Biopolymer Networks

Question 1

Biopolymer Networks Inside Cells

Answer 1

• intermediate filaments
• microfilaments
• microtubules

Question 2

Cytoskeleton

Answer 2

• composed of intermediate filaments, microfilaments and microtubules
• defines the cells mechanical properties

Question 3

Biopolymer Networks Outside Cells

Answer 3

• extracellular matrix which defines tissue mechanical properties
• composed of collagen and hyaluronan

Question 4

Collagen

Answer 4

• self-assembles hierarchically
• long triple stranded helix, tropocollagen, is its primary feature measuring 300nm x 1.5nm
• most collagens form fibrils and then networks of fibrils,
• fibrils are ~100nm in diameter and a few micrometres long

Question 5

Hyaluronan

Answer 5

• present in every body tissue, it has structural and biochemical functions
• produced in enzymes in cell membrane then extruded directly into the ECM
• linear, regular, hydrophilic polymer and negatively charged when dissolved in water
• typically large with variable contour length, 1-10μm
• in water it forms random coils ~100nm in diameter, an open structure that randomly wiggles around

Question 6

Forming Networks From Collagen

Answer 6

• collagen fibrils self-assemble on their own

- fibril organsiation varies with collagen type

Question 7

Forming Networks From Hyaluronan

Answer 7

• HA and other filaments (e.g. actin) DO NOT self-assemble, they stay out of solution in pure form
• cross-link via accessory proteins
• cross-link stability and filament connectivity can vary

Question 8

Hierarchical Organisation

Answer 8

• salient feature of filament networks

- molecule combine to form filaments which combine to form networks

Question 9

Composite Networks

Answer 9

• different filaments co-assemble into a single network

- multiple networks can interpenetrate

Question 10

Strain Stiffening Behaviour

Answer 10

-e.g. skin is soft to touch but resistant to stretching at larger deformations, this is due to collagen networks

Question 11

Mucus

Answer 11

• large glycoproteins that become physically cross-linked / entangled
• keeps lungs clean - ‘gel on brush’ organisation provides lubrication for mucus transport and pathogen/dust clearance
• helps snails/slugs move - thixtropy, time-dependent shear property

Question 12

Elasticity of Biopolymer Filaments - Bending

Worm Like Chain Model

Answer 12

-persistance length, Lp, quantifies the correltation length of filament axi orientation upon thermally activated fluctuations
= exp(L/Lp)
-the most general model, can be used for stiff, flexible and semi-flexible polymers:
stiff: Lp&raquo_space; Lc
semi-flexible: Lp ~ Lc
felxible: Lp &laquo_space;Lc
-where Lc is the length of the chain

Question 13

Elasticity of Biopolymer Filaments - Bending

Freely Jointed Chain Model

Answer 13

-for flexible filaments
-model parameterises local stiffness by the Kuhn length, b, where:
Lp = b/2
-model movement of the chain as a random walk

Question 14

Elasticity of Biopolymer Filaments - Bending

Isotropic Elastic Rod

Answer 14

-for less flexible filaments
-bending modulus, κ, determined by filament radius and Young’s modulus (E):
κ = 4π r^4 E
-where:
Lp = κ / kbT

Question 15

Elasticity of Biopolymer Filaments - Stretching

Flexible and Semi-Flexible Filaments

Answer 15

-stretching opposed by spontaneous bending to maximise filament conformational entropy

Question 16

Elasticity of Biopolymer Filaments - Stretching

Straight and Straightened Filaments

Answer 16

-stretching opposed by intra-filament connectivity (quantified by Young’s modulus)

Question 17

Measuring Filament Elastic Properties

Answer 17

• stretching force analysis by AFM
• noise analysis of clamped filaments
• statistical shape analysis of immobilised filaments

Question 18

Elasticity of Cytoskeleton Fiaments

Answer 18

• microtubules - stiff, Lp>1mm
• microfilaments - semi-flexible, Lp=3-17μm
• intermediate filaments - flexible, Lp=0.2-1μm

Question 19

Elasticity of ECM Fiaments

Answer 19

• collagen - semi-flexible, Lp~9μm

- hyaluronan - felxible, Lp~4nm

Question 20

Hyaluronan Elasticity

Answer 20

-the combination of local stiffness (b=2Lp»a) and global flexibility (b<

Question 21

Affine Network Model

Answer 21

-model for networks of flexible polymers
-assumes that all cross-links are displaced affinely with the whole network
-this means that changes on an individual filament level are the same as global network changes
-shear modulus:
G = nkbT / V = ρkbT / Mx ~ kb*T / ξ²
-where:
n = number of filaments in network
V = total volume of network
ρ = network density (total mass over total volume)
Mx = average mass per network strand
ξ = correlation length, so ξ³ is the volume per strand

Question 22

Volume Fraction

Answer 22

φ = r²L/ξ³

-where ξ³ is the volume of the network of unit and r²L is the volume of a filament

Question 23

Gaussian Chain and Strain-Stiffening

Answer 23

-the affine network models chains as Gaussian which negelcts finite chain extensibility - when stretching approaches the contour length, lc, stiffening occurs and eventually the chain would snap

Question 24

Contour Length and End-to-End Distance

Answer 24

lc = bN
-where b is the Kuhn length and N is the number of segments
lo = b N^(0.5)

Question 25

Freely Jointed Chain Regimes vs Gaussian Chains

Answer 25

- linear extension: l ≤ 0.4 lc - linear elastic stretching: 1 ≤ l/lo ≤ 0.4 N^(0.5) - so Gaussian model is valid for l ≤ 0.4 lc, the linear regime

Question 26

How to make ultrasoft hydrogels?

