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Covalent interactions (bonds) hold the atoms of
biopolymers and small molecules together. Covalent bond energies are on
the order of 100 kcal/mole (400 joule/mole). Covalent bonds don't break
when proteins fold (or unfold), when RNA folds, when DNA anneals, or
when membranes assemble.
Those processes (protein folding/unfolding, etc) are controlled by
non-covalent interactions. Each given non-covalent interaction is
relatively weak, often with a free energy of less than RT. But
cumulatively the energies are huge. Count the number of intramolecular
pair-wise interactions between the atoms of a globular protein, or
between protein and water atoms in an unfolded protein. Huge numbers of
small non-covalent forces drive the spontaneous folding or unfolding of
proteins and nucleic acids. The folded/unfolded equilibrium of a
proteins is generally small. The protein is held in 'delicate balance
between powerful countervailing forces'. Large forces drive a protein to
fold. Large forces drive it to unfold. It is the small difference
between these large numbers that determines direction of the reaction. A
small change in pH or temperature can change the balance. Unfortunately
these forces are numerous, complex and poorly understood.
How to denture protein in your own kitchen? When you heat an egg to around 60 deg C, the proteins denature and aggregate (and that
turns them white). Or add lemon juice to your milk.
Noncovalent forces were discovered by
van der Waals during analysis of deviations from the ideal gas law. He noticed that molecules are sticky.
There are many different ways of classifying noncovalent interactions. For our purposes these
are:
(B) charge-charge interactions
(E) hydrogen bonding interactions
A. Short Range Repulsive Interactions
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Atoms need space. Overlap of the occupied
orbitals of two non-bonded atoms results in electrostatic repulsion
between the electrons of those atoms. This repulsive energy acts over a
very short range, but goes up very sharply when that range is violated.
The repulsion goes up as 1/R^12. It is important only when atoms are in
very close proximity, but then it becomes very important. Because this
repulsive term rises so sharply as distance decreases it is reasonable
to think of atoms as hard spheres, like small pool balls, defined by van
der Waals radii and surfaces. When two atoms approach each other their
van der Waals surfaces make contact when their distance reaches the sum
of their van der Waals radii (here we are assuming a lack of bonding
interactions). The same principle applies to interactions between
molecules although the shapes are more complex. The smallest distance
between two non-bonded atoms is the sum of the van der Waals radii of
the two atoms.
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The van der Waal radius of carbon is evident from the spacing between the layers in graphite. Those coordinates are contained here [coordinates]. The distance between atoms in different layers of graphite is never less than twice the van der Waals radius of carbon (2 x 1.7 = 3.4 Å). The atoms within a graphite layer are covalently linked and so are in violation of the van der Waals radius. vdw surfaces are also violated by hydrogen bonds and 1,3 contacts.
How to detect short range repulsion? Try compressing a liquid.B. Charge-Charge Interactions.
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These are interactions between cations and anions, which are functional groups with formal charge. Charge-charge interactions are very strong, and cause the vapor pressure of sodium chloride, for example, to be very low. The coordinates of sodium chloride are located here [coordinates]. These interactions, the attractive force between the sodium cation and chloride cation in sodium chloride, are frequently called 'electrostatic interactions'. However ALL molecular interactions are inherently electrostatic in nature, so here we will avoid the term 'electrostatic interaction' when we are describing a 'charge-charge interaction'. For example the amino acids aspartic acid (an anion) and lysine (a cation) engage in charge-charge interactions. Charge-charge interactions can be either attractive or repulsive, depending on the charges of the interacting species. The electrostatic force between two point charges is given by:
where k = 9.0 x 10^9 nt-meter^2/coul^2
q = -1.6 x 10^-19 coulombs for an electron.
r = distance between the point charges (meters)
e = the dielectric constant of the medium (unitless).
e reflects the tendency of the medium to shield one charge from another. e is 1 in a vacuum, around 4 in the interior of a protein and 80 in water. The problem of calculating electrostatic effects in proteins is complex in part because of non-uniformity of the dielectric environment. The dielectric micro-environment is variable, with less shielding of charges in regions of hydrocarbon sidechains and greater shielding in regions of polar sidechains.
One can crudely estimate the energetics of a charge-charge interaction in a protein. The energy of an amine (charge +1) and a carboxylic acid (charge -1) separated by 4 Å in the interior of protein is given by:
This rough approximation is around 10-fold greater than the values determined experimentally.
Charge-charge interactions fall off slowly with distance (1/r).
