1.0 Introduction
The atomistic computational methods of chemistry and physics have evolved into very informative and sophisticated tools which with the advent of supercomputers now allow one to model many of the fundamental biochemical problems at the heart of biology and pharmacology. Thus, it is becoming possible to predict, quantitatively, the three-dimensional structure and dynamics of bio-macromolecules and to analyze their interactions with other molecules at appropriate levels of computational resolution ranging from electronic, to molecular, to the mesoscale cellular level. This is making it possible to address the complex issues of biological systems in terms of chemical/molecular level atomistic descriptions that provide quantitative simulations of the fundamental processes. Such atomistic computer simulations allow one to obtain static and dynamic molecular models for complex bio-systems that describe the properties of the real physical system or process in terms of concepts that emphasize the essence of the phenomena. The atomistic description of the interactions in atoms and molecules can be used to describe how the precise shape and exposed surface of the protein determines its function, which is critical to a drug design. This atomic level description of the dynamical structure of proteins should be valuable for understanding:
 active site reaction chemistry in enzymes,
 binding energetics and rates of small molecules or ligands (eg, retinal to the signal receptor protein rhodopsin) ,
 binding of antibody to the antigens,

 conformational changes in proteins and how these modify function (relevant to neurological diseases),
 changes in binding of proteins and other ligands to DNA (relevant to expression),
 correlation of structure and function of proteins and nucleic acids used in drug design industry and
 multitudes of problems at the heart of biology.
The heirarchical strategy for computational modeling of biochemical systems is illustrated in the previous chapter. In this chapter we describe the first two levels of simulations and the attempts to coarsen the simulations to the next level in the heirarchy namely the mesoscale. At the foundation of all models lies quantum mechanics
(QM) which solves the Schrodinger equation to describe the electronic wavefunctions and properties. The complexity of QM limits the applications to systems with up to 10 to
200 atoms (depending on the accuracy), leading to distance scales of less than 20 Å and time scales of femtoseconds. The quantum mechanical forces are then translated to a set of parameters describing the bonded and non-bonded forces in a molecule. This set of parameters is known as forcefield (FF). Newtonian equations are solved with atoms as basic entities using the forces calculated with the FF, a process referred to as molecular dynamics (MD). With MD one can now consider systems with up to 1 million atoms, allowing practical simulations on systems as large as small viruses (say 300 Å and 10 nanoseconds). The fundamental unit of MD is atoms (not the electrons of QM), allowing us to interpret the chemistry of the systems. A set of parameters that describe forces between assemblies of atoms (collection of atoms or domains of a protein) of a molecule can be derived from the output of MD simulations. These parameters are further used in mesoscale dynamics.
However, for larger systems (the virus or the study of bacterial chemotaxis), it is useful to use a coarser level of description where collections of atoms are described as a single entity. Here the interactions between the molecules would be derived from the atomic level calculations (MD), but the assembly would be the fundamental unit rather than atoms. In this chapter we have given a flavor of some of the initial attempts to do
2

coarse grain dynamics. We describe the fundamental theory involved in atomic level simulations starting from quantum mechanics and quote the relevant applications in the field of computational molecular biology.
2.0 Molecular Dynamics (MD)
QM methods solve the Schrodinger equation to obtain the electronic wavefunction that describes all properties of a system (atom, molecule or solid). There is a hierarchy of levels of approximation involved in QM, depending on the size of the system and the accuracy required. The most accurate (Schaefer 1984, Parr 1989) are ab initio QM methods (known by names like Hartree-Fock, Density Functional Theory,
Configuration Interaction, MP2, etc.). For many problems semi-empirical methods
(Pople 1970) (based partly on comparison to experiment) are useful (these are known by names such as MNDO, Extended Huckel, AM1, etc.). QM methods are essential for describing systems where the natures of the bonds change, for example, chemical reactions, excited states of molecules, and electron transfer.
For describing the structures and dynamics of such large molecules as proteins and DNA, the nature of the bonds between the atoms is relatively insensitive to the environment. It is mainly the packing and conformation that is significant. Thus, it is accurate to replace the electrons of QM with the FF. Here the choice of parameters in the
FF is critical. They should give a description close to QM; however, it is usually difficult to include polarization, charge transfer, and changes in bond order into the FF. As a result direct theoretical descriptions of chemical reactions have been the domain of QM.
2.1 The Force Field
The choice of the FF is essential to the accurate predictions of the properties for a system. For biomolecules the FF can be described in terms of two types of interaction energies
E tot

