Introduction to Modeling in Biophysics

Joel Ireta

Cell Theory

The cell is the fundamental unit of a living system

Organism Organs Tissues

Cells The activity of the organism as a whole is just the sum of the activities and interactions of its individual cells

http://www.biosci.uga.edu/almanac/bio_103/notes/may_15.html

Cell Structure

Organelles:identifiable structures inside a cell that perform a particular function.

Membrane:a semipermeable covering that encloses the cellular contents

Composition

90% water

10% The dry weight :

50% protein15% carbohydrate15% nucleic acid 10% lipid 10% miscellaneous

Composition by element 60% H 24% O 10% C 5 % N 1%

SPIons (Na, K, Ca, Fe)Trace elements

Biological Macromolecules

Biological MacromoleculesNucleic acids

DNA is the major store of geneticinformation

RNA translates the information storedin the DNA into proteins

Proteins Built up from amino acids , they are the working parts of the cell. Enzymes, receptors …...

Carbohydrates The main source of cellular energy, and also structural components of cells.

LipidsBipolar molecules whose configuration accounts for many of the biological membrane's properties.

(biomolecules)

Biophysics

The complexity of biomolecules ultimately derives from the information contained in the sequence of nucleotide bases in DNA.

The task of biophysics is to decode this message or at least to describe how the phenomena of biology at different levels emerge from this kernel of information.

1

1. H. Frauenfelder, P. G. Wolynes, R. H. Austin, Rev. Mod. Phys. 71, S419 (1999)

Biomolecules

Lipids

Carbohydrates

Proteins

Biomolecules are non-branched polymers

Nucleic Acids

•rigidity covalent bonds

•repulsion Steric effect, i.e. due to Pauli exclusion principle

•attractions van der Waals, hydrogen bonding

Interactions

Protein Structure

Proteins are built from 20 different amino acids

The Peptide Bond

C HC

ON

C

Rn

Rn-1 The peptide bond has a partial doublebond character

Peptide group characteristics

Planar

RigidPeptide group

The resonant model, theoretical model proposed by L. Pauling

R1 R1

C N

O

HCα

Cα

C N

O

HCα

Cα

R2 R2

-

+

Singlebond

doublebond

Singlebond state

Doublebond state(zwitterion)

α-helix: The Success Of a Theoretical Prediction

Antecedents:

X-ray diffraction spectra of fibrous proteins(α-keratin, β-Keratin found e.g. in hair)

Pauling-Corey Model (1950): a helical conformation where planar peptidesare connected by hydrogen bonds

D. A. Eisenber, “The discovery of the α-helix and β-sheet, the principal structural features of proteins”, Proc. Natl. Acad. Sci. USA 100, 11207 (2003)

Steric Effect

R

φψ

Dihedral AnglesAllowed regions where repulsionamong atoms is negligible(theoretical prediction)

Allowedconformation Repulsive overlap

Not allowed conformation

R R

ϕ

ψ

Ramachandran-Diagramm

C HC

ON

C

Rn

Rn-1

Secondary Structure of proteins

The α-helix conformation is the most common secondary structure

α-Helixβ-Sheet

ϕ

ψ

secondary structure(β-sheet)

Protein Structure

Primary structure(amino acid sequence)

Native conformation random coil

unfolding(denaturation)

folding

Protein Folding Problem

Anfisen’s thermodynamic hypothesis: Native conformations of proteins areare conformations at the global minimaof their accessible free energy surface

Levinthal Paradox: If an unfolded statesearches randomlyfor the global itwill take years even for small proteins toreach the native state(global minimum). However they fold inseconds

Random state

ative state

Free energy (1960’s)

(1960’s)

1

4

Helix-Coil Transition

Random coilDenaturation( unfolding )

helix

TemperatureSolventPressure

The formation of a helix can be divided in two steps:

1. helix nucleation:

2. helix propagation:

2

3

12

34

5

Helix-Coil TransitionThe Lifson-Roig Theory:

A residue is classified by its location in (φ,ψ) space

c: coil

h: helical

( )∫ −=nonhelical

iikTiG

i ddeu ψφ1'

( )∫ −=helical

iikTiG

i ddev ψφ1'

''iii vuz += conformational integral over (φ,ψ) space

i

ii z

vv'

=

i

ii z

uu'

=

Probability to find residue i in helical conformation

Probability to find residue i in random-coil conformation

but:''ii vu >> then: '

'

i

ii u

vv ≈ 1≈iuPartitionfunction ∏=

rN

iizZ

Assuming the residue conformationsare independent

(1960’s)

Helix-Coil Transition

Statistical weight of a residue that is both helical and form hydrogen bonds

( ) ( )( )∫ +−+−=helical

iikTiiiGiG

i ddew ψφ1,,1' 31

=

110000110000

vv

vw

cchcchhh

M

cchcchhh

'