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In a molecule with unlike atoms,
electrons are not shared equally. The tendency of any atom to pull
electrons away from other atoms is characterized by a quantity called
electronegativity.
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In a molecule composed of atoms of various electronegativities the atoms with lowest electronegativities hold partial positive charges and the atoms with the greatest electronegativities hold partial negative charges.
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The extent of charge separation within a molecule is characterized by the dipole moment (m). To express dipole moments, charges are expressed in esu's and distances in centimeters. The dipole moment of an electron and a proton separated by 1 Å is given by:
The orientation of the dipole moment of a peptide is approximately parallel to the N-H bond and in magnitude is around 3.7 Debye. That is a relatively large dipole moment.
The dipole moment of HCl is 1.0 Debye, that of CH3Cl is 1.9 and that of HCN is 2.9. The large dipole moment of a peptide bond should lead one to expect that dipolar interactions are important in protein conformation and interactions. The large dipole of a peptide bond can be attributed in part to resonance.
A dipole can interact with point charges (called Charge-Dipole Interaction), other dipoles (called Dipole-Dipole Interaction), and can induce charge distribution in surrounding molecules (called Dipole-Induced Dipole Interaction). We will discuss each of these interactions separately.
Dipole-dipole Interactions. The interaction energy between two dipoles can be either positive or negative and can be calculated with Coulomb's Law. Listed below are the energies of interaction for two dipoles with moments of 1 Debye at a distance of 5 Å in a medium of e = 4.
Dipole-dipole interactions fall off with 1/r^3.
Dipole-Induced Dipole. Interactions. A molecule with a permanent dipole can induce a dipole in a second molecule that is located nearby in space. The strength of the interaction depends on the dipole moment of the first molecule and the polarizability of the second. Molecules with pi electrons, such as benzene and phenyl alanine are more polarizable that molecules without pi electrons.
Dipole-induced dipole interactions are always attractive and can contribute as much as 0.5 kcal/mole to stabilization.
Dipole-induced dipole interactions fall off with 1/r^4.
D. Fluctuating Dipole (Dispersive Interaction, London Forces)
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Molecules behave like oscillating dipoles. In molecules that are located nearby to each other the oscillators are coupled. Coupled fluctuating dipoles experience favorable electrostatic interaction known as dispersive interactions. The strength of the interaction is related to the polarizabilities of the two molecules.
Dispersive interactions are always attractive and occur between any pair of molecules, even non-polar molecules. Dispersive interactions provide the only attractive force between molecules in liquid N2, which boils at 77K. For a given atom-atom contact the energy of stabilization provided by dispersive interactions is very small (0.05 kcal/mole). However the total number of contacts within a protein is generally enormous, so that dispersive interactions can make a large contribute to stability.
Fluctuating dipole interactions fall off with 1/r^6.
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The idea that a single hydrogen atom could bond simultaneously to other two atoms was proposed in 1920 by Latimer and Rodebush and their advisor, G. N. Lewis. M.L. Huggins also describes the hydrogen bond in his 1919 dissertation.
An acceptor atom (A) with a basic lone pair of electrons can interact favorably with a donor atom (D) that bears an acidic proton. A strong hydrogen bond requires that both atoms A and D are electronegative atoms. The most common hydrogen bonds in biological systems involve oxygen and nitrogen atoms. Sulfur can also engage in hydrogen bonds. Hydrogen bonds where atom D is a carbon atom have been observed although these are relatively weak interactions. Hydrogen bonds are essentially electrostatic in nature, although the energy can be decomposed into additional contributions from polarization, exchange repulsion, charge transfer, and mixing. In traversing the Period Table, increasing the electronegativity of atom D strips electron density from the proton, increasing its partial positive charge, and increasing the strength of the hydrogen bond. Two different ways of representing a hydrogen bond are shown below.
A hydrogen bond is not an acid-base reaction. In an acid/base reaction, but not in a hydrogen bond, the proton is transferred from D to A. But the acidity of the proton bound to D and the basicity of the lone pair of A both correlate roughly with the strength of the hydrogen bond.
Water. Water is an excellent hydrogen bonding solvent. A water molecule linked by hydrogen bonds to two other water molecules is located here [coordinates]. Notice that the hydrogen-bonding distance from H to O is around 1.8 Å, which is less than the sum of the O and H van der Waals radii (O, 1.5 Å; H, 1.0 Å). Also notice that the hydrogen-bonding distance from O to O is around 2.8 Å, which is less than twice the van der Waals radius of oxygen (1.5 Å).
