Molecular modeling of localized collective motions and dynamics in proteins

1.2 Structure basis for protein motion and dynamics The motions of the atoms in a protein tend to share certain characteristics that can be explained in terms of the basic structure of

Chapter 1

Introduction

1.1 Protein motion and dynamics vs protein function

Protein motions and dynamics are involved in a variety of biological processes in living organisms Though in some of these processes the static structure of a protein determines its function (e.g collagen in tissues or α-keratin in hair), protein motions and dynamics are crucial in others cases Examples are metabolism, transport and synthesis of biomolecules etc In fact, all dynamic biological processes can find the origin in protein motions and dynamics Muscle contraction, for example, is based on the combined motion of actin and myosin Other examples are the molecular motors kinesin and F1-ATPase Motion and dynamics also play important roles in many other proteins whose primary function is not mobility itself Conformational change is actually essential to the function of many transport proteins, enzymes, and those proteins involved in signal transduction, immune protection or gene expression[1]

It has been noticed that the biological function of most of the globular proteins often includes an interaction with one or more different molecules on appropriate occasions, such as small ligand, substrate, peptide, a fragment of nucleic acids, even another protein

In many enzymes, conformational changes serve to enclose the substrate, thereby

preventing its release from the protein and ideally positioning it for the protein to perform its function, as in lysozyme For example, immunoglobulins are highly flexible in order to

be able to interact with a large range of ligands Generally, functional interactions of flexible ligands with protein binding sites often require conformational adjustments in both the binding ligands and the host protein The structural changes in some proteins regulate the interaction between ligands and protein through induced fit and allosteric effects [2,3]

The “induced fit” theory by Koshland [4] proposed that the original structure of active site in enzymes does not fit substrate exactly, but the presence of the substrate induces structure changes in the active site to fit for substrate binding It is expected that each intermediate step of the whole cycle of enzyme catalysis requires the enzyme molecule, especially the active site region, to be in a specific conformation different from another

Allosteric effect is found in a special class of proteins, so-called allosteric proteins Substrate binding to one subunit of these multimeric proteins triggers conformational changes in proteins which alters the substrate affinity of the other subunits, thereby sharpening the switching response of these proteins [5,6]

Moreover, the importance of motion and dynamics to protein function has been further confirmed by various experimental studies Two major sources of evidence come from X-ray crystallographic analysis and Nuclear Magnetic Resonance (NMR) One example from X-ray analysis is myoglobin In order to capture, bind and release oxygen (O2) freely, myoglobin has been found to have more mobile character toward the periphery of the molecule although the core surrounding of its heme group is compact [7] A similar X-

ray analysis of lysozyme has produced the intriguing observation that the enzyme’s active

site cleft undergoes an ~1 Å closure upon substrate binding [8] On the other hand,

NMR study revealed that several sub-states often exit for one protein or enzyme These sub-states, which each have slightly different atomic arrangements, randomly interconvert

at rates that increase with increasing temperature [9]

Instead of being stationary at fixed positions, the atoms in a protein molecule are rather

in a state of constant motion The “static” view of a protein structure from X-ray analysis

is at best a representation of its average structure The atoms in each protein molecule exhibit sizable high-frequency fluctuations about this average The atomic fluctuation affects various bond interactions in proteins, especially those relatively weaker non-covalent interactions For example, hydrogen bonds break when the partner atoms fluctuate out of a certain limit of distance, while alternative hydrogen bonds reform if the new partners come closer Hydrogen bonds keep breaking and reforming during protein motion which gives protein molecule extra dynamic features In summary, proteins are constantly changing the details of their conformation Therefore any attempt to understand the function of proteins requires a scientific investigation of protein motion and dynamics

1.2 Structure basis for protein motion and dynamics

The motions of the atoms in a protein tend to share certain characteristics that can be explained in terms of the basic structure of proteins Each protein is made up of a specific number of small units, the amino acids Each of the 20 different amino acids is characterized by a side chain, a distinctive chemical group that ranges in complexity from

a hydrogen atom in the simplest amino acid, glycine, to elaborate rings of atoms in the most complex amino acid, tryptophan The side chain of each amino acid can vary not only in size and shape, but also in charge, reactivity and ability of forming hydrogen bonds

