Single Molecule Dynamics & Crossing Membranes Flashcards
Single Molecule Dynamics
Overview
- diffusion of single molecules is a key process in biological materials
- thermal energy, kb*T, is effective at keeping the objects in motion and in randomising their direction
How can there be well-ordered structures and controlled processes in this random environment?
- strong interactions, interactions»_space; kb*T, but this would then require a mechanism to break the strong bonds in order for rearrangment to occur
- non-equilibrium processes, system is maintained out of equilibrium via external fields or chemical energy (ATP) which allows them to avoid reaching thermal equilibrium
- statistical physics, each particle moves randomly and is subject to unpredictable randomisation by kb*T, BUT the collective statistical behaviour is entirely predictable
Why is diffusion important in cells?
- diffusion allows molecules to move without doing work
- it allows molecules to find each other e.g. enzymes-substrates, proteins-proteins, communication through ions/signalling molecules/proteins, drug-target sites
Single Molecule Diffusion
- molecules move because other molecules keep bumping into them
- individual molecules follow a random walk due to these collisions
- Brownian motion
- diffusion is the collective motion of many molecules, many random walks by many molecules
Random Walks
Definition
-at each time step, randomly pick a direction and move one unit in that direction
Random Walks
Mean Displacement
E[x(N)] = 0
Random Walks
Mean Square Displacement
E[x(N)²] = NL²
- where L is the step distance
- as becomes large, the distribution approaches Gaussian with mean 0 and variance NL²
Diffusion as a Function of Time
N = t/Δt
-where N is the number of steps, t is the total time and Δt is the time step
=>
E[x(N)²] = NL² = t/Δt * L²
Diffusion in Cells
-extend random walk to 2D and 3D = 2mDt -where is the mean square displacement -and D is the diffusion constant -and m is the spatial dimension (1D,2D,3D)
Diffusion Examples for Different Dimensions
1D - proteins along DNA
2D - lipids / proteins in membranes
3D - ions / proteins within and outside cells
Stokes-Einstein Equation
D = kb*T / [6π η(T) rH]
- where η(T) is the viscosity
- and rH is the hydrodynamic radius of the molecule
Diffusion Inside Cells vs in Water
- diffusion is slower inside cells
- the viscosity of the cytoplasm is non-uniform
- cells are highly crowded meaning there is a large excluded volume
Composition of Cells
-in weight: 70% water 15% proteins 6% RNA 4% small molecules 2% phospholipids 2% polysaccarides 1% DNA
Mean Separation Between Proteins in a Crowded Environment
= [N/V]^(-1/3)
- N is the number of molecules
- V is the total cell volume
Typical Protein Size
3nm-20nm
Factors Affecting Protein Diffusion
- collisions with crowders
- hydrodynamic interactions
- attractive interactions
- immobile barriers (membranes)
- sieving effects
- spatial heterogeneity (different areas of the cell with differnet densities of different proteins)
Molecular Crowding - Excluded Volume
- excluded volume is the space that the centre of the probe molecule can’t reach
- this depends on the radius of the probe molecule, Rp, and the radius of background molecules, Rb
How do we measure the diffusion constant?
- track proteins tagged with fluorescent molecules or track fluorescent molecules directly
- three main techniques:
- -fluorescence correlation spectroscopy (FCS)
- -fluorescence recovery after photobleaching (FRAP)
- -single particle tracking
Fluorescence Correlation Spectroscopy (FCS)
- excites volume of space with laser
- as molecule enters this region it is excited and fluoresces
- molecules move in and out of the region by diffusion, this can be measured by measuring the intensity of the fluorescing molecules
- apply autocorreltation
- use τD = r²/4D to find D
Fluorescence Recovery After Photobleaching (FRAP)
- quickly bleach the fluorecent dyes in a given area by shining high intensity light ‘turning off’ molecules
- monitor diffusion into the bleeched area over time (recovery) due to diffusion of bleached (so non-fluorescing) molecules out and fluorescing molecules in
- using I(t) = 1 - mf*exp(t/τD)
- where mf is the mobile fraction, the fraction of the molecules that can move
Single Particle Tracking
- tag molecules at low enough concentrations to be able to follow the motion of single molecules over time
- calculate D from the observed random walk mean square distance: =2mDt
- in real biological systems often observe anomalous diffusion
Annomolous Diffusion
- non-linear diffusion behaviour, ∝ τ^α
- free diffusion is α=1
- super diffusion is α>1, MSD increases faster than expected for random motion due to external energy input possibly caused by molecular motots or strong intracellular flow
- sub diffusion is α<1, MSD increases slower than expected for random motion due to interactions with surrounding molecules possible caused by molecular crowding and confinement
What use is the diffusion constant?
Rates
-looking at reaction rates between molecules:
rate = kon * [mol1] * [mol2]
-where [X] indicates the concentration of X
What use is the diffusion constant?