Oxygen is highly electronegative, and gains partial negative charge by withdrawing electron density from the two hydrogen atoms to which it is covalently bonded, leaving them with partial positive charges. Water has a balanced number of hydrogen bond donors and acceptors. In ice, every water molecule acts as a donor in two hydrogen bonds and an acceptor in two hydrogen bonds. A very small ice cube, viewed at atomic resolution is located here [coordinates]. For additional information on water, see the section on water and the hydrophobic effect.
Ammonia.
Hydrogen bond strengths form a continuum. Strong hydrogen bonds of 20-40 kcal/moll, generally formed between charged donors and acceptors, are nearly as strong as covalent bonds, Weak hydrogen bonds of 1-5 kcal/mol, sometimes formed with carbon as the proton donor, are no stronger than van der Waals interactions. Moderate hydrogen bonds, which are the most common are formed between neutral donors and acceptors are from 5-15 kcal/mol.
The geometry of a hydrogen bond can be geometrically described by three quantities, the D to H distance, the H to A distance, and the D to H to A angle. Hydrogen bonds are not necessary, or even generally, linear. In fact hydrogen bonds can be two-centered (as shown above) and three-centered and four-centered as shown below.
Hydrogen atoms are not observable by x-ray crystallography as applied to proteins and nucleic acids. So a geometric description of hydrogen bonding that is dependent on proton position is not practical in protein and nucleic acid structures. In these cases one is usually limited to analysis of the D to A distance. It is common to ascribe a hydrogen bond if a distance between A and D is less than the sum of their van der Waal radii. However this limit is probably too conservative.
In biological systems, hydrogen bonds are frequently cooperative. For example in the hydrogen-bonded system below, hydrogen bond 1 increases both the acidity of the hydrogen, and the basicity of the oxygen, in hydrogen bond 2.
F. Solvent, Counter Ion, and Entropic Effects
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DNA. Nucleic acids in aqueous solutions can be considered rod like polyanions surrounded by inorganic cations (Na+, K+ and Mg2+) and polyamines. The high axial density of negatively charged phosphate groups on DNA causes strong radial electric fields. These electric fields lead to steep radial gradients of the surrounding cation concentrations. Theoretical considerations (counterion condensation) predict that the local concentration of a monovalent cation such as K+ near the surface of DNA is around 2 Molar. This local concentration (i.e., near the DNA) of K+ is largely independent of K+ concentration in bulk solution.
Release or uptake of local cations accompanies processes such as helix-coil and other conformational transitions that change the axial charge density. Release or uptake of local cations also accompanies binding of proteins, and intramolecular compaction or collapse of DNA. These effects explain the dramatic salt dependencies of DNA melting and DNA-protein complexation.
DNA condensation. Genomic DNAs are very long molecules. The 160,000 base pairs of T4 phage DNA extend to 54 microns. The 4.2 million base pairs of the E. coli chromosome extend to 1.4 millimeters. In biological systems, long DNA molecules must be compacted to fit into very small spaces inside a cell, nucleus or virus particle. The energetic barriers to tight packaging of DNA arise from decreased configurational entropy, bending the stiff double helix, and intermolecular (or inter-segment) electrostatic repulsion of the negatively charged DNA phosphate groups. Yet extended DNA chains condense spontaneously by collapse into very compact, very orderly particles. In the condensed state, DNA helixes are separated by one or two layers of water. Condensed DNA particles are commonly compact toroids. DNA condensation in aqueous solution requires highly charged cations such as spermine (+4) or spermidine (+3). Divalent cations will condense DNA in water-alcohol mixtures. The role of the cations is to decrease electrostatic repulsion of adjacent negatively charged DNA segments. The source of the attraction between nearby DNA segments is not so easy to understand. One possible source of attraction are fluctuations of ion atmospheres in analogy with fluctuating dipoles between molecules (London Forces).
Proteins. When thinking about the free energy of protein folding (folding is the transition from denatured state to native state) it is important to understand that the stability of a folded state in a biological system can only be understood as a difference between the free energies of the native and denatured states.
The molecular interactions described in the previous sections (charge-charge, dipole-dipole, hydrogen bonding...) are observed in both states. In the native state many of these interactions are intramolecular, for example one part of the protein will form a hydrogen bond with another part of the protein. In the denatured state, interactions are intermolecular, (i.e., between the protein and water molecules, cations, & anions). A intramolecular hydrogen bond observed in the native state will be replaced in the denatured state by several hydrogen bonds between the protein and water molecules. The same is essentially true for charge-charge, dipole-dipole, dipole-induced, and fluctuations dipole interactions.
