coined tensegrity as

“islands of compression inside an ocean of tension”

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“islands of compression inside an ocean of tension”

� Common cellular interpretation

� Actin microfilaments – tension elements

� Microtubules – compression elements

Additional components

3

� Additional components

� ECM – compression element

� Intermediate filaments – tension elements

� Cross-linked actin bundles – compression elements

� Central to tensegrity is the amount of pre-loaded

tension in the structure (prestress).

� Dictates the stiffness and mechanical response

� Cells have prestress that is balanced by ECM

attachments

4

attachments

� Detach the cell from culture

dish and it rounds up

� Akin to cortical tension in

lipid drop models

Trypsin added to detach cell

5

Here, B-B’ �C-C and C-C’ � B-B

� Element dimensions

� sx sy sz strut-to-strut separation

� l1 l2 l3 tension lengths

� LA LB LC strut lengths (=1)

Recall vector notation:

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� Recall vector notation:

� Vector

� Distance

� Direction

2 2 2

ˆˆ ˆ

ˆ /

i

i

i i i

l xi yj zk

l l x y z

l l l

= + +

= = + +

=

�

�

�

� Based on model geometry

l

( ) ( ) ( )

2 2 2

22 21 1 1 1

1 2 2 2 2

2 212 2

i

x y

l x y z

l s s

l s s s

= + +

= + − +

= + − +

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� Similarly½sx

½½

l12 2

1

12 2

2x y y

l s s s= + − +

½sy

2 2

2

2 2

3

12 2

2

12 2

2

y z z

z x x

l s s s

l s s s

= + − +

= + − +

� Balance of forces on A-A

T

F3

F1

( ) ( )

,

3 1

3 3 1 1

0

ˆ ˆ4 4 0

ˆ ˆˆ ˆ4 4 0

x AAF

T F i F i

T F l i F l i

=

+ ⋅ + ⋅ =

+ ⋅ + ⋅ =

∑� �

ˆi i i

F Fl=�

A

By symmetry,

2 of 8 tension

forces shown

By symmetry,

2 of 8 tension

forces shown

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½sx

½½

l1

½sy� Similarly

( ) ( )1 1 1

2 2 2

3 1

3 1

4 4 0x xs s

T F Fl l

−+ − =

( ) ( )

( ) ( )

1 2

,

1 2

32

,

2 3

0 1

0 1

y BB y y

z CC z z

F FF s s

l l

FFF s s

l l

= ⇒ − =

= ⇒ − =

∑

∑

� Struts are slender with zero transverse loading

� Cables are linear elastic springs

where k is spring stiffness and lR is unstressed cable length

( )i RF k l l= −

9

R

� Define prestress (strain)

where l0 is uniform initial length of each cable for an initially

isotropic structure

0

0

Rl l

lξ

−=

� At rest, structure is isotropic, therefore

0 1 2 3

1

1/ 2

0.375

A B C

x y z

L L L

s s s

l l l l

= = =

= = =

= = = =

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� For given applied force T and prestress (ξ , k, lR)

� Have 6 unknowns (∆sx ∆sy ∆sz ∆l1 ∆l2 ∆l3)

� Have 3 geometric length equations and 3 force

balance equations

� Can solve…

� Structural stiffness

� High prestress (ξ), stiffness is constant

x

TE

s≡

∆

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constant

� Tensegrity structure is Hookean

� Low prestress, stiffness increases with applied force

� Strain hardening seen by Petersen (1982) and Radmacher (1996)

� Stiffness increases with prestress

� Prestress:� Traction forces for cells on gels

embedded with fluorescent microbeads

� Young’s Modulus� Cell stiffness measured with magnetic

twisting cytometry

� Tensegrity predict linear relationship between prestress and effective

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between prestress and effective stiffness

� At top, slope is 0.4 but tensegrity predicts slope to be 1

� At bottom, dashed-line slope is 1.04.� Accounting for 14% loading in MTs not

measurable by traction forces

Adapted from Wang, N. (2001) PNAS, 98:7765

Adapted from Wang, N. (2002) Am. J. Physiol.

Cell Physiol, 282:C606

� Class Discussion� Mechanotransduction shows that cells adapt to stress

� Tension only, where is compression? Not only MT, could be ECM.

� Cells are fluid filled, how does that relate to tensegrity?

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tensegrity?

� Dampening consideration

� What about AFM? Discrete points have high stiffness at struts, low stiffness at springs

� FEM is more powerful to model cell mechanics

� But disregards the mechanical nature of each protein (filaments)

� Tensegrity: local perturbation results in global rearrangement� Local load should cause distal disturbance

� Continuum models show local response � Dissipate inversely with distance from point

of load

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of load

� Examples� Mantiotis (1997) pulling on ECM coated bead

causes nuclear distortion

� Hu (2003) used MTC and MitoTracker and observed discrete displacements at distal pts

� Both observations were actin-CSK dependent