Answer 26

- need to reach a concentration where individual extended coils can overlap, this threshold is the overlap concentration, c* - for given polymer size, networks soften with increasing Kuhn length (b) as long as b<

Question 27

End-to-End Distance for Flexible Chains

Answer 27

R ∝ Lc ^(0.5)

Question 28

End-to-End Distance for Semi-Flexible Chains

Answer 28

R ∝ L

Question 29

End-to-End Distance for Rigid Rods

Answer 29

R ∝ L

Question 30

Modelling Connective Tissue

Answer 30

- connective tissues require an adaptive reponse to strain, they need to be: - -soft at small deformations for homeostasis and migration / proliferation of cells - -stiff at large deformatoins to maintain tissue structure and prevent rupture - networks of semi-flexible flaments are strain-stiffening so fit these requirements - studying real tissues is challenging as they are made of many different molecules - in-vitro reconstituted networks made with defined composition (purified) are easier to analyse than real tissues

Question 31

Networks of Semi-Flexible Filaments

Answer 31

- deformation is non-affine, network mechanics are effected by strand mechanics AND network reorganisation - this makes it difficult to develop simple analytical descriptions

Question 32

Networks of Semi-Flexible Filaments | Computer Simulation With Rigid Rods

Answer 32

- with non strain, ε=0, the system is in its normal state, zero deformation - at low strain, ε=0.08, filaments bend - at high strain, ε=0.24, filaments start to align in the strain direction and stretch

Question 33

Networks of Semi-Flexible Filaments | Bending/Stretching Regime

Answer 33

- transition from a bending to a stretching dominated regime is a plausible mechanism of strain-stiffening - filament orientation is the dominant mechanism for this (undulations postpone stiffening onset), this would predict k∝σ^(1/2) - but in real, collagen rich tissues dependence of stress on strain is given by k∝σ so transition from bending to stretching regime can't be the full explaination of collagen network behaviour

Question 34

Differential Elasticity

Answer 34

-one way to define strain stiffening: k = dσ/dε -where ε is strain and σ is stress

Question 35

Coordination Number

Answer 35

- the number of fibre segments meeting at a junction | - isostatic stretching requires z≥6 (in 3D) or z≥4 (2D)

Question 36

Networks of Semi-Flexible Filaments | Importance of Connectivity

Answer 36

- experiments realised that connectivity must be important for strain stiffening - collagen networks have low connectivity with z between 3 & 4, so sub-isostatic - sub-isostatic means: - -extended strain stiffening regime with k∝σ - -k independent of concentration ρ - -in the linear elastic regime (K=G) G∝κρ² - these three characteristic fit experiment well

Question 37

Three Elasticity Regimes

Answer 37

1) linear elasticity (K=G∝σ^0) 2) bending dominated strain stiffening (K∝σ^1) 3) stretching dominated strain stiffening (K∝σ^0.5)

Question 38

Dimensionless Bend-Stretch Stiffness

Answer 38

``` the dimensionless bend-stretch stiffness (κ~) is a key determinant for elastic behaviour across the three elasticity regimes κ~ = κ/μL² -where L is the segment length -for isotropic elastic rods: κ = r²/L² & κ∝ρ ```

Question 39

Composite Biopolymer Networks

Answer 39

- in biology, fibrous collagen networks in tissues are imbedded in a soft hydrated matrix of hyaluronan and proteoglycans - the HA matrix effects collagen network elasticity by: - -increasing the linear elastic modulus from HA matrix elasticity - -the two regimes of strain-stiffening shift to larger strains due to internal pre-stress

Question 40

Computational Models for Composite Biopolymer Networks

Answer 40

-network models can be extended to composite systems to study the physical mechanisms behind hyaluronan matrix effects -can use a two component lattice based model -set the proportion of HA to collagen -programme different bending/stretching properties for the two different filament types -

Question 41

Mapping Composite Network Models to Single Component Models

Answer 41

- composite network reponse can be mapped onto an effective single component model: - -enhanced stiffness, κeff~, accounts for the hindrance of filament bending by HA matrix - -pre-compressed network accounts for internal stresses - internal stress generation is a powerful control variable to tune the onset of strain stiffening -> this could be exploited in synthetic materials with pH or temperature responsive components