Discussion topic for week 1

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