E nonbond

E valence
(1.1) where E valence
describes interactions involving covalent bonds and E nonbond describes the nonbonded interactions.
3

The nonbond energy is separated into electrostatic (Coulomb) and van der Waals
(vdW) components. Each atom has associated with it an atomic charge, q
I
, leading to an electrostatic energy of the form
E elec

I ,

J q
I

R q
IJ
J (1.2)
Here q
I is the charge on atom I, R
IJ
is the distance between atoms I and J, and

accounts for the units and dielectric constant of the medium. The two most common forms for the vdW energy are Lennard-Jones 6-12
E
LJ

AR
IJ

12 
BR
IJ

6 (1.3) and Buckingham exponential-6
E exp 6

Ae

CR
IJ

BR
IJ

6 (1.4) where the A , B , and C parameters are usually defined by comparing to experimental data or to accurate QM calculations on small molecules. More recently, Morse functions
E
IJ
= A {[e -C(R-Ri) -1] 2 -1} (1.5) are becoming popular (Brameld 1997, Gerdy 1995).
Because the nonbond terms require calculation for all pairs of atoms (scaling as the square of the number of atoms), it is the bottleneck for simulations of very large systems. To reduce these costs it is common to ignore interactions longer than some cutoff radius (using a spline function to smoothen the potential at the cutoff radius).
More recently fast multipole techniques [the cell multipole method (CMM); Ding 1992;
Lim 1997, Figuerido 1997] have been used to obtain accurate nonbond energies without cutoffs but scaling linearly with the size of the system. These fast and accurate methods are being used in MD simulations for large-scale biological systems (Vaidehi 1997). For biological systems hydrogen bonding (e.g., between the amide hydrogen and the carbonyl oxygen) is particularly important in determining structure and energetics. This is included in (1.2) through the electrostatics (van Gunsteren 1987; Jorgensen 1988; Hagler
1974, Hermans 1984) and sometimes a modification of the vdW potential (Mommany
1975; Levitt 1983; Hodes 1979; Brooks 1983; Weiner 1986; Mayo 1990; Cornell 1995).
The valence energy is usually described as
4

E valence

E bond

E angle

E torsion

E inversion
(1.6) where bond describes the interaction between two bonded atoms, angle describes the interaction between two bonds sharing a common atom, torsion describes the interaction between a bond IJ and a bond KL connected through a bond JK, and inversion is used to describe non-planar distortions at atoms with three bonds.
Since the covalent bonds are expected to remain near equilibrium the bond stretching and angle bending are taken as harmonic:
E bond

1
2
K
IJ
( R
IJ

R
0
)
2 (1.7)
E angle

1
2
K
IJK
(
  
0
)
2
or (1/2) C (cos

- cos

0
)
2
(1.8)
Here R
0 is the equilibrium bond distance, K
IJ is the bond force constant,

is the bond angle between bonds IJ and JK,

0
is the equilibrium bond angle, and K
IJK
(or C) is the angle force constant.
The torsion energy is described in terms of the dihedral angle,

, between bonds
IJ and KL along bond JK. This is periodic and can be written as
E torsion

1
K

, n
( 1
 d cos( n

))
2 n
6 

1 where K 
,n
is the torsion energy barrier for periodicity n,
(1.9) d
 
1 describes whether the torsion angle (

=0) is a minimum (d=+1) or a maximum. Torsion potentials are essential in describing the dependence of the energies on conformation.
Finally an inversion term (improper torsion) is needed to describe the distortions from planarity of molecules(e.g., in aromatic amines or amides). For cases where the equilibrium geometry is planar (e.g., amine N or C) we use
E inversion

K inv
( 1
 cos

) (1.10)
For cases where the equilibrium geometry is non-planar (e.g., amine) we use
E inversion

1
2
C
I
(cos
  cos

0
)
2 where
C

I
K
(sin inv

0
)
2
(1.11)
(1.12)
5

2.1.1 Standard Force Fields
Given the functional forms, the FF is defined by the particular choices for force constants and equilibrium geometries. It is these parameters which determine the quality and usefulness of the FF. To be most useful we want a simple FF with very few parameters and simple rules for predicting the parameters. The simplest such generic FF is Dreiding (Mayo 1990), where there is only one bond force constant (bond order times
700 kcal/mol Å) and one angle force constant (100 kcal/mol rad). Equilibrium bond distances are based on atomic radