Chapter 22, Macromolecules and aggregates

Ideality and reality

Simplicity of small systems and complexity of real systems

Entropy and order

Dealing with large systems

Proteins

DNA

Amyloids and Cellucoses

Liposome

Artificially Synthetic macromolecules

Common synthetic polymers

ContentsStructure and dynamics 22.1 different levels of structure 22.2 Random coils 22.3 The structure of proteins 22.4 The structure of nucleic acids 22.5 The stability of biological polymersDetermination of shape and size 22.6 Mean molar mass 22.7 Mass spectroscopy 22.8 Laser light scattering 22.9 Ultracentrifugation 22.10 Electrophoresis 22.11 Size exclusion chromatography 22.12 ViscositySelf-assembly 22.13 Colloids 22.14 Micelles and biological membranes 22.15 Surface films

Assignment for chapter 22

• 22.1(b)

• 22.5(a)

Structure and dynamicsConfiguration

Structural features elated toa given arrangement of chemical bonds

Spatial arrangement of the differentparts of a chain.

Conformation

It can be changed only by breakingChemical bonds and forming new ones.

It can be changed into another byRotating one part of a chain around a bond.

Different levels of structure

Primary structure (sequence)

Secondary structure (helix, sheet etc.)

Tertiary structure (overall 3D structure)

Quaternary structure (formedby different 3D structures)

Random coils

All bond angles are arbitrary.Free rotation.

Random coils (a more realistic model)

Bond angle is fixed.Free rotation.

Freely jointed chain model

Measure of size

Contour length: NlRC l

Measure of size

N

n

eN

P 22

1 2

2

1D chain, N units, each with length l,

The probability that the ends are nl apart:

ProofNumber of chains pointing to the right: NR

Number of chains pointing to the left: NL

!nN!nN

!N

!N!N

!NW

RL

2

1

2

1

Effective length: (NR-NL)l=nl N=NR+NL

The number of ways forming a chain with end-to-end distance nl:

N

N

RL nNnN

N

NNN

N

P

2!21

!21

!2/

)!(!

!

bonds of tsarrangemen ofnumber totalright the tobonds N with polymers ofnumber R

With Stirling’s approximation:

The probability that the ends are nl apart:

N

n

eN

P 22

1 2

2

3D casel

nl

222

3

214 RaeRa

f

21

22

3

Nl

a

P(RR+dR)=fdR

Measure of size

Root mean square separation:

lN

fdRRRrms

21

21

20

(classroom exercise)

222

3

214 RaeRa

f

2

1

22

3

Nl

a

0

2

134

22

a

xa dxex

Measure of size

Radius of gyration:

21

2

2

11

ijijg R

NR

Rij: separation of atoms i and j

lN

Rg

21

6

2/3

:length of rod uniform thin longA

21lR

l

g

RRR g

21

5

3 : radius with sphere uniform Solid

Exercise• Calculate the mean separation of the ends of a freely

jointed chain of N bonds pf length l.

RfdRR 0

fdRRR nn0

0

2/138233 )()(4 2/1

2

2/1 ldReRR Na

aRa

0

213

4

22

a

xa dxex

Conformational entropy

Nn

kNS

/

11ln2

1 11

!nN!nN

!N

!N!N

!NW

RL

2

1

2

1

WkS ln

2/})()ln{(

ln)2/1(2ln)1()2ln(

})!(ln{})!(ln{!ln/

11

2/1

21

21

nNnN nNnN

NNN

nNnNNkS

The most probable conformation is the onewith n=0:

NNkS ln2ln)1()2ln(/ 212/1

Constrained chains

21

1

1

cos

cosF

lNRrms 21

2 lN

Rg

21

3

For tetrahedral bonds,

)5.109( 3/1cos 0

222

3

214 RaeRa

f

FeR

af Ra 222

3

214

For polyethylene with M=56 kg/mol,N=4000, l=154 pm (C-C bond),

nm 6.5,nm 14 grms RR

The structure of proteins

Corey-Pauling rules

Classroom question:How many standard amino acidsare most commonly found in living things?

α helix

Conformational energy

22

1estretchstretch RRV

22

1ebendbendV

3131 cosBcosAVtorsion

r

qqV jicoulomb 4

612 r

D

r

CVLJ

1012 r

F

r

EV bondingH

Bond stretching:

Bond bending:

Bond torsion:

Coulomb forces between partial charges:

Dispersion and repulsive forces:

Hydrogen bonding:

anglesbonds

b KbbkRU 20

20 )(

2

1)(

2

1)(

ij

ji

ij

ij

ij

ijij r

qq

rr 0

612

pairs nonbonded 4)()(

)(2

1)cos(1

2

1 20

improperimp

dihedrals

KnxKx

Potential Energies

bonds

b bbk 20 )(

2

1

Kb :bond force constants

b : bond length

vibration

b

• Kθ :angle force constant

• Θ: bond angle

angles

K 20 )(

2

1

bending

Θ

• KX :dihedral angle force constant

• X: dihedral angle

dihedrals

nK cos12

1

torsion

• Kimp : improper dihedral angle• Φ: improper torsion angle (e.g., angle between ab and acd)

2

02

1 impropers

imp K

Improper torsion

a

b

c

d

• εij : Lennards-Jones well depth

• σij : Distance at the Lennards-Jones minimum between atoms i and j.• qi : Partial atomic charge• ε0 : Dielectric constant• r ij : The distance between atoms i and j

ij

ji

ij

ij

ij

ijij r

qq

rr 0

612

pairs nonbonded 4)()(

Intermolecular forces

(van der Walls forces)

Lower energy means more stable

Typical distribution of torsion angles(Ramachandran plot)

Glycyl residue

Alanyl residue

βsheet

Parallel and anti-parallel sheets

Higher-order structure

Four-helix bundle

Higher-order structure

β-barrel

The structure of nucleic acids

Secondary structure of DNA

Higher-order structure of DNA

The stability of biological polymers

Denaturation and renaturation

Determination of shape and size

22.6 Mean molar mass

22.7 Mass spectroscopy

22.8 Laser light scattering

22.9 Ultracentrifugation

22.10 Electrophoresis

22.11 Size exclusion chromatography

22.12 Viscosity

Mean molar masses

i

iin MNN

M1

i

iiw Mmm

M1

iii

iii

w MN

MNM

2

iii

iii

Z MN

MNM

2

3

Number-average molar mass:

Weight-average molar mass:

Z-average (mean cubic) molar mass:

Viscosity-average molar mass

MALDI principle

21

2

zeEd

mlt

2

2

l

teEd

z

m

A typical MALDI mass spectrum

Laser light scattering

2

2

0 sin

r

I

IR

wCpMKR

A

Cpr,r

N

d/dnVnK

4

220

24

Determining molecular size with light scattering

2

222

22

321

sin161

3

11

g

g

RRsP

Ultracentrifugation

Electrophoresis

f

zeEs

Isoelectric point of a protein

Size-exclusion chromatography

Viscosity

c 10

cc limlimcc

10

00

0

0

000

t

t

aVMK

Molar mass from viscosity

Self-assembly

22.13 Colloids

22.14 Micelles and biological membranes

22.15 Surface films