Biopolymer Interfaces & Cell Mechanics Flashcards
Flory Radius
- the end-to-end distance
- for an ideal chain: Rf ∝ b N^0.5
Radius of Gyration
- mean distance of all segments from the centre of mass
- for an ideal chain: Rg ∝ b N^0.5
Hydrodynamic Radius
- effective size with which the polymer diffuses
- for an ideal chain: Rh ∝ b N^0.5
Polymer Brushes in Biological Systems
- arrays of polymer molecules end-attached to an interface (planar or curved)
- brush thickness is given by H and the distance between polymer molecules is s giving grafting density σ = 1/s²
- polymer brushes are common in biological systems where the polymer is typically wither a hydrophobic polysaccaride (well-disolved and flexible) OR an intrinsically disordered protein
- they have various functions
Flexible Polymer Chains in Solution - Scaling Laws
Ideal Chain
- also FJC or random walk
- contour length: Lc = bN
- Flory radius: Rf = bN^(1/2)
Flexible Polymer Chains in Solution - Scaling Laws
Real Chain
- self-avoiding random walk
- Flory radius: Rf ~ bN^(3/5)
Flexible Polymer Chains in Solution - Scaling Laws
Long vs Short Range Interactions
- short range interactions, e.g. local stiffness, only effects b
- long range interactions effect the Flory exponant
Flexible Polymer Chains in Solution - Effective Excluded Volume
Definition
-chain segments in a polymer can repel or attract, this effect is described by an effective volume, v, per chain segment
Flexible Polymer Chains in Solution - Effective Excluded Volume
v > 0
-excluded volume repulsion (good solvent)
-chain segments repel
-polymer coil stretches as a real chain
-most common for biological systems
Rf ~ [v/b³]^(1/5) b N^(3/5)
Flexible Polymer Chains in Solution - Effective Excluded Volume
v = 0
-Θ-condition / Θ-solvent
-attraction between segments balances repulsion of chains
-polymer behaves as an ideal chain
Rf ~ bN^(1/2)
Flexible Polymer Chains in Solution - Effective Excluded Volume
v < 0
-excluded volume attraction (poor solvent)
-attraction between segments is strong
-dense globule forms
Rf ~ b² |v|^(-1/3) N^(1/3)
Effect of Grafting on Polymer Configuration
-when grafting one end to a surface, flexible polymers in good solvents adopt a distinct conformation depending on surface density
-when s≥Rf there is no overlap of chains, polymers retain their random coil conformation and look the same as they would in solution, ‘mushrooms’
H ~ Rf
-when s
Alexander de Gennes Polymer Brush Model
Formula
H ~ σ^(1/3) b N
-where σ is the grafting density
Alexander de Gennes Polymer Brush Model
In Terms of v
H ~ v^(1/3) σ^(1/3) N ∝ N
Alexander de Gennes Polymer Brush Model
Pros. & Cons.
- correctly predicts scaling of brush thickness: H ∝ N
- but as it assumes all chain ends are in the same plane
- cannot provide accurate numerical pre-factor (as it is a scaling approach)
Self-Consistent Field Theory Polymer Brush Model
-additionally to the Alexander de Gennes Polymer Brush Model, it provides pre-factors and correctly predicts the density profile and distance of chain ends
Brushes of Locally Stiff Polymers
-locally stiff chains (a«b></b>
Polyeclectrolyte Brushes
-brushes of charged polymers are sensitive to salt and pH changes
–without added salt they are highly swolen due to the osmotic pressure of trapped counterions
–salt leads to charge screening and reduces brush thickness
–reponse to pH is not well understood
-for moderate & high concentrations, effect of salt can be accoutned for by an added excluded volume:
v ~ vo + α²/4cs
-where vo is the exlcuded volume without charges, α is the fractional charge per monomer and cs is the salt concentration
Elasticity of Polymer Brushes
- non-linear response to compression
- approximating as homogeneous, an elastic modulus can be derived for the initial regime of linear compression
- for small deformations (lineare elastic regime), and ignoring pre-factors, the elasticity of simple polymer gels is recovered
Hyaluronan Brushes
-hyaluronan is locally stiff and charged both of which promote highly swollen and soft brushes
Interpentration of Brushes
-it is NOT entropically favourable for two brushes to interpentrate
Mechanobiology
- how physical forces and changes in mechanical properties of cells and tissues contribute to development, cell differentiation, physiology and disease
- ECM elasticity directs stem cell lineage specification - cells sense the mechanical properties of the ECM in order to direct specialisation to the appropriate cell type
Mechanosignalling in Soft Connective Tissues
- cell and ECM are pre-stressed to facilitate mechanosensing - cells pull on ECM to determine its mechanical properties and vice versa
- at homeostasis the mechanical stresses are balanced
- these interactions are mediated by a transducer which both the cell and ECM are connected to
Mechano-Transducers at Cell Surface
- focal adhesions are large dynamic protein complexes that sense and transduce forces from ECM
- transmembrane proteins (integrins) bind to ECM and connect to intracellular protein complex
