lecture 16 – m olecular interactions
DESCRIPTION
Lecture 16 – M olecular interactions. Summary of lecture 1 4. - PowerPoint PPT PresentationTRANSCRIPT
Lecture 16 – Molecular interactions
Properties of ideal and non-ideal (interacting) gases can be calculated from the partition function and therefore macroscopic observables such as P, V, T can be related to microscopic properties such as the interatomic potential describing interactions between gas molecules
The equation of state of a real monatomic gas reflects intermolecular interactions. Deviations from ideal gas behavior can generally be expressed as a infinite power series in the density =N/V:
Summary of lecture 14
...)()( 32 TCTBkTP
The pressure and other properties can be calculated from the partition function or the configuration integral as follows:
Summary of lecture 15
The first term describes ideal gas behavior, while the configurational integral Z(V,T) describes interactions between molecules.
TVQkTP
ln
3 / 2 ln ,ln 1ln 2 ( , )!
N Z V TQP kT kT mkT Z T V kTV V N V
Under the assumption that the density is low enough that all intermolecular interactions are pair-wise, i.e. between two molecules only, the energy of the system that can be written as:
Summary of lecture 15
In many (but not all) cases, the pair-wise interactions are a function only of the inter-molecular distance r and not of any direction. Then:
E K Upm
UTi
i
N
iji j
N
2
1 2
N
jiijij rUU )(
rdeTVZ NrU ijij )(...),(
The interaction term U describes molecular interactions are in general very complex, but the configurational integral and therefore the partition function can be evaluated explicitly for certain functional forms of the interaction potential
At low densities, where only pairs of molecules are likely to interact, the equation of state is
Summary of lecture 15
...)()( 32 TCTBkTP
)(TBkTP
Where the second virial coefficient B(T)=-b2
drreb kTrU 2
0
)/)(2 )1(2
Intermolecular Potentials
Interaction between biological molecules and between atoms within each molecules are complex and are only partially understood because of their complexities; however, the nature of forces holding together biological structures or allowing biological molecules to interact is the same as the forces between molecules in a gas
Thus, when we study how a drug binds a receptor, a protein folds or a protein binds DNA, we describe interactions between individual atoms that comprise the molecules using terms that are derived from the fundamental properties of matter as studied in simple gases and solids
Intermolecular Potentials
Often, we partition the overall interaction energy between molecules or within molecules into terms that describe individual interactions as follows:
...612int ij
ij
ij
ij
ij
jiramolecula r
ArB
rqq
CUU
Where the first term reflects interactions between atoms within a molecule (which will be described in the next lecture); the second term is the Coulomb interaction between charged particles; the third reflects the strong repulsion between atoms and molecules at short distance and the fourth weak attractive forces between molecules (London dispersion forces)
Electrostatic interactions
A very important interaction between biological molecules is of course the electrostatic potential expressed by Coulomb’s law, for two charges separated by a distance r:
It is convenient when dealing with atomic charges and distances to express r in Angstrom (0.1 nm) and the charges in multiples or fraction of the electron charge.
rqq
rU4
)( 21
molekJrqq
rU /1389)( 21
Here is the permittivity of the medium and reflects its dielectric properties; =1 for vacuum, =2 for non-polar hydrocarbons and =80 for water. The higher the permittivity, the more electrostatic interactions are ‘screened’.
Electrostatic interactions
The higher the permittivity, the more electrostatic interactions are ‘screened’. The interior of a protein usually contains many hydrophobic amino acids, so electrostatic repulsion can be much larger than in water; as a consequence, burying a charged group is very costly unless it is interacting with a charge of opposite sign, which is very often seen in proteins
molekJrqq
rU /1389)( 21
Electrostatic interactions
Although charge is of course quantized and an electron charge is the unit charge, complex molecules have delocalized electron systems so that the effective charge on a single atom is equivalent to a fraction of the electron charge
When charges are distributed asymmetrically in a molecule, then an electric dipole is generated which is responsible for other more complex forms of electrostatic interactions that are of fundamental importance in biology
The electric dipole is a vector given by the sum of all charges and their locations:
i
ii rq
Electrostatic interactions
Charges can be calculated by quantum chemistry methods (calculating the electron wave function for the molecule, for example nucleic acid bases) and the dipole moment for a given molecule can be calculated this way
Interaction between molecules with dipolar moments depend on the inverse third power of the distance but, because of their vectorial character, also depend on the orientation of the two dipoles and therefore are highly directional.
i
ii rq
Hard sphere and van der Waals repulsions
All atoms and molecules repel at short distance, just like billiard balls that cannot penetrate each other. Perhaps the simplest description of molecular interactions is to treat interacting molecules as hard spheres, which only interact when the centers come within the sum of the radii, just like billiard balls.
