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Covalent bonds In simple terms a covalent bond exists between two atoms if they share electrons between them. In contrast, an ionic bond is formed if an electrons are transferred between atoms (e.g., in sodium chloride an electron is given up by the sodium atom to form a Na+ ion and accepted by the chlorine atom to form a chloride, Cl-ion. A single bond is formed when one pair of electrons is involved and a double bond when two pairs are involved. In quantum chemical terms such a picture is overly simplistic as the although a bonding orbital results in an increase in electron density between the atoms it also spreads over the rest of the molecule. This is particularly in the case of delocalized bonding. The "classical" example of delocalized bonding is the benzene molecule - which can be described as resonance hybrid between a number of alternate structures: The standard way to approximate the potential energy for a bond in a protein and most other molecules is to use a Hooke's law term where r is the length of the bond (i.e., the distance between the two nuclei of the atoms between which the bond acts), r_eq is the equilibrium bond length and K_r is a spring constant. This basically represents the bond as a spring linking the two atoms. Graph of the potential energy dependence for a C=O Bond Atom pair r_eq in Å K_r in kcal/(molÅ2) C=O C-C2 C-N C2-N N-H 1.229 1.522 1.335 1.499 1.010 570 317 490 337 434 O a carbonyl oxygen C a sp2 carbon (such as that attached to an O) N a main chain nitrogen atom H a hydrogen atom attached to the N C2 a "united atom" group(CH2) Bond angles A bond angle between atoms A-B-C is defined as the angle between the bonds A-B and B-C: Angle Θeqin degrees Kein kcal/(mol.degrees2) C-N-H 119.8 35.0 C2-N-C 121.9 50.0 C2-N-H 118.4 38.0 C-C2-N 110.3 80.0 C2-C-O 120.4 80.0 C2-C-N 116.6 70.0 O-C-N 122.9 80.0 Potential energy curve for the N-C-O Bond Angle Dihedral angles Vn gives the energy barrier to rotation, n the number of maxima (or minima) in one full rotation and γ determines the angular offset. The use of the sum allows for complex angular variation of the potential energy. Barriers for dihedral angle rotation can be attributed to the exchange interaction of electrons in adjacent bonds. Steric effects can also be important. Potential energy curve for the omega dihedral angle Interazioni elettrostatiche Non-bonded Interactions Electrostatic interactions where qi and qj are the magnitude of the charges, rij is their separation, ε0 the permittivity of free space and εr the relative dielectric constant of the medium in which the charges are placed The dielectric constant of selected materials Material Dielectric constant Water (20oC) Water (0oC) Ice (-10oC) Methanol Liquid H2S(-85.5 oC) Beeswax Paraffin Liquid Argon(-191oC) Vacuum 80.3 87.7 ~98 33.6 9.3 2.9 2.0-2.5 1.5 1.0 (by definition) The strictly correct way to use the law would be to consider every nucleus and electron separately, plug it into the Schrödinger equation and apply quantum chemical methods to solve the equation for the spatial configuration of nuclei we are interested. As already mentioned this is completely impractical for biomolecular systems. So instead we wish to develop a useful model for the interactions between nuclear centres (commonly called "atoms") without having to explicitly deal with the electrons in a system. The simplest approach is to just to consider the formal charges of the protein. Formal charges show whether chemical groups are ionized i.e., whether an atom or set of atoms has lost or gained an electron. Isolated amino acids (in neutral solution) are zwitter ionic - this means that although the molecule has no overall charge it carries both a negatively charged group and a positively charged group: In practice salt bridges are relatively rare in proteins and in practice they normal occur on the surface as opposed to internally. An exception is when an internal salt bridge is involved in the catalytic mechanism of an enzyme such as in the asp-his-ser triad of serine proteases (a classic example of the structural basis of enzyme activity): The reason for this is that although an internal salt bridge is a strong interaction in comparison to having the isolated residues widely separated in a vacuum it is normally destabilizing for a protein. This apparent paradox is due to that fact that when considering the effect of an interaction one must consider the difference in the (free) energy between the folded and unfolded but solvated states. In the unfolded state the residues involved in a salt bridge would be widely separated but each making very favourable interactions with water molecules (there is an entropic contribution to this). These interactions are lost when the same residues are buried in the largely hydrophobic core of the protein. Similar arguments apply to practically all considerations of elucidation the energetic contributions to protein folding or ligand binding - normally a small overall free energy advantage arises from the balance between large but cancelling contributions. Hydrogen bonds 2.8 Å 6kcal/mol Partial Charges We have seen that electrostatic interactions are of fundamental importance to proteins. We shall now briefly examine the manner in which they are normally treated in computational studies. The most common approach is to place a partial charge at each atomic centre (nucleus). These charges then interact by Coulomb's Law. The charge can take a fractions of an electron and can be positive or negative. Charges on adjacent atoms (joined by one or two covalent) bonds are normally made invisible to one another - the interactions between these atoms being dealt with by covalent