Diffusion Limited Reaction
-assumes that every encounter leads to a reaction i.e. that rate is limited by diffusion ONLY
kon,diff = 4π (D1 + D2) (R1 + R2)
Saffman-Delbruck Model
- answers the question ‘does diffusion of bodies in the membrane scale with 1/R as proposed in the Stokes-Einstein equation for free bodies?’
- predicted logarithmic dependence of proteins differentiation coefficient on it’s hydrodynamic radius
Why is movement across membranes important?
- vital to compartmentalisation and communication between cells
- controlling differences in concentrations creates electrochemcial gradients crucial for electrical signalling, energy production and sensing
- all viruses and drugs must cross the membrane in order to enter cells
What are the four methods of entry into cells?
1) transmembrane diffusion (small molecules slip between the lipid molecules)
2) poration (free diffusion through the hole in the membrane)
3) ion channels/transporters (specifically mover certain molecules or open in response to stimuli)
4) vesicle trafficking (vesicle fuses to membrane then releases molecules on the other side)
Important Ions
- hydrogen - pH
- potassium - movement and cognition
- calcium - signalling, muscles
- magnesium - activity of ATP, enzymes, muscles
Anions
-chloride - pH, nerve impulses
Complex Ions
OH-, NH4 +, SO4 2-, PO4 3-, HPO4 2- (Pi)
Transmembrane Diffusion
Semi-permeability
NO
-large, uncharged polar molecules (glucose, sucroese)
-ions
YES
-small, uncharged polar molecules (H2O, ethanol, glycerol)
-gases (CO2, N2, O2)
Hydrophobic, Hydrophilic and Amphipathic
Definition
- hydrophobic - polar groups
- hydrophillic - no polar groups
- amphipathic - both polar and non-polar groups (e.g. lipid molecules)
Transmembrane Diffusion
Solubility-Diffusion Model
P = KD/d P = permeability K = oil/water partition coefficient, how much the molecule wants to sit in the membrane D = solute's diffusion coefficient in the oil slab, how fast the molecule moves once it's in the membrane d = membrane thickness
Dependence of Membrane Permeability on Membrane Properties
- membrane permeability also depends on membrane packing properties:
- lipid species
- temperature
- cholesterol content
- higher area per lipid results in higher permeability
Transmembrane Diffusion
Flick’s Law Applied to Membranes
J = -D dc/dx = -D (c1-c2)/d = -D/d Δc D = diffusion coefficient d = membrane thickness Δc = concentration difference across membrane
Transmembrane Diffusion
Permeation Constant
J = -D/d Δc = Ps * Δc
=>
Ps = D/d
Transmembrane Diffusion
Timescale of Leakage
Δc = Δco exp(-t/τ)
-where τ is the timescale of leakage:
τ = V / Ps*A
Transmembrane Diffusion
Red Blood Cells
- bioconcave disk shape
- high area to volume ratio to increase O2 concentration and interaction with the haemoglobin proteins inside
Transmembrane Diffusion
Osmotic Pressure
-derived based on Gibbs free energy taking into account water free energy and mixing entropy
Osmotic Pressure Equation
Δp = Ns/V * kb * T Ns = number of molecules V = volume T = temperature kb = Boltzmann constant
High Salt Concentration to Preserve Food
- salt causes a hypertonic environment and plasmolysis (shrinkage)
- imperfect method as some bacteria are halophilic -> can thrive in a high salt environment
- certain membrane proteins can open in response to protein allowing rapid flow of molecules in/out of cell
Ways to Deform a Membrane
- 4 ways:
- -stretch (e.g. pressure)
- -bend (e.g. creating new vesicles)
- -expand
- -shear
- each has its own eleastic response that depends linearly on the extent of deformation
Membrane Mechanics
Free Energy Due to Stretching
Fstretch = Ka/2 ∫ (Δa/a)² dA Ka = area expansion modulus
Membrane Mechanics
Surface Tension
λ = Ka (Δa/a)
-surface tension is isotropic, a force per unit length
Ka = area expansion modulus
Membrane Mechanics
Young-Laplace Equation for Spherical and Non-Spherical Geometries
-spherical:
λ = RΔp/2
-non-spherical
Δp = λ [1/R1 + 1/R2]
Membrane Mechanics
Rupture
-lipid bilayers rupture at 1-25mN/m
Membrane Trafficking
- transport of materials can be facilitated by lipid vesicle fusion to bring molecules into cells (endocytosis) or out of cells (exocytosis)
- viruses can exploit these mechanics to bring genetic information into the cell to replicate and spread
- used for communication between cells (e.g. nerve cells) by transport of communication molecules (e.g. neurotransmitters)
Membrane Trafficking
Energy Cost
-use the free energy of bending equation:
Fbend = Kb/2 ∫ (Δa/a)² dA
= Ka/2 (Δa/a)² 4πR²
= 8πKb
-so energy cost is independent of vesicle radius!
-there are also proteins which aid in this process as the energy cost is generally too high for spontaneous formation
Viral Entry Mechanisms
- membrane fusion - virus membrane fuses with host membrane to release virus cell contents into host cell
- endocytosis (without membrane fusion) virus enters cell with its own membrane intact
- injection via loaded spring
