Chapter 22, Macromolecules and aggregates Ideality and reality Simplicity of small systems and complexity of real systems Entropy and order Dealing with.
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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
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
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
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