interactions. Note that the concept of a partial charge is only a convenient abstraction of reality. In practice many electrons and nuclei come together to form a molecule - partial charges give a crude representation of what a neighbouring atom will on average "see" due to this collection. The standard modern way to calculate partial charges is to perform a (reasonably high level) quantum chemical calculation for a small molecule which is representative of the group of interest (e.g., phenol is considered for tyrosine). The electrostatic potential is then calculated from the orbitals obtained for many points on the molecular surface. A least squares fitting procedure is then used to produce a set of partial charges which produce potential values most consistent with the quantum calculations. The normal treatment for partial charges is to assume they are fixed. In practice the electric field caused by other atoms and molecules will polarize an atom effecting its electron distribution and thus its partial charge. In turn the partial charge produces an electric field which affects neighbouring charges and thus fields. The process of polarization has an energetic effect. In practice it is difficult to find adequate parameters to treat systems as complex as proteins. Induction effects can be shown to decay by a r-6 relations so they can normally be regarded as implicitly corrected for when the dispersion term is fitted. Dispersion The Dispersion interaction can be shown to vary according to the inverse sixth power of the distance between the two atoms. The factor Bij depends on the nature of the pair of atoms interacting (in particular their polarizability) The factor Bij depends on the nature of the pair of atoms interacting (in particular their polarizability). It is normal to parameterize the dispersion empirically using structural and energetic data from crystals of small molecules. It is not possible to use simple quantum chemical calculations to find parameters. In this each electron is solved independently keeping the other orbitals frozen (in a self consistency). This effectively means that electrons only experience a time averaged picture of other electrons - so that dispersion cannot come into effect. More advanced methods in quantum chemistry introduce methods to tackle "electron correlation" to avoid this. Repulsion When two atoms are brought increasing close together there is a large energetic cost as the orbitals start to overlap. In the limit that the atomic nuclei where coincident the electrons of the two atoms would have to share the same orbital system. The Pauli exclusion principle states that no two electrons can share the same state so that in effect half the electrons of the system would have to go into orbitals with an energy higher than the valence state. For this reason the repulsive core is sometimes termed a "Pauli exclusion interaction". The Lennard-Jones potential and van der Waals Radii The equation can be rewritten in an equivalent more instructive form (choosing the case for an interaction be two atoms of the same type): The minimum of the function is at r = 2R* and has an energy of minus E*. The distance R* is know as the van der Waals radius for an atom and E* is its van der Waals well depth. atom type van der Waals radius in Å C (aliphatic) O H N P S 1.85 1.60 1.00 1.75 2.10 2.00 van der Waals well depth in kcal/mol 0.12 0.20 0.02 0.16 0.20 0.20 It is important to note that the Lennard-Jones interaction between uncharged atoms (such as CH3 groups) is less attractive than that between charged groups such as oxygens. The difference is that the contribution from electrostatics will dominant the L-J interactions. In cases where uncharged groups form compact structures van der Waals energies are often cited as stabilizing the conformation. Although partly true very often the major contribution comes rather from hydrophobic exclusion. Solvent effects and the hydrophobic interactions The fact that proteins normally function in an aqueous environment considerably complicates the understanding of the interactions between different groupings. We have already seen that when thinking about the effect that various interactions (such as hydrogen bonds) have on the overall stability of a protein one must compare the free energy contribution to the folded state with that of the random coil. In this section we will examine two important contributions that solvent makes to protein interactions. • every dipole lines up so that its positive end points toward the negative charge and vice-versa. This means that the electric field caused by the dipoles will oppose the original electric field at all places. This reduction in field causes a reduction in electric potential and thus a reduction in the interaction energy. • the electric field between charges permeates the whole of space - it does not only depend on what is immediately in between the charges. So we have seen that the dielectric constant of materials is caused by microscopic dipoles in the material. These have two sources: 1. Electronic polarizability. As we have seen in both the sections on induction and dispersion when an electric field is applied to an atom a dipole is induced. This is because the electron cloud surrounding the nucleus tends to be displaced by the field. This induced dipole contributes to the dielectric constant of any material. 