Biopolymer Networks Flashcards
Biopolymer Networks Inside Cells
- intermediate filaments
- microfilaments
- microtubules
Cytoskeleton
- composed of intermediate filaments, microfilaments and microtubules
- defines the cells mechanical properties
Biopolymer Networks Outside Cells
- extracellular matrix which defines tissue mechanical properties
- composed of collagen and hyaluronan
Collagen
- 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
Hyaluronan
- 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
Forming Networks From Collagen
- collagen fibrils self-assemble on their own
- fibril organsiation varies with collagen type
Forming Networks From Hyaluronan
- 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
Hierarchical Organisation
- salient feature of filament networks
- molecule combine to form filaments which combine to form networks
Composite Networks
- different filaments co-assemble into a single network
- multiple networks can interpenetrate
Strain Stiffening Behaviour
-e.g. skin is soft to touch but resistant to stretching at larger deformations, this is due to collagen networks
Mucus
- 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
Elasticity of Biopolymer Filaments - Bending
Worm Like Chain Model
-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»_space; Lc
semi-flexible: Lp ~ Lc
felxible: Lp «_space;Lc
-where Lc is the length of the chain
Elasticity of Biopolymer Filaments - Bending
Freely Jointed Chain Model
-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
Elasticity of Biopolymer Filaments - Bending
Isotropic Elastic Rod
-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
Elasticity of Biopolymer Filaments - Stretching
Flexible and Semi-Flexible Filaments
-stretching opposed by spontaneous bending to maximise filament conformational entropy
Elasticity of Biopolymer Filaments - Stretching
Straight and Straightened Filaments
-stretching opposed by intra-filament connectivity (quantified by Young’s modulus)
Measuring Filament Elastic Properties
- stretching force analysis by AFM
- noise analysis of clamped filaments
- statistical shape analysis of immobilised filaments
Elasticity of Cytoskeleton Fiaments
- microtubules - stiff, Lp>1mm
- microfilaments - semi-flexible, Lp=3-17μm
- intermediate filaments - flexible, Lp=0.2-1μm
Elasticity of ECM Fiaments
- collagen - semi-flexible, Lp~9μm
- hyaluronan - felxible, Lp~4nm
Hyaluronan Elasticity
-the combination of local stiffness (b=2Lp»a) and global flexibility (b<
Affine Network Model
-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
Volume Fraction
φ = r²L/ξ³
-where ξ³ is the volume of the network of unit and r²L is the volume of a filament
Gaussian Chain and Strain-Stiffening
-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
Contour Length and End-to-End Distance
lc = bN
-where b is the Kuhn length and N is the number of segments
lo = b N^(0.5)
Freely Jointed Chain Regimes vs Gaussian Chains
- 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
How to make ultrasoft hydrogels?
- 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<
End-to-End Distance for Flexible Chains
R ∝ Lc ^(0.5)
End-to-End Distance for Semi-Flexible Chains
R ∝ L
End-to-End Distance for Rigid Rods
R ∝ L
Modelling Connective Tissue
- 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
Networks of Semi-Flexible Filaments
- deformation is non-affine, network mechanics are effected by strand mechanics AND network reorganisation
- this makes it difficult to develop simple analytical descriptions
Networks of Semi-Flexible Filaments
Computer Simulation With Rigid Rods
- 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
Networks of Semi-Flexible Filaments
Bending/Stretching Regime
- 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
Differential Elasticity
-one way to define strain stiffening:
k = dσ/dε
-where ε is strain and σ is stress
Coordination Number
- the number of fibre segments meeting at a junction
- isostatic stretching requires z≥6 (in 3D) or z≥4 (2D)
Networks of Semi-Flexible Filaments
Importance of Connectivity
- 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
Three Elasticity Regimes
1) linear elasticity (K=G∝σ^0)
2) bending dominated strain stiffening (K∝σ^1)
3) stretching dominated strain stiffening (K∝σ^0.5)
Dimensionless Bend-Stretch Stiffness
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
Composite Biopolymer Networks
- 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
Computational Models for Composite Biopolymer Networks
-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
-
Mapping Composite Network Models to Single Component Models
- 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
