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Discussion topic for week 1
• Eukaryotes (multi-cell organisms) evolved into very large sizes
whereas prokaryotes (single-cell organisms) remained quite small
(about 1 micrometer).
What has prevented prokaryotes from growing to larger sizes?
Weekly discussion topics are listed in the web page:
www.physics.usyd.edu.au/~serdar/bp/bp.html
Reminder: please look at the statistical physics notes in the web page
and make sure that you have the necessary
background.
Basic properties of cells (Nelson, chap. 2)
• Fundamental structural and functional units
• Use solar or chemical energy for mechanical work or synthesis
• Protein factories (ribosome)
• Maintain concentration differences of ions, which generates a
potential difference with outside (-60 mV)
• Sensitive to temperature, pressure, volume changes
• Respond to changes in environment via sensors and motility
• Sense and respond to changes in internal conditions via feedback
and control mechanisms (extreme example: apoptosis--cell
death)
Two kinds of cells:
• Prokaryotes (single cells, bacteria, e.g. Escherichia coli)
Size: 1 m (micrometer), thick cell wall, no nucleus
The first life forms. Simpler molecular structures, hence easier to study
Flagella: long appendages used for moving
• Eukaryotes (everything else)
Size: 10 m, no cell wall (animals), has a nucleus,
Organelle: subcompartments that carry out specific tasks
e.g. mitochondria produces ATP from metabolism (the energy currency)
chloroplast produces ATP from sunlight
Cytoplasm: the rest of the cell
Structure of a typical cell
Plasma membrane
Molecular parts
Electrolyte solution:
water (70%)
ions (Na, K, Cl,…)
Organic molecules
Hydrocarbon chains
(hydrophobic)
Double bonds
Functional groups in organic molecules
Polar groups are hydrophilic. When attached to hydrocarbons,
they modify their behaviour.
Four classes of macromolecules:
polysaccharides, triglycerides, polypeptides, nucleic acids.
(sugars) (lipids) (proteins) (DNA)
Simple sugars (monosaccharides): e.g. ribose (C5H10O5),
Glucose is a product of photosynthesis
Glucose and fructose have the same formula (C6H12O6) but
different structure
Disaccharides are formed when two monosaccharides are chemically
bonded together.
Lipids (fatty acids) are involved in long-term energy storage
Saturated fatty acids
Unsaturatedfatty acid(C=C bonds)
Phospholipids are important structural components of cell membranes
Phosphatide:
At normal pH (7), the oxygens in
the OH groups are deprotonated,
leading to a negatively charged
membrane.
Phospatidylcholine (PC):
The most common phospholipid
has a choline group attached
….PO4CH2CH2N+(CH3)3
Proteins (polypeptides) perform control and regulatory functions
(e.g. enzymes, hormones, ).
The building blocks of proteins are the 20 amino acids.
-
-
pH
pH
pH
COOCNH
COOCNH
COOHCNH
2
3
3
10
210
2
Formation of polypeptides
0HCOOCNHCOCNH
COOCNHCOOCNH
23
33
-
--
In water:
Protein structure
3.6 amino acids per turn, r=2.5 Å pitch (rise per turn) is 5.4 Å
-helix
-sheet
Nucleic acids are formed from ribose+phosphate+base pairs
The base pairs are A-T and C-G in DNA
In RNA Thymine is substituted by Uracil
Adenosine triphosphate (ATP) has three phosphate groups.
In the usual nucleotides, there is only one phosphate group
which is called Adenosine monophosphate (AMP)
Another important variant is Adenosine diphosphate (ADP)
B-DNA (B helix)
ROM (Read-Only Memory) contains1.5 Gigabyte of genetic information
Base pairs per turn (3.4 nm): 10
Primary structure of
a single strand of DNA
Primary structure of
a single strand of RNA
Hydrogen bonds
among the base
pairs A-T and C-G
Local structure of DNA
Dynamic and flexible
structure
Bends, twists and knots
Essential for packing 1 m
long DNA in 1 m long
nucleus
Central dogma
Tools of Molecular Biology
• X-ray diffraction
• Nuclear magnetic resonance (NMR) spectroscopy
• Electron microscopy
• Atomic force microscopy
• Mass spectrometry
• Optical tweezers (single molecule exp’s)
• Patch clamping (conductance of ion chanels)
• Computational tools (molecular dynamics, bioinformatics, etc.)