- the protein complex (including talin, vincilin) sits at cell periphery and binds actin stress fibres
- actomyosin stress fibres generate tensile stress in cell
Pre-Stress in Cells
- mechanical elements of cells are pre-stressed
- generated by pulling of actin filaments by myosin molecular motors
- this is an active process requiring ATP
- pre-stress in the resting state needs to be accounted for when considering cell response to external mechanical stress
Signal Propagation to Nucleus
- for specialisation as result of mechanical sensing at membrane
- there are two possible mechanisms:
- -direct force transduction to nucleus via cytoskeleton to nucleoskeleton (purely physical)
- -force dependent signal transduction via cytosolic signalling cascades possibly involving ion channels (biochemical pathway)
- these pathways are NOT mutually exclusive, it is likely that both play some role in mechanoregulation of gene expression - still an active area of research
Methods to Characterise Cell Mechanical Properties
- atomic force microscopy
- magenetic bead microrheometry
- micropipette aspiration
- laser tweezer microrheology
Methods to Characterise Cell Mechanical Properties
Atomic Force Microscopy
- maps local mechanica properties (nm-μm scale)
- colloidal probe run over cell which is attached to surface, laser bounced off of probe arm to optical position detector
- force vs distance curve analysed with contact mechanics model (e.g. Hertz) to extract mechanical properties (e.g. Young’s modulus)
- can map properties
- varying probe size (10nm-10μm) allows probing over a spectrum of length scales
Methods to Characterise Cell Mechanical Properties
Magnetic Bead Microrheometry
- measures local mechanical properties at μm scale
- micron sized magnetic bead functionalised with fibronectin and attached to apical syrface of cell adhered to cover slip
- magnet brought in proximity of cell and B field used to generate force on bead
- measure displcacement response of cell to applied magenetic forces
Methods to Characterise Cell Mechanical Properties
Micropipette Aspiration
- measures cell mechanical properties (μm scale)
- suction pressure used to some of cell down the length of a pipette
- applying known pressure and measuring the length of cell in the pipette and normalising with respect to pipette radius allows for effective measure of elasticity
Methods to Characterise Cell Mechanical Properties
Laser Tweezer Microrheology
- local mechanical properties inside cell
- optical trap applies defined force on microbeads injected into the cell then measure concomitant displacement to infer viscosity of cytoplasm / cell elasticity
- allow for direct analysis of mechanical properties inside cells
- best for soft materials since the force range is limited to very small
Models of Cell Mechanics
- cell cortex and cytoplasm model
- tensegrity model
Cell Cortex & Cytoplasm Model
Description
- thin, elastic cortex lining attached to cell membrane, thickess ~100nm
- cytoplasm is modelled as homogeneous either viscous, viscoelastic or elastic
Cell Cortex & Cytoplasm Model
Lumped Parameter Viscoelastic Model
- model cell propeties using a viscoelastic circuit
- the viscoelastic cell response involves a fast elastic response, viscous relaxation regime and viscous flow regime
- in the cell cortex - cytoplasm model elastic properties are assigned to the cortex and viscous ones to the cytoplasm
Cell Cortex & Cytoplasm Model
Experiment
- experiments on pure networks of actin and crosslinker allow us to test if the actin cortex alone can reproduce cell behaviour
- if you take purified actin and crosslinkers can create invitro constituted cortex
- this actin cortex recapitulates the strain stiffening of collagen in the ECM (K∝σo)
Tensegrity
Definition
- tensional integrity - interaction of set of isolated compression elements with set of continuous tension elements with aim of providing a stable form in space
- in cells actin stress fibres are the main tension elements
- microtubules and ECM fibres are the main compression elements
Tensegrity Model
Description
- depicts entire cell as mechanical network of interconnected parts with local mechanical perturbation generating global deformation
- this means that nucleus deforms with the rest of the cell, a simple way to propagate signals and modulate gene expression & downstream cell responses
- pre-stress is implicit in tensegrity, in tensegrity structures elasticity depends linearly on pre-stress
- the same dependence of elasticity on (pre) stress as actin and collagen networks on their own (E~σo)
Tensegrity Model
Pros & Cons
- a cell cannot be a fully-autonomous tensegrity structure since we known that cells sense and respond to their environment to change shape
- good evidence for actin stress fibres being under tension
- some evidence for microtubles under pressure BUT ECM also plays a role in balancing tension from actin stress fibres
- cells in suspension tend to be spherical as mechanical external cues required for complex cell shape changes