Proteins are commonly consisting of from 50 to more than 500 amino acids, which corresponds to some 500 to 5,000 non-hydrogen atoms The precise sequence of amino acids determines the average structure and other properties of the protein In particular, the balance of the attractive and repulsive forces between the individual atoms of which the protein are composed causes the peptide to fold in a characteristic way essential to its motions and its functions

In a protein, the amino acids are linked together into a polypeptide chain by peptide bond, as being indicated in Fig 1.1 The peptide bond itself is rigid, because it is involved into tautomerization that gives it considerable double bond character, as Fig 1.2 shows However, there are many other strong bonds in the main chain of the polypeptide free to twist For instance, the N-C and C-C bonds relatively are free to rotate These rotations are represented by the torsion angles phi (φ) and psi (ψ), respectively (shown in Fig 1.3) Phi and psi can vary to certain extent within the Ramanchandran Plots as Fig 1.4 shows [10] This allows the protein to potentially adopt many different conformations Therefore, the twisting, or bond rotation, allows one part of the polypeptide chain to move with respect to another As the polypeptide chain twists and turns, its various side chains move with it The side chains themselves have rotatable bonds which imparts additional flexibility Fig 1.5 shows the rotatable bonds in peptide

The flexibility of the polypeptide backbone and of the side chains is what enables each protein to fold into its characteristic native, or average, structure These sites also facilitate the fluctuations of protein atoms around their average positions Even in the folded protein, however, the thermal energy corresponding to the atomic velocities at room temperature is sufficient to allow twisting motions

Fig 1.1 Peptide bond of amino acids

The rectangle line is the part of the peptide bond

Fig 1.2 Tautomerization of peptide bond

Fig 1.3 Torsion angles in the main chain of protein

φ and ψ are free to rotate

Fig 1.4: The Ramachandran plot

In the above diagram the white areas are sterically disallowed regions for all amino acids except glycine which lacks a side chain The red regions correspond to conformations where there are no steric clashes, i.e these are the allowed regions namely the alpha-helical and beta-sheet conformations The yellow areas show the allowed regions if slightly shorter van der Waals radi are used in the calculation, i.e the atoms are allowed to come a little closer together

Fig 1.5 Sites of flexibility, rotatable bonds in peptide chain

The drawing depicts only the principal atoms of a polypeptide chain The backbone of the chain (black bonds) consists of carbon and nitrogen atoms; the linkage called peptide bonds is rigid, whereas the intervening bonds allow rotations (curved arrows) The side chains shown in detail also contain rotatable bonds

1.3 Classification of protein motion

Protein motion involves a wide variety of conformational change ranging from very subtle, local fluctuations to large, global movements These may be motions of individual atoms, groups of atoms, or whole section of the molecule Generally they can be classified into three broad categories for convenience according to their coherence, displacement amplitude and time-scale, as shown in Table 1A [11] Methods used to study them are given in the table

Table 1A: Types of motions found in protein

Motion Spatial

dis-placement (Å)

Characteristic time (second)

Energy source

Method of observation Atomic

fluctuations

0.01 to 1 10-15 to 10-11 kBT Computer

simulation, X-ray diffraction Collective

motions

0.01 to >5 10-12 to 10-3 kBT NMR,

fluorescence, hydrogen exchange, simulation, X-ray Triggered

an analysis of the spreading of atomic electron density produced by such motion The energy for these motions comes from the kinetic energy inherent in the proteins as a

function of temperature They are normally driven by collisions with solvent molecules or

by random collision with neighboring atoms in the protein Although many of them individually may not be important for protein functions, they contain information that is of considerable significance There may be a correlated directional character to the active-site fluctuations that play a role in enzyme catalysis Furthermore, the small amplitude fluctuations are essential to all other motions in proteins They serve as the “lubricant” which makes it possible for large-scale protein motions to happen on a physiological time scale

The second category contains collective motions, such as the movements of groups of atoms that are covalently linked in such a way that the group moves as a unit Noncovalently interacting groups of atoms may also move collectively The size of the group ranges from a few atoms to many hundreds of atoms, or even entire structural domains, as in the case of the flexible Fc portion of immunoglobulins [12] There are two types of rapid collective motion: those that occur infrequently (like internal aromatic ring-flips), and those that occur with high probability (many collective motions of small groups

of neighboring atoms, bonded or nonbonded, are in the pico-second time regime.) Collective motions can also be very slow, as local unfolding of a polypeptide segments The energy for collective motions also derives from the thermal energy inherent in a protein as a function of temperature The time scale of collective motions (from picoseconds to nanoseconds or slower) allows some of them to be studied by techniques such as NMR and fluorescence spectroscopy