See, Methods in Molecular Biophysics by Serdyuk et al. for detailed
discussion of these methods
Mass spectrometer
Charged biomolecules are accelerated
and injected to the velocity selector
which has transverse E and B fields.
Only those which have velocity v= E/B
will pass through.
In the next chamber, there is only a B
field, which bends the beam by
r = mv/Bq.
The mass is accurately determined from
the measured radius of gyration.
Velocityselector
Optical tweezers
Single molecule experiment using optical tweezers. Increasing the force
on the bead triggers unfolding of RNA (Bustamante et al, 2001).
Patch clamping in ion channels
(Neher & Sakmann)
Using a clean pipette and suction, enable accurate measurement of
picoamp currents in ion channels.
X-ray diffraction
Basics
1. Accelerating charges emit radiation
223
2
sin4
ac
eddP
Where a is the acceleration of the charge and is the angle between
the acceleration and radiation vectors.
• Maximum radiation occurs in the direction perpendicular to a.
• The only way to increase the intensity of radiation is via a.
Generic x-ray tubes use bremstrahlung (breaking radiation)
Isotropic, only selected wavelengths, low intensity
Synchrotrons accelerate electrons around a circular path (relativistic)
Directional, continuous, intense (one is operating in Melbourne now!)
Larmor’s formula, non-relativistic
2. Charged particles scatter incident radiation
X-rays are electromagmetic radiation with nm
ckktiEE ,/2)],(exp[0 k.r
Where E is the the electric field amplitude, k is the wave vector
and is the frequency.
An EM wave scattered by a charged particle has the amplitude
sin'2
2
0mc
qEE
Where is the angle between incident and scattered radiation.
Because nuclei are much heavier than electrons, they can be ignored.
Note the q dependence; light atoms (e.g. H, He) are much harder to see.
Scattering from a collections of atoms is descibed using form factors
dVif ]exp[)()( q.rrq
Where is the charge density, q is the momentum transfer in
the scattering, i.e. q = k-k'.
Thus form factor is just the Fourier transform of the charge density
X-ray scattering provides information on f, which is then inverted
via inverse Fourier transform to find the electron density maps
dVif ]exp[)()2(
1)(
3q.rqr
X-ray scattering from a single atom
Atom in space 1D cut in FT 2D cut in FT
X-ray scattering from two atoms
Braggs law: n = 2d.sin()
Atoms in space 1D cut in FT 2D cut in FT
Atoms in space 1D cut in FT 2D cut in FT
X-ray scattering from 5 atoms in a row
Atoms in space reciprocal space
X-ray scattering from a lattice of atoms
Atoms in space reciprocal space
X-ray scattering from a monoclinic lattice (75 degrees)
Atoms in space reciprocal space
X-ray scattering from a square box
Atoms in space reciprocal space
X-ray scattering from a circular box
Random Walks and Diffusion (Nelson, chap. 4)
Friction: when an object moves faster than its fair share (i.e. Ekin>3kT/2)
its kinetic energy is degraded by the surrounding molecules.
Examples of kinetic energy:
a) 1 kg ball with speed 1 m/s: Ekin= 0.5 J ≈ 1020 kT
Average speed after equilibration:
b) 1 ng cell with speed 1 mm/s: Ekin= 0.5 x 1018 J ≈ 100 kT
Average speed after equilibration:
At that speed, the cell could move 10 times its size in 1 second!
Mesoscopic objects in liquid execute a random motion called Brownian
(Dr Robert Brown, 1828).
Brownian motion arises from random kicks of molecules (Einstein, 1905)
m/s 1010/6.1 mkTv
mm/s m/s 1.010 4 v
Random walk in 1D
Toss a coin and take a step (of length L) to the right if it is heads,
and to the left if it is tails.