2. Orientational polarizability. If the molecules composing the material have an intrinsic dipole moment and they are free to rotate these will have a tendency to rotate so as to oppose the external field. The larger the dipole moment the larger the induced field and so the larger the dielectric constant. In the early days of protein simulation (1980's) it was often assumed that the effective dielectric "constant" between two charges at a separation of R angstroms varied as R. This had the fairly dodgy justification that one could expect electronic polarizability to dominate when two atoms are in proximity, when R is around 3 (not necessarily true as the medium surrounding the charges as well as between them has a dielectric effect). At large distances the dielectric would get larger reflecting the fact that increasing amounts of water would tend to surround the charges. The real justification for the model was that it save a lot of computer time when this was a really scare commodity. It changes Coulomb's law into a R-2 dependence so that it avoids the necessity of finding using the square root to work out the distance between two atoms. Recent work has found that using a more sophisticated function to represent "dielectric constant" (usually with a sigmodial form) has yielded good results. A more normal approach today is to actually include a large number of water molecules in a molecular dynamics simulation. A dielectric constant of 1 or 2 (to represent electronic polarizability) is used. An alternative approach is to use a "reaction field". The hydrophobic effect This is in essence a simple observation that a apolar molecules tend to aggregate in the presence of water. It is known to be entropically rather than enthalpically driven. For this reason it is not usual to incorporate a term for hydrophobic energy into potential energy functions. In fact simulations can be performed using molecular dynamics methods showing hydrophobic aggregation for small molecules using a conventional potential energy function Potential energy functions We have briefly reviewed the variety of interactions which are important in protein interactions and seen suitable simple mathematical forms for their representation. These are drawn together to form a potential energy function: This function can be used to calculate a value for the potential energy (PEF(R)) for any conformation of a given protein defined by the (normally Cartesian) coordinate vector R. A number of important points can be made: * We called the function a "potential" energy function as it does not contain contributions made to the total energy made by the motions of the atoms involved. It is possible to calculate these using molecular dynamics methods. •The function aims to give reasonable values for the difference in "microstate" energies between two different conformations. The absolute value for the energy given does not mean anything (certainly NOT the free energy of formation). Only differences have meaning. The above equation also does not allow the examination of any process which involves the change in chemical bonding, e.g., one cannot simulate chemical reactions in an enzyme active site with it. •This kind of function is normally of little use in estimating whether a protein adopts a particular fold To be able to calculate the potential energy of a protein using the above equation a large number of parameters (equilibrium bond lengths beq, bond stretching constants Kb ...) is required. The process of finding these is arduous and in there are only around four potential energy functions in common usage for proteins (CHARMm, AMBER, GROMOS and ECEPP). •Although results obtained with current potential energy functions are only approximate they have one great advantage - they are computationally cheap. This allows the introduction of realstic representation of environment - such as having large numbers of explicitly modelled water molecules surronding a protein. • It also allows the calculation of the potential energy for many different conformations of the same molecule. This facilitates the use of techniques such as molecular dynamics which allows the thermal motions of a system to be explored. • This can be contrasted with quantum chemical methods which even for small systems are so expensive that only a limited number of calculations can be made but produce very accurate energies. Energy minimization This is in many ways the simplest simulation procedure. The basic idea is that starting from some structure (R) we find its potential energy using the potential energy function given as equation (1) above. The coordinate vector R is then varied using an optimization procedure so as to minimize the potential energy PEF(R). Very often these methods are used if a distorted structure is produced - e.g. a homology based model. Energy minimization can then relieve short interatomic distances while maintaining important structural features. Energy minimization can be used to help to solve experimental structures: * In X-ray crystallography measure a set of intensities I(h,k,l) for a large number of reflections (h,k & l are Miller indices listing the reciprocal lattice points of the crystal). These are proportional to Fobs(h,k,l)2 the observed structure functions. Once there is an approximate idea of the structure (an initial model) the model's electron density can be used to calculate expected structure factors Fcalc(h,k,l). The structure can be refined by minimizing: The program XPLOR is commonly used to do this In NMR (nuclear magnetic resonance) experiments give approximate distances between hydrogen atoms (NOE's) and some dihedral angles. To obtain a 3D structure a similar process is performed in which the objective function is the sum of PEF(R) and restraint terms for the distances and dihedral angles. Molecular dynamics simulated annealing is used to optimize the function to obtain a set of 3D structures consistent with the experimental data. * Denaturazione delle proteine Temperatura Pressione Acido Base Agenti caotropici Salting out Detergenti