The third category contains motions which can be described as triggered conformational changes These are the motions of groups of atoms (i.e individual side

chains) or whole sections of a protein (i.e loops of chain, domains of secondary and tertiary structure, or subunits) that occur as a response to a specific stimulus The distance moved can be as much as 10 Å or more The time scale can be estimated from the rate of binding or turnover reactions The energy for triggered conformational changes comes from specific interactions, such as electrostatic attractions or hydrogen bonding interactions The best-known example of a triggered conformational change is the transition in tertiary and quaternary structure that occurs when ligands bind to the iron atoms of hemoglobin [13]

Category one and two define atomic motions that, whether individual or correlated, involve random excursions about the equilibrium conformation In contrast, category three

is used to classify motions that involve a transition from one equilibrium conformation to another At any given time, a typical protein exhibits a wide variety of motions described above However, the effective dynamical units in proteins are those collective motions that behave nearly as half-rigid bodies under physiological temperature Examples include range from small chemical groups in the side chains of residues, to a fragment of peptide chain Most of the time the small chemical groups display only relatively small internal motions owning to the high energy cost associated with deformations of bond lengths, bond angles or dihedral angles about multiple bonds The more functionally important collective motions, stimuli triggered or not, involve displacements of larger groups associated with torsional motions with rotationally permissive single bonds in main chain [11] Comparing with the short-time relatively small amplitude motions, the substantial displacement of this class of localized collective motions occur over longer time intervals,

and result in concomitant displacements of the different fragments of protein, like domain

or intra-domain motion in biological process

1.4 Features of functionally important collective motions in protein

Relatively large inter- or intra-domain movements provide spectacular examples of collective motions, and thus have been under extensive investigation because of their important functional roles They normally occur in proteins with half-rigid domains, or constrained sub-domains, or functional groups, linked by short flexible linker regions, which have fewer packing constraints and are free to undergo conformation change An illustrative example is opening and closing of the binding site of HIV-1 protease (HIV Pr) that results from conformational changes in the region covering the reactive site Based on the different open and close even half-close states, analysis of protein crystal structures has shown that protein inter-domain motion can take two basic forms: hinge motions that are not constrained by tertiary packing interactions and shear motions between close-packed segments of polypeptide[14]

The characteristic of these two forms of mechanisms of protein flexibility are summarized in Fig 1.6 and Table 1B According to analysis from Gerstein, motions of close packed segments of polypeptide can be divided into those that are perpendicular to

an interface and those that are parallel Generally, hinge motions produce a motion perpendicular to the plane of an interface, so that the interface exists in one conformation but not in the other, as in the opening and closing of a book or a door Shear motion tends

to be similar to scissors’ trimming It is parallel to the plane of the interface, which is limited by the packing contacts involving the interdigitation of side chains These two basic mechanisms can be applied in a great variety of protein motions[14]

Fig 1.6 Hinged and shear mechanism for domain closure See table1B for a summary of the characteristics of both mechanisms

Table 1B Scheme showing the difference between shear (sliding) and hinge motions

The essential characteristics of the various motions are summarized below

Well-packed interfaces MAINTAINED, throughout

motions

NOT MAINTAINED; rather created, burying surface Motion at interface Parallel to plane of interface

(shear)

Perpendicular to interface

angles

Large changes in a few torsion angles

1.4.1 Feature of hinge domain motion

The most basic motion of a protein is a few large changes in main-chain torsion angles

in the localized region, i.e., at flexible inter-domain linker regions The deformation of an extended strand is the simplest hinge motion because its only constraint is that the torsion angles of the strand remain in the allowed regions of the Ramachandran diagram Consequently, its torsion angle changes can be very large and the resulting motion can rotate the conjoint chain up to 60˚ For example in lactate dehydrogenase, two adjacent torsion changes rotate a strand by ~35˚ in a direction not accessible by a single change [15] If the torsional changes fall in the more constrained alpha-helices, the deformation

of helices will spread over to more residues than the deformation of sheets Because residues in the helices are subject to more severe hydrogen-bonding and steric constraints than those in sheets, their torsion angles are restricted to a smaller region of the Ramachandran diagram Thus, if residues are to remain on a helical conformation, the possible changes in their torsion angles are correspondingly smaller than those of resides

in an expanded conformation Such spread-out helical deformations can produce bending

motions For instance, changes in eight torsion angles between 9˚ and 15˚ in the terminus of a helix in a mutant lysozyme bend its end to produce a shift of 3.3 Å [16]