One of the primary driving forces for protein folding is solvent release. In the transition from denatured to native state, a large number of solvent molecules are released. Release is defined as the conversion from bound state to free state.
G. Water and the Hydrophobic Effect
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Most living organisms are around 80% by weight water. Water is a reactive substance with unusual properties and is a crucial determinant in the structure and properties of other cellular components. Water is a uniquely powerful solvent power for ions and of equal importance is a uniquely weak solvent for non-polar substances.
The forces of attraction between water molecules in the liquid state are unusually high. The melting point, boiling point, heat of vaporization, heat of fusion, and surface tension of water are higher than those of similar substances. For example the heat of vaporization of water (540 cal/g) is over twice that of methanol (263) and nearly ten times that of chloroform (59).
Electrostatic interactions are highly attenuated in water. The attractive force between two oppositely charged ions in solution is inversely proportional to the dielectric constant of the solvent. The dielectric constant of water (80.0) is over twice that of methanol (33.1) and over five times that of ammonia (15.5)
Molecular Structure of Water in the Crystalline State. A water molecule is tetrahedral in shape with either a hydrogen atom or a lone pair of electrons at each apex of the tetrahedron. Oxygen, which is highly electronegative, withdraws electron density from the hydrogen atoms. The charge distribution of a water molecule is shown below.
X-ray and neutron diffraction of crystalline ice shows each water molecule engaged in four hydrogen bonds with intramolecular oxygen-oxygen distances of 2.76 Å. Each oxygen atom is located at the center of a tetrahedron formed by four other oxygen atoms. Each hydrogen atom lies on a line between two oxygen atoms and forms a covalent bond to one oxygen (bond length: 1.00 Å) and a hydrogen bond to the other (hydrogen bond length: 1.76 Å). The hydrogen atoms are not located midway between oxygen atoms. For additional information see the section on hydrogen bonding interactions
There are many degrees of freedom in hydrogen bond donor/acceptor relationships which are interconverted by cooperative rotations. Ice is rather disordered in this respect. The water molecules in the crystalline state are not closely packed and there is empty space within the crystal. The loose packing arrangement is the reason that water decreases in density upon freezing (i.e., ice floats).
Molecular Structure of Water in the Liquid State. In the liquid state, water is not nearly as ordered as the diagram above might indicate. At O deg C a time-averaged water molecule is involved in around 3.5 intermolecular hydrogen bonds. Some of them are three- and four-centered.
The Hydrophobic Effect. Oil and water do not mix because of the hydrophobic effect.
Hydrophobic substances are those which are highly soluble in non-polar solvents but only slightly soluble in water. This definition excludes substances which are generally insoluble because of strong intermolecular cohesion. Hydrophobic substances are non-polar and non-hydrogen bonding.
The schematic diagram above illustrates what happens when a hydrophobic substance (cyclohexane in this case) is converted from vapor to neat liquid to aqueous phase. In the first step from vapor to neat liquid there is an increase in intramolecular interactions and a decrease in rotational and translational degrees of freedom. Therefore one expects, and sees, a favorable enthalpy contribution (negative delta H) and an unfavorable entropy contribution (negative delta S) for the condensation. In the second step from neat liquid to dilute aqueous phase, the change in stability conferred from intramolecular interactions is a wash, no gain or loss. But the water loses entropy. Somehow water is more highly ordered in the vicinity of a cyclohexane molecule. Therefore, for this step, delta S is negative and so is delta G.
As illustrated below, in the aqueous phase a region of relatively low entropy (high order) water forms at the interface between the aqueous solvent and a hydrophobic solute.
The decrease in entropy at the interface arises from the strong intermolecular forces between water molecules. In bulk water, these forces are isotropic (extending in all directions). At the interface these forces are anisotropic because the cyclohexane molecule does not form hydrogen bonds. Thus water at the interface is rotationally and translationally constrained.
When isolated octane molecules aggregate in aqueous solution, the total volume of interfacial water decreases. Thus the driving force for aggregation of hydrophobic substances arises from an increase in entropy of the aqueous phase. The driving force for aggregation does not arise from intrinsic attraction between hydrophobic solute molecules.
If one considers the entropy of the cyclohexane molecules alone, a dispersed solution appears to be of greater entropy, and more stable, than an aggregated state. Similarly, a protein may appear to have greater entropy in a random coil than in a native state. Only when the entropy of the aqueous phase is factored into the equation can one understand the separation of water and oil into two phases, and the folding of a protein into a native state.
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