'

i

ii u

ww ≈

Statistical weight for the central element of all eight triplets

Helix-Coil Transition

bMaZ N 2−=Partition function for a chain with N residues

( )11vva =

=

1

1v

v

bwhere

Average number of helical hydrogen bonds per molecule w

Znh lnln

∂∂

= Fractionof helix

h

h

Nn

H. Qian, J. A. Schellman J. Phys. Chems 96, 3987 (1992)

D. S. Kemp, Helv. Chim. Acta 85, 4392 (2002)

Reviews on helix-coil theory

Protein Folding ProblemPathway solution to the proteinfolding problem

Funnel landscape, solution tothe protein folding problem

Not accepted nowadays

hydrophobic

polar

K. A. Dill and H. S. Chan Nature Struc. Biol. 4, 10, (1997).

Funnel landscape correspondingto slow-fast kinetics

Protein Folding Problem

Funnel landscape correspondingto complex kinetics

Evolution has chosen protein sequences (primary structure) that minimizeroughness

Proteins obey two distinct set of principles

Force FieldsClass I Force-Fields:

( ) ( ) ( )[ ] ∑∑∑∑

−+++++−+−=

pairsnonbond ij

ij

ij

ij

ij

ji

torsionsanglesbondsb r

CrA

rqq

nkkbbkrV 6122

02

0 1cos)( δϕθθ ϕθ

Two-body interaction

Three-body interaction

Four-body interaction

Two-body interaction

1

2 3

4

5b

θ

ϕ rij

Two-body interaction

Three-body interaction

Force constants adaptedto match normal-modesfrequencies for a numberof peptide fragments

ChargesObtained from ab-initioCalculations, usuallyHF/6-31G*

Lennard-JonesParameters

Fitting to reproduce densitiesand heats of vaporization inliquid simulations

Four-body interaction

12

3

4 Fitting to reproduce ab-initio (HF or MP2)potential energy surfaces

BLYP/TZVP RAMACHANDRAN PLOT1

Alanine dipeptide

C7eq(-83.8, 75.1)Ground State

C5 (-1550.0, 158.8)1.77 kcal/mol

C7ax(70.8, -56.6)2.36 kcal/mol

β2 (-119.6, 15.3)3.36 kcal/mol

αL(68.7, 23.3)5.02 kcal/mol

α’(-161.8, -47.6)6.88 kcal/mol

α

1. R. Vargas et al J. Phys. Chem. A 106, 3213 (2002)

Ramachandran Plot: Force-Fields Results

Free energy for solvatedAlanine-dipeptide

J. W. Ponder and D. A. Case Adv. Protein. Chem. 66, 27 (2003)

Protein Folding Problem: Force-Fields Results

Problem of MD simulations: sample the phase-space efficiently

Free-energy surface for folding a helix

Sampling different regions of theconformational space using methodsof high-temperature molecular dynamics

Fragment of hydrogen bonds formedRadi

us o

f gy

rati

on

Folding following asingle pathway

Funnel-like potential energy surfaceC. L. Brooks III, Acc. Chem. Res. 35, 447, (2002)

V. A. Daggett, Acc. Chem. Res. 35, 422, (2002)

Weak Interactions

hydrogenbonding interaction

a) strong > 10 kcal/mol ( 0.43 eV)

b) moderate 3 kcal/mol (0.13 eV) to 10 kcal/mol (0.43)

c) weak < 3 kcal/mol (0.13 eV)D = donor atom A = acceptor atom

rhb

r2

θσ

BD

H

Aα

δ+

δ- µ1

µ2

Hydrogen bonds are predominantlyelectrostatic interactions. However...

-3

-2.5

-2

-1.5

-1

-0.5

0

0 2 4 6 8 10

r2(Å)

E (K

cal/m

ol)

Hydrogen bonds are directional : σ usually ranges from 140° to 180°

Hydrogen bonds are cooperative: they strongly interact with each othermodifying its bond strength

For small r2 multipole expansion ofthe electrostatic interaction doesnot converge properly

Full electrostaticinteraction energy

R-3

R-4

R-5

R-6

S Scheiner, Hydrogen bonding a theoretical perspectiveOxford University Press (1997)

Techniques accounting for the electroniccorrelation are needed for an accurate description of the hydrogen bonds

Dispersion energies contributes significantly to theHydrogen bond energy

( ) ( ) ( ) ( )( )( ) Bluer

Yellowrrrrr BAAB

;0;0

<∆>∆

−−=∆

ρρ

ρρρρ

H

Attractive part : electrostaticinduction an dispersion energies (charge transfer ?)

Repulsion part: electronic exchange interaction

Hydrogen Bond NatureO

H

N

Projection of the electrostatic potential on a charge density isosurface.System: alanine peptide dimers forming a hydrogen bond

O N

Hydrogen Bond Nature

Water dimer

-6

-5

-4

-3

-2

-1

0HF MP2 CCSD CCSD(T)

Ener

gy (K

cal/m

ol)

At least MP2 is needed to accuratelydescribe the hydrogen bond interactionJ. E. del Bene, Hydrogen Bonds. Encyclopedia

of Computational Chemistry Vol. 2. Schleyer, D. Ed. in Chief.(John Wiley, Chichester U. K. 1998).