C-1.4.2 Feature of shear domain motion

Large shifts of close-packed segments of polypeptide would require switching between different interdigitating configurations Small shear motions (Figure 1.7) that do not involve repacking of the interface are commonly seen in domain closure The interdigitating side chains normally accommodate shear motions, mostly, by small (<15˚) changes in side-chain torsion angles The main chain of each segment in a shear motion does not deform appreciably The segment shift and rotate relative to each other by no more than 2 Ǻ and 15˚, amounts likely to be the limits of low-energy conformational adjustment Except at very small interfaces, larger movements than these require the combination of several shear motions

Fig 1.7: Small shear motion in citrate synthase

Representative shear motions between close-packed helices Note how the mainchain only shifts by a small amount and the sidechains stay in the same rotamer configuration

1.4.3 Summary

In general, proteins that can undergo predominantly hinge domain motion usually have two domains connected by flexible linker regions that are relatively unconstrained by packing A few large torsional changes are sufficient to produce almost the entire domain motion The rest of the protein rotates essentially as a rigid body, with the axis of the overall rotation passing through the flexible linker regions

Since an individual shear motion is small, a single one is usually not sufficient to produce a large domain motion Usually, a number of shear motions combine to give a large effect In other words, the peptides that link the shearing segments have small main-chain torsional changes to accommodate the relative movements

It is important to realize that hinge and shear motions are only ideal paradigms to describe large inter-domain motions A real protein motions often has a combination of both motions, i.e., hinges in one part of the protein and shearing interfaces elsewhere Nevertheless, many protein large-scale collective motions can be described as occurring predominantly by a hinge or a shear mechanism, or none of them

In the smaller intra-domain protein motions, hinge and shear mechanism are also involved in the collective motion, when individual loops or helices move relative to each other For example, in Trp repressor, shear mechanism between 2 helices exists to adjust position of helix-turn-helix reading head domain to enable it to bind to DNA[17]

1.5 Computational methods to study motions and dynamics in protein

As computer hardware has become faster and computational methods become more sophisticated, computer modeling has been developed and widely applied in studying protein collective motions and dynamics in the past decades Protein motion can be derived from atomic interactions based on knowledge of the structural fluctuations that occur as a result of thermal motion Such fluctuations can be obtained in various ways, such as molecular dynamics, harmonic dynamics, and stochastic dynamics, and others

Molecular Dynamics (MD) techniques has so far provided the most detailed and direct results on protein motions In MD modeling, insights into molecular flexibility and activity are sought by numerically following molecular configurations in time according to Newton’s law of motions [18,19,20] One of the main tasks in MD simulation is to derive and analyze the overwhelming amount of trajectory that describes the time-dependant changes in atomic coordinates Several kinds of methods have been applied to facilitate the analysis in order to reveal the concerted fluctuations with large amplitudes Examples are principal component analysis (PCA), Monte Carlo methods[21,22,23,24] essential dynamics analysis (EDA) [25] or molecular optimal dynamic coordinates analysis [26] These aspects of Molecular Dynamics modelling have been covered recently in several reviews [27,28,29,30]

In theory, MD modeling can bridge spatial and temporal resolution and thus capture molecular motion over a wide range of thermally accessible states In practice, the

numerical time-step problem has limited most applications to straightforward integration with very small time-steps compared to the motion of major interest With current state-of-art methods and super-computers, a microsecond protein simulation has been described for the 36-residues subdomain of a small protein, vallin, via Cray T3E [31] However, a typical protein of hundreds of amino acids can only be simulated for time-scales of at most nanoseconds [32] Comparing to the most conformational changes taking place in proteins (time range from 10-12 to 102 s ), such simulations may not grasp the essential motions related to biological function which occur at much longer time-scales[33]. Thus a clear gap exists between time scales that can currently be obtained by molecular dynamics and the time scale required for most of biological processes

To study the longer time-scale and more complex functionally related motions in proteins, it is generally necessary to eye on other than the straightforward molecular dynamics simulation method Harmonic analysis of protein dynamics differs from straight molecular dynamics in that they provide a concise and compact description of the concerted motions in protein molecules After modification and improvement, harmonic analysis can also be applied to investigate protein dynamics of bond breaking such as hydrogen bond disruption In early attempts, harmonic approximation had been used to examine dynamical properties of proteins or their fragments, and now it has developed into one of the key computational tools to study protein motion and dynamics

