Transcript Introduction and review of Matlab

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)
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Fundamental structural and functional units
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Use solar or chemical energy for mechanical work or synthesis
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Protein factories (ribosome)
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Maintain concentration differences of ions, which generates a
potential difference with outside (-60 mV)
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Sensitive to temperature, pressure, volume changes
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Respond to changes in environment via sensors and motility
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Sense and respond to changes in internal conditions via feedback
and control mechanisms (extreme example: apoptosis--cell death)
Two kinds of cells:
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Prokaryotes (single cells, bacteria, e.g. Escherichia coli)
Size: 1 mm (micrometer), thick cell wall, no nucleus
The first life forms. Simpler molecular structures, hence easier to study
Flagella: long appendages used for moving
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Eukaryotes (everything else)
Size: 10 mm, 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
Unsaturated
fatty 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
….PO4-CH2-CH2-N+-(CH3)3
Proteins (polypeptides) perform control and regulatory functions
(e.g. enzymes, hormones, ).
The building blocks of proteins are the 20 amino acids.
pH  2
10  pH  2
pH  10
NH3+ - C - COOH
NH3+ - C - COONH2 - C - COO-
Formation of polypeptides
In water:
NH 3+ - C - COO - + NH 3+ - C - COO  NH 3+ - C - CO - NH - C - COO - + H2 0
Protein structure
3.6 amino acids per turn, r=2.5 Å
pitch (rise per turn) is 5.4 Å
 -helix
b -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 mm long
nucleus
Central dogma
Tools of Molecular Biology
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X-ray diffraction
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Nuclear magnetic resonance (NMR) spectroscopy
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Electron microscopy
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Atomic force microscopy
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Mass spectrometry
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Optical tweezers (single molecule exp’s)
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Patch clamping (conductance of ion chanels)
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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.
Velocity
selector
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.
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
dP
e2 2 2

a sin 
3
d 4c
Larmor’s formula, non-relativistic
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!)
2. Charged particles scatter incident radiation
X-rays are electromagmetic radiation with l  10 nm
E  E0 exp[ i (k.r - t )], k  2 / l ,   ck
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
q2
E '  E0 2 sin 
mc
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
f (q)    (r) exp[iq.r ] dV
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
 (r) 
1
(2 )
3
 f (q) exp[ -iq.r ] dV
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: nl = 2d.sin()
Atoms in space
1D cut in FT
2D cut in FT
X-ray scattering from 5 atoms in a row
Atoms in space
1D cut in FT
2D cut in FT
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
Atoms in space
reciprocal space
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: v  1.6  kT / m  10-10 m/s
b) 1 ng cell with speed 1 mm/s: Ekin= 0.5 x 10-18 J ≈ 100 kT
Average speed after equilibration: v  10-4 m/s  0.1 mm/s
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)
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
x  nL - ( N - n) L  (2n - N ) L
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
N!
P ( n) 
p n q N -n
n!( N - n)!
Moments of the binomial distribution can be obtained using the binomial
theorem (see the stat. phys. notes)
S ( p, q )  ( p + q )
N
N

n 0
N!
p n q N -n
n!( N - n)!
N
N
P
(
n
)

(
p
+
q
)
1

n 0
N
n   nP(n)  pN
n 0
N
n 2   n 2 P ( n)  n
2
+ Npq
n 0
var( n)  n 2 - n
2
 Npq
Average position in 1D random walk after N steps
x  2n - N L  2 n - N L  (2 p - 1) NL  ( p - q) NL
Spread in the position is given by the variance


x 2  4n 2 - 4 Nn + N 2 L2  4 n 2 - 4 N n + N 2 L2

x
2
 4n
var( x)  x 2 - x
2
 4 n2 - n
If
p  q  1 / 2,

2

- 4 N n + N 2 L2
2
L
2
 4 var( n) L2  4 NpqL2
x  0, var( x)  x 2  NL2
Hence rms ( x)  N L
Connection with the molecular world:
Molecular collisions occur randomly. Nevertheless we can still define
a mean collision time (Dt) and a mean free path (L), which allows
us to introduce time via t  NDt
or
N  t / Dt
2
L
x 2 (t )  NL2  t
Dt
We define D  L2 2Dt as the diffusion coefficient
The mean-square displacement becomes
x 2  2 Dt
Generalisation to 2D and 3D is straightforward
2D :
r 2  x 2 + y 2  x 2 + y 2  4 Dt
3D :
r 2  x 2 + y 2 + z 2  6 Dt
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.
x 2  2 Dt
Example of 2D random walk
Perrin’s experimental data for
Brownian motion of a colloid
particle (size: 0.075 mm)
t=300
t=300
N=300
N=7500
Dt=1
Dt=1/25
L=1
L=1/5
D=0.5
D=0.5
Computer simulation of random motion in 2D
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.)
P ( n )  P ( n )e
where
-( n - n ) 2 2 Npq
1
-( n - n ) 2

e
2 
2 2
1
1
n  Np, P(n ) 

,   Npq
2 
2Npq
For the position variable x  (2n - N ) L, we have
 x  Npq 2 L
x - x  ( n - n ) 2 L,
P( x) 
1
2  x
e
-( x - x ) 2 2 x2
x  (2n - N ) L  (2 p - 1) NL  ( p - q ) NL
Other examples of random walk:
1. Polymer conformations
They have a random coil structure
Single step size
3L
rms distance for N links:
3L
rrms  3 N L
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
-0.57 (fit to exp)
-0.5
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:
dv
m  F - v
dt
dv
F
 0  vter 
dt

(terminal velocity)
For a spherical object, the friction coefficient is given by Stokes formula
  6R
where R is the radius of the object and  is the viscosity of the medium.
Typical values for  (kg/ms): air: 10-5, water: 10-3, oil: 0.1
For a cell in air, vter ≈ 5 cm/s
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
1 f 2
Dxi  vi Dt +
Dt
2m
Dx  Dv Dt +
in one step
1 f 2 average over many steps
Dt
2m
Since v is randomly oriented Dv=0. Introduce the drift velocity as
Dx Dt
vd 

f
Dt 2m
L2
Combine  with D 
,
2Dt
2m
  
Dt
D  m
L2
Dt
2
 m v 2  kT
Einstein relation