ABBA →+

BAABbindinghb EEEEE −−==

DFT Accuracy and the Hydrogen Bond Directionality

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

110 120 130 140 150 160 170 180

θ(°)

PBE

erro

r per

hb

(kca

l/mol

)rhb

θBD

H

A

C

O

N

H H

H

2formamide

H

CC

C

H H

HH

H

H

N O

2N-Methyl Acetamide

C

C

CN

O

H

HH

H

H

H

H

2N-N dimethylformamide

With increasing deviation from a linear arrangementof the hydrogen bonds, the accuracy of the DFT-PBEdecreases.J.Ireta, J. Neugebauer, M. Scheffler J. Phys. Chem A, 108, 5692 (2004)

1

2

3

4

5

6

7

Helix axis

After the second turn the hydrogen bond strength increases smoothly

10

The hydrogen bond strength difference between long finite chains and the infinite one is due to the large electric field at the ends of the finite chains

9

8

-5

-4

-3

-2

-1

0

2 4 6 8 10 12 14 16 18 20Number of peptide units

∆Eh

b (k

cal/m

ol)

cooperativityPolyGlyPolyAla

First turn

second turn

third turn

+ -

Electrostatic potential

Helix axis

Ending Effects

-10

-9

-8

-7

-6

-5

-4

-3

2 4 6 8 10 12 14 16 18 20

Number of peptide units

Ehb

(kca

l/mol

)

PolyGlyPolyAla

-5.4 kcal/mol, N=7

Ehb, ∞

Ehb , ∞ ~ 1 kcal/mol

Cooperativity: Force-Field Results

M. John et al to be published

Model

Helixaxis

Zr

r

θ

LCarbon

Oxygen

α− Carbon

Nitrogen

α− Carbon

zyxn nZeenrenrR ++= )sin()cos( θθ

One dimensionalcrystal

Unitcell

Nmo360=θ

M turns per unit cell

N peptide units per unitcell

o57.99exp =θ

Twist

(Polyalanine crystal)

θ can not be sampledcontinously

n m θ

4 1 90.0º15 4 96.0º11 3 98.18º7 2 102.86º

Compression/stretching

Z can be sampled continously

Helix Twist (θ)

Stability with respect to FES(kcal/mol)

π-helix (hbs between i and i+5 peptide units)

α-Helix(hbs between i and i+4peptide units)

310-helix(hbs between i and i+4peptide units)

0

-3

-2

-1

α-helix

Compression/stretching

Twist

Potential Energy Surface of Polyalanine

),,( RZfET θ=For every (θ,Z) the helixgeometry (R) is fully relaxed

Helix Length(Z)

707580859095

100105110115120

1 1.2 1.4 1.6 1.8 2 2.2 2.4

Helix length per pu (Å)

Twis

t (°)

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

1 1.2 1.4 1.6 1.8 2 2.2 2.4

Helix length per pu (Å)

Stab

ility

(kca

l/mol

)

Helixaxis

Zr

r

θ

LCarbon

Oxygen

α− Carbon

Nitrogen

α− Carbon

The helix releases stress by changing its twist, which leads to an structural transition

Structural transition

π-helix α-helix 310-helix

transition

1.3 kcal/mol

2 kcal/mol

0.5 kcal/mol 0.4 kcal/mol

Structural Transitions: Force-Field Results

Amber 99

Helix-length per pu

Energy

M. John et al to be published

PhononsUnit Cell

0 1-1

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=∂

∂

−

−

−−

−

−

cartcartcart

cartcartcart

cartcartcart

cart

uF

uF

uF

uF

uF

uF

uF

uF

uF

uF

1

1

0

1

1

1

1

0

0

0

1

0

1

1

0

1

1

1

x

y

F

Fy

Fxx

yFy

Fx

Fr

Fθ

Fr

Fθ

r

θ

rθ

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=∂

∂

−

−

−

cilindcilindcilind

cilindcilindcilind

cilindcilindcilind

cilind

uF

uF

uF

uF

uF

uF

uF

uF

uF

uF

0

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

0

Transforming to cylindricalcoordinates

Cartesian

Cylindrical

Phonon Dispersion Spectrum of Polyalanine

Dotted linesunscaledFrequencies(factor 1.02)

Solid linesscaled frequencies

Amida A band (N-H stretching)

Amida 1 band (C=O stretching)

Amida 2 band (C-N stretchingN-H bending)

L. Ismer et al submitted to PRE

[1] B. Fanconi, W. E. Small, and W.L. Peticolas, Biopolymers 10, 1277 (1971)[2] V.K. Datye, and S. Krimm, J. Chem. Phys. 84, 12 (1986)[3] L. Ismer, J. Ireta, S. Boeck and J. Neugebauer submitted to PRE[4] M. Daurel, P. Delhaes, and E. Dupart, Biopolymers 14, 801 (1975)

Heat Capacity of Polyalanine

Force-field results [1, 2]

PBE results [3]

Experiment [4]

An accurate description of the cooperativity by DFT lead to a good agreement withexperiment (at low temperatures)