1.5.1 Normal mode analysis (NMA) for protein collective motion

Harmonic analysis was motivated by vibrational spectroscopic studies[34,35,36], in which the calculation of normal mode frequencies from empirical potential functions has

long been a standard step in the assignment of infrared and Raman spectra [37,38] One assumes that the vibrational displacements of the atoms from their equilibrium positions are small enough that the potential energy can be approximated as a sum of terms which are quadratic in the displacements The coefficients of these quadratic terms form a matrix

of force constants which, together with the atomic masses, can be used to set up a matrix equation for the vibrational modes of the molecule

A straightforward computation of normal modes in Cartesian coordinates involves a numerical diagonalization of a matrix of size 3N*3N for a molecule with N atom A set of harmonic vibrational “modes” will be generated by the calculation, among which 3N-6 eigenvalues will provide the internal vibrational frequencies of the molecule The associated eigenvectors to respective eigenvalues give the directions and relative amplitudes of the atomic displacements in each normal mode There is a close connection between mode frequency and the collective character of the protein motions A clear drop-off in collectivity has been found as the frequency increases, with large collectivity only found in modes below 200 cm-1[39]

Each single mode comprises the concerted motions of many atoms which are useful in characterizing fluctuations from a stable equilibrium structure With present-day computers, it

is not difficult to study proteins up to a few hundreds of amino acids with an all-atom model [39] For example, computation of the normal modes for the 159 amino acid protein dihydrofolate reductase, required about 4 hours on a Silicon Graphics R10000 workstation [40]

The standard application of NMA to large protein molecule is computationally expensive; however, several sophisticated numerical techniques can be applied to extract the lowest frequency modes of large molecules [41] A common approximation for larger systems is that bond lengths and angles are fixed This can reduce the size of the matrix involved by about an order of magnitude Calculations can be carried out by direct construction of the potential and kinetic energy matrices in (curvilinear) internal coordinates [42] or through matrix partitioning techniques that start from Cartesian derivatives [43] In general, reduction of the dimensionality of the expansion space has noticeable but not overwhelming effects on the resulting normal mode description of the dynamics The directions of the lower-frequency modes are largely preserved, but frequencies in general are higher in the lower-dimensional space [44], suggesting that small fluctuations in bond lengths and bond angles allow the dihedral angles to become more flexible Many practical aspects of computing modes for large molecules are available elsewhere [45,46].

In brief summary, Normal Mode Analysis uses harmonic approximation of the atomic force-field, for which a single experimental structure suffices Although the harmonic model may not provide the complete description of the motional properties of a protein because of the contribution of anharmonic terms to the potential energy, normal mode analysis has become an increasingly active area of research on protein collective motions over the past years [39] For instance, domain motions within the regulatory and catalytic chains of aspartate transcarbamylase were analyzed by NMA [47,48] NMA has also recently been applied to a-lytic protease in an attempt to understand the substrate specificity [49] Many other studies[50,51] have suggested that NMA is the most

full-suitable theoretical methods in determining functionally relevant motions[52] Therefore, NMA is adopted in this thesis as the theoretical method to study localized collective motions in protein

1.5.2 Modified self-consistent harmonic approach for H-bond breaking dynamics

For protein systems that are strongly bonded and well below their melting temperature, atom displacements are small and the simple harmonic approximation works very well in studying the dynamic motion However, there are often situations where the displacements are not small enough and the anharmonic terms influence the mean-square displacements Examples are motions around melting temperature where the bonds dissociate, or hydrogen bond breaking during protein fluctuation or conformational change

In these cases, the displacements of even near neighbor atoms are necessarily large and the simple harmonic approximation fails When applying harmonic analysis to hydrogen bond disruption of protein dynamics, a modified self-consistent harmonic approach (MSHA) for the analysis of melting is necessary to be developed

Modified self-consistent harmonic approach (MSHA), an important extension of normal modes analysis, has been derived using quantum statistical methods [53] It has been applied into systems where large inter-atomic displacements arise from thermal activation rather than quantum uncertainty [54,55] In this approach, statistical distribution

of random thermal fluctuational motions leading to occasional disruption of individual bonds is derived from that of self-consistently determined vibrational normal modes H-