If we get n heads after N throws, the position will be
Repeating this experiment many times, we will get a distribution of
positions in the range [-NL, NL]. Since x and n have a 1-to-1
correspondence, the same distribution applies to that of heads & tails.
This is given by the binomial distribution: Given that the probability of
throwing a head is p and tail q (p+q=1), that of n heads out of N trials is
LNnLnNnLx )2()(
nNnqpnNn
NnP
)!(!
!)(
Moments of the binomial distribution can be obtained using the binomial
theorem (see the stat. phys. notes)
Npqnnn
NpqnnPnn
pNnnPn
qpnP
qpnNn
NqpqpS
N
n
N
n
NN
n
N
n
nNnN
22
2
0
22
0
0
0
)var(
)(
)(
1)()(
)!(!!
)(),(
Average position in 1D random walk after N steps
Spread in the position is given by the variance
NLqpNLpLNnLNnx )()12(22
LNx
NLxxxqp
)(
)var(,0,2/1 22
rms Hence
If
2222222
2222
2222222
4)var(44)var(
44
4444
NpqLLnLnnxxx
LNnNnx
LNnNnLNNnnx
Connection with the molecular world:
Molecular collisions occur randomly. Nevertheless we can still define
a mean collision time (t) and a mean free path (L), which allows
us to introduce time via
We define as the diffusion coefficient
The mean-square displacement becomes
Generalisation to 2D and 3D is straightforward
t
LtNLtx
2
22 )(
Dtx 22
ttNtNt /or
tLD 22
Dtzyxr
Dtyxyxr
6
4
2222
22222
:3D
:2D
Examples of 1D random walk
Squared displacement Mean-square displacement
for a single random walk for 30 random walks
In both graphs, the lines describe the diffusive motion,
It is satisfied only for the ensemble average.
Dtx 22
Example of 2D random walk
Perrin’s experimental data for
Brownian motion of a colloid
particle (size: 0.075 mm)
Computer simulation of random motion in 2D
t=300
N=300
t=1
L=1
D=0.5
t=300
N=7500
t=1/25
L=1/5
D=0.5
Mean collision times of molecules in liquids are of the order of picosec.
Thus in macroscopic observations, N is a very large number.
Large N limit of the binomial distribution is Gaussian (see stat. phys.)
where
For the position variable we have
222 2)(2)(
2
1)()(
nnNpqnn eenPnP
NpqNpq
nPNpn
,2
1
2
1)(,
NLqpNLpLNnx
exP
LNpqLnnxx
xxx
x
x
)()12()2(
21
)(
2,2)(
22 2)(
,)2( LNnx
Other examples of random walk:
1. Polymer conformations
They have a random coil structure
Single step size
rms distance for N links:
Mass is proportional to N and
diffusion coeff. is proportional to
1/r N-1/2 (for close packing, N-1/3)
2. Stock market
3. Gambling
L3
LNrrms 3
L3
-0.5
-0.57 (fit to exp)
Gambling as an example of biased random walk
Biasing is worst in poker machines
Roulette provides one of the least
biased form of gambling
Chances of winning with red or odd
100 x 18/37 ~ 49%
Friction
Macroscopic observation: motion of an object in a viscous medium is
damped by a force proportional to its speed:
For a spherical object, the friction coefficient is given by Stokes formula
where R is the radius of the object and is the viscosity of the medium.
Typical values for (kg/ms): air: 105, water: 103, oil: 0.1
For a cell in air, vter ≈ 5 cm/s
velocity) (terminalter
Fv
dt
dv
vFdt
dvm
0
R 6
Microscopic interpretation: motion of the object is modified by random
molecular collisions. We model this motion via 1D random walk subject
to an external force f. In between collisions, the object moves by
in one step
average over many steps
Since v is randomly oriented v=0. Introduce the drift velocity as
2
2
21
21
tmf
tvx
tmf
tvx ii
kTvmt
LmD
tL
D
tm
fmt
tx
v
22
22
,2
22
with Combine
d
Einstein relation
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