Hard sphere and van der Waals repulsions
In this model, the 2 atoms or molecules have zero interaction when the distance between their centers is greater than the sum of their radii
When the distance equals the sum of the radii the repulsive potential is infinite, which prevents the spheres from inter-penetrating Although a very crude description, this very simple model is nonetheless effective and attractive because of its simplicity
This intermolecular potential may be represented as follows:
21
21
RRrRRr
)(0)(
rUrU
Hard sphere and van der Waals repulsions
Example : Let us calculate the second virial coefficient and the equation of state assuming a gas of hard sphere molecules.
We have to evaluate the integral
21
21
RRrRRr
)(0)(
rUrU
23
32 R
kTP
R
RRkTrU Rdrrdrredrredrreb
0
3220
0
22
0
)/)(2 3
22)1()1(2)1(2
drreb kTrU 2
0
)/)(2 )1(2
Hard sphere and van der Waals repulsions
If the radii of the molecules are equal, then that term is simply four times the volume of a sphere of radius R. B(T) reflects the volume excluded or occupied by the molecule
A more accurate representation of repulsive forces is given by the equation:
21
21
RRrRRr
)(0)(
rUrU
23
32 R
kTP
U rBrij
ij
ij
( ) 12
This potential falls off rapidly, but not as rapidly as the hard sphere approximations and is a more accurate description of molecular repulsions when atoms and molecules come into close contact.
London Attractive Potential
Molecules that do not come too close experience an attractive force. The origin of these attractions may be from electric monopoles or dipoles associated with the molecules. In the case of dipolar interactions, the attractive potential has the form
This is called the London attractive potential and is purely quantum mechanical in origin.
6
ij
ijij r
ArU
The 6-12 Potential
Attractive and Repulsive terms can be combined to yield a fairly realistic representation of intermolecular interactions at short to medium distances
This potential (often referred to as Lennard-Jones) is very often used to describe weak attractive and strong repulsive forces between molecules.
612
4)(ij
ij
ij
ij
rA
rB
rU
The 6-12 Potential
For a given atom or molecular pair, the potential can be written as:
612
4)(rR
rRrU
Here is the depth of the attractive well and R is the distance at which the potential is zero, thus changing sign from repulsive (>0) to attractive (<0). That is also the position at which the two molecules will be found (potential minimum) if there is no other force acting on them
The 6-12 Potential
For a given atom or molecular pair, the potential can be written as:
612
4)(rR
rRrU
The 6-12 Potential
To evaluate the second virial coefficient using the 6-12 potential requires completing a rather difficult integral
We can elect to approximate the 6-12 potential with a more artificial expression that nonetheless capture the essential features of the 6-12 potential, the square well potential:
The 6-12 Potential
We can elect to approximate the 6-12 potential with a more artificial expression that nonetheless capture the essential features of the 6-12 potential, the square well potential:
)(rU
r<R
0 r>nR
R<r<nR
The 6-12 Potential
Calculate the equation-of-state of a gas of molecules that interact according to a square-well potential.
)(rU
r<R
0 r>nR
R<r<nR
2 / 2 0 22
0
( ) 2 ( 1) ( 1) ( 1)R nR
e kT
R nR
b B T e r dr e r dr e r dr
The 6-12 Potential
11
332 3/
33
2 neRRb KTe
From which we can calculate the expression of state to be:
2 / 2 0 22
0
( ) 2 ( 1) ( 1) ( 1)R nR
e kT
R nR
b B T e r dr e r dr e r dr
3 3
/ 3 22 1 13 3
e KTP R R e nkT
The 6-12 Potential
If /kT<<1 then e
kTkT / 1
Therefore, if the depth of the well is much smaller than the thermal kinetic energy kT:
3 3
/ 3 22 1 13 3
e KTP R R e nkT
3 3
3 2 22 1 1 13 3
P R R an bkT kT kT
The 6-12 Potential
Therefore, if the depth of the well is much smaller than the thermal kinetic energy kT:
3
123
3 Rna
32 3Rb
3 3
3 2 22 1 1 13 3
P R R an bkT kT kT
The 6-12 Potential
3
123
3 Rna 32 3Rb
The parameter a reflects attractive energy terms and is related to the depth of the attractive well () and its width (nR)
As a increases, the pressure P decreases as molecular attractions diminish the force exerted on the container walls.
On the other hand, as excluded volume effects increase, b increases, which in turn increases the pressure.
3 3
3 2 22 1 1 13 3
P R R an bkT kT kT
The 6-12 Potential
3
123
3 Rna 32 3Rb
Essentially repulsive interactions between molecules drive collisions against the container walls thus increasing the pressure. If b=a/kT, then repulsive and attractive forces balance and the equation of state reduces to the ideal gas law.
3 3
3 2 22 1 1 13 3
P R R an bkT kT kT
The 6-12 Potential
3
123
3 Rna 32 3Rb
The expression above is equivalent to the van der Waals equation of state, a phenomenological equation that has the following functional form (see homework).
NRTNbVVNaP
2
2
3 3
3 2 22 1 1 13 3
P R R an bkT kT kT