H-bond disruption probability due to such vibrational motions can then be derived from this statistical distribution

This approach and its algorithm of H-bond parameters have been used to compute thermal fluctuational disruption probability of individual H-bonds in DNA and drug-DNA systems in both pre-melting and melting temperature regime [56,57], with results in fair agreement with observations It is expected that MSHA can also be applied to protein dynamics probe of H-bond breaking In the second part of this thesis, protein dynamics of hydrogen bond disruption is studied by MSHA

1.6 Outline of this thesis

The second chapter of this thesis is concerned with the classical molecular mechanics

in relation to protein dynamics Covalent and Non-covalent interactions are introduced and the common potential functions for each of them are presented

The third chapter presents an extension of theoretical methodology, harmonic approximation in research of protein motion and dynamics Hamonic analysis is based on the vibrational motions of harmonic oscillator Over the range of thermal fluctuations, the displacements of each atom in proteins are small enough to be approximated by simple harmonic oscillator Thus each item of potential energy function will take the form of harmonic potential This chapter presents the mathematical formulation of vibrational analysis first, followed by application issue of normal mode analysis (NMA) in study of protein collective motion, and modified self-consistent harmonic approach (MSHA) in study of hydrogen bond disruption dynamics Not only the parameters used in our

calculation, but also the advantages as well as the disadvantages of the MSHA method are discussed in this chapter

Chapter 4 is about implementation of normal modes analysis to localized collective motions Protein collective motions have been the subject of extensive research because of their fundamental implication to protein function Especially those localized in flexible linker region of protein structure, a few torsional change can initiate the domain or sub-domain motions Nevertheless, the previous NMA studies on protein collective motions so

motions are highly anharmonic and collective that most of the residues are involved In order to find out those functionally import``ant motions localized in flexible linker regions, computed normal modes are scanned for 10 structures of 5 proteins Possible relationship between normal modes frequency range of 20 cm-1 to 200 cm-1 and torsional collective motions localized in protein flexible linker regions has been identified

We also apply harmonic approximation into case of hydrogen bond breaking caused by thermal fluctuation in chapter 5 Hydrogen bond is a ubiquitous feature of protein They are important forces in stabilizing the protein structures and were proposed as part of

“stereochemical code” for protein folding A modified self-consistent harmonic approach

is employed to investigate how easily thermal fluctuation can lead to hydrogen bond breaking Based on Boltzman relationship, H-bond disruption probability can be roughly compared to experimentally estimated free energy of H-bond deletion in proteins As such, proteins with available experimental data of H-bond deletion provide good examples to test the usefulness of MSHA Proteins with hydrogen bonds of folding code are also under investigation It is expected that these H-bonds will have higher level of stability

In the end, Chapter 6 finishes the thesis with some concluding remarks on theoretical approaches in the field of protein dynamics and an outlook to the future

on a rather simple model of molecular interactions within a system with contributions from processes such as bond stretching, bond angle bending, rotation of torsion angle and other interactions For convenince to describe the model we first divide these underlying forces in proteins into consideration of covalent interactions (namely bonds stretching, angle bending and torsion) and non-bonded interactions (electrostatic interactions, hydrogen bonding, van der waals interactions), then summarize them into a potential energy function (PEF)

2.1 Covalent interactions

2.1.1 Covalent bonds and bond stretching

In simple terms a covalent bond exists between two atoms if they share electrons between them 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 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 true

in the case of “delocalized” bonds or “resonance” Delocalized bonding is important in protein structure: it is why the peptide bond is rigid and it also occurs in phenylalanine, tryptophan, glutamic acid, arginine side chains

Atomic motions in proteins as well as in other molecules stretch or compress the length

of covalent bonds in proteins The standard way to approximate the potential energy for a bond stretching is the harmonic approximation via Hooke's law term:

Where r is the length of the bond (i.e., the distance between the two nuclei of the atoms of

the bond), r_eq is the equilibrium bond length and K_r is a spring constant This equation

basically represents the bond as a spring linking the two atoms Typical values for bond

constants and equilibrium bond lengths taken from the AMBER potential energy function are listed in Table 2A:

Table 2A: Typical values for bond stretching from AMBER

Atom pair r_eq in Å K_r in kcal/(mol^2)

C = O 1.229 570

C - C2 1.522 317

C - N 1.335 490 C2 - N 1.449 337

N - H 1.010 434

The shape of the potential energy well is parabolic and the motion, therefore, tends to

be harmonic especially when atoms are fluctuating around equilibrium positions (see Fig 2.1)

This kind of approach does not attempt to reflect the energy of bond formation - it only seeks to reflect the energy difference on a small motion about the equilibrium value A much more accurate representation is based on the application of the Morse potential which has an anharmonic potential energy well Function of Morse potential will be introduced in section of 2.2.3.3

Fig 2.1 Graph of the potential energy dependence for a C=O Bond This graph shows the potential energy for a C=O bond using a harmonic potential in Eq 2.1 with the parameters given in the AMBER potential energy function Note that when the bond is at its equilibrium

length i.e., r = r_eq the potential energy is assigned to be zero and when r approaches large values

(i.e the bond breaks) the energy goes infinite

2.1.2 Bond angle bending

A bond angle θ between atoms A-B-C is defined as the angle between the bonds A-B and B-C:

Fig 2.2 Bond angle

As bond angles are found (experimentally and theoretically) to vary around certain value, it is sufficient in most applications to use a harmonic representation (in a similar manner to the bond stretching potential), as being shown in Eq.2.2):

-Eq.2.2 Table 2B shows the typical values for and equilibrium bond angles and bond angle constants from the AMBER potential energy function:

Table 2B: Typical parameters for bond angle bending from AMBER

Angle in degrees in kcal/(mol.degrees^2)

To illustrate the shape of potential energy well for bond angle bending, an

example of Fig 2.3 shows the potential energy for a N-C-O bond angle using the

harmonic potential with the AMBER parameters given in Table 2B The dashed

line indicates an energy of 0.29 kcal/mol which is equal to 1/2RT at a

temperature of 300K This is the energy that an individual degree of freedom can

expect at this temperature and indicates that this bond could be expect to be

undergoing vibrations of the order of 4 degrees at room temperature

Fig 2.3 Potential energy curve for the N-C-O Bond Angle

rotation, n is the number of maxima (or minima) in one full rotation and determines the

angular offset The use of the sum allows for complex angular variation of the potential energy (in effect a truncated Fourier series is used)

Fig 2.5 shows the potential energy curve for H-N-C-O dihedral angle The parameters used are from the AMBER potential energy function (Table 2C):

Table 2C: AMBER parameters used in H-N-C-O dihedral angle

n (in kcal/mol) (in degrees)

Fig 2.5 Potential energy curve for the omega dihedral angle The AMBER potential energy function calculates the torsion energy for every set of four atoms bonded in a chain This contrasts with other PEF's (such as CHARMm), where only one set of four atoms is considered to be the source of the energy for each torsional degree of freedom

2.2 Non-bonded interactions

In late 1960's, it was found that the potential energy function of covalent interaction is insufficient to provide a full representation of the total energy in proteins Apart from the above described interactions, non-bonded interactions have been found to play essential roles in stabilizing protein structure

The non-bonded interactions represent the pair-wise sum of the energies for all possible interactions between non-bonded atoms in protein molecule The non-bonded energy accounts for electrostatic, van der Waals interactions, and hydrogen bond interactions

2.2.1 Electrostatic interactions

2.2.1.1 Coulomb’s Law

Atoms in proteins carry partial charges Electrostatic interactions between these atoms

can be described by the Coulomb’s law (as shown in Eq.2.5):

where and are the partial charges of atoms, is the distance between the pair of atoms,

is the permittivity of free space, and is the relative dielectric coefficient of the

Essentially, Coulomb's law states that the interaction between two charged particles is inversely proportional to distance separated them (as being indicated in Fig 2.6) At a distance of 2.0Å and in the absence of any intervening atoms, the attraction of a positive charge to a negative charge yields a stabilization (indicated by the negative sign of the energy) of about 46 kcal/mol In other words, nearly 48 kcal/mole are required to separate the opposite charges to an infinite distance Apparently, this force is a strong non-covalent force comparing to other non-covalent forces which normally range from less than 1 to tens of kcal/mole This energy is about half of the strength of a C-C covalent bond

Therefore, we may estimate that electrostatic attractions, as well as repulsions, can provide substantial forces that impact protein structures

We must be cautious in these estimates, however, because the charges in a carboxylate anion and an ammonium cation are spread out over many atoms Such delocalization of charge will decrease the magnitude of the electrostatic attraction by increasing the average separation of the charges

2.2.1.2 Electrostatics in protein

In proteins the individual amino acids are polymerized by peptide bond This results in

a peptide backbone which is electrically neutral with the exceptions of the ends of the

carboxyl end carries a positive charge (-CO2-) In some cases (e.g., a number of membrane polypeptides) the ends are chemically modified to avoid these charges (for instance by acetylation of the amino end group)

Most of the standard amino acids found in proteins have uncharged side chain groups However, there are a number of basic residues which are positively charged at normal pH

as in Lysine and Argine (Fig 2.7) :

Fig 2.7 Charges of Arginine and Lysine

In addition, histidine is normally charged at neutral pH (the charge normally residing

on the delta carbon but sometimes on the epsilon) When the residue is placed in a basic environment it loses a proton and becomes uncharged, as shown in Fig 2.8:

Fig 2.8 histidine charges in different PH

There are two standard residues which normally carry a negative charge: glutamic and aspartic acids (Fig 2.9):

Fig 2.9 Charges of Asp and Glu residues

Many proteins are bound by inorganic ionic species such as metal ions Such ions can play important roles in the mechanism of a protein An example of this is the enzyme

substrate, polarizing it and stabilizing the transition state in the reaction of xylose-xylulose conversion[58]

One might expects that, a positively charged lysine or arginine residue can form a strong interaction with a negatively charged asp or glu group In proteins, this interaction

is referred to as a salt bridge In real situations, salt bridges are relatively rare in proteins

and they are normally found on the surface as opposed to internally An exception is the case of the asp-his-ser triad of serine proteases (a classic example of the structural basis

of enzyme activity) In aqueous solution, salt bridges on protein surface are highly solvated, so that the free energy of solvation of two separated ions is roughly equal to the free energy of formation of their unsolved ion pairs Electrostatic interactions therefore contribute little stability towards a protein’s native structure

In summary, electrostatic interaction, such as ion pair and salt bridge in protein, are

strong but do not greatly stabilize proteins [59] This account for the observation that

although ~75% of charged residues occur in ion pairs, very few ion pairs are buried (unsolvated) and that ion pairs that are exposed to the aqueous solvent tend to be but

poorly conserved among homologous proteins [60]

2.2.2 Van der Waals interaction

Atoms are attracted to each other by weak dispersion forces (also called London forces), linked to small oscillating dipoles created by electron cloud fluctuation Atoms are also repelled to each other due to the result of the electron-electron repulsion that occurs as two clouds of electrons begin to overlap Therefore, there are both attractive and repulsive components in van der Waals forces between groups of atom Fortuitously, attractive forces act over a longer distance (becoming stronger linearly as 1/r to the sixth power

increases), while repulsion forces only act over a short distance (becoming more repulsive linearly as 1/r to the twelfth power increases.) This observation is expressed in the Lennard-Jones potential (Eq.2.6), which calculates the energy of interaction as a combination of attractive and repulsive forces:

12 6

r

B r

A

The factors A and B depend on the nature of the pair of atoms interacting (in particular their polarizability) which can be determined experimentally for each atom or functional group The distance between two atoms or atom groups at which E = 0 defines the van der Waals radii for those two groups The van der Waals radius of an atom is an expression of the positive (repulsive) branch of the curve, which is very steep due to the inverse 12th power dependency

Since atoms tend to interact with one another at a distance that gives the most negative energy of interaction, Rm (the distance that corresponds to the minimum energy of interaction) is often somewhat longer that the van der Waals (vdW) radius The large reciprocal exponents mean that the force falls off rapidly with distance, effectively zero above 5 or 6 angstroms, as shown in Fig 2.10

Comparing to ionic force and hydrogen bonds, van der Waals interactions are normally much weaker and it was once been neglected [61] However, there are two points we need

to be aware of One is that the long-range electrostatic interaction is considerably been diminished because of the dielectric strength of aqueous media and the effects of solvation

-Eq.2.6

In contrast, protein crystal structures reveal that the interior are packed at the same density

as solids, implying that a high number of close contacts exist in the folded protein The total van der Waals interactions of a protein molecule could therefore amount to hundreds

of kcal/mol

Thus the collections of the huge number of individual atom-atom van der Waals interactions that occur in large protein molecules make them a significant role in maintaining protein structures

Van der Waals energies are cited as part of forces stabilizing the conformation of protein Very often another major non-bonded force comes from hydrogen bond

Fig 2.10 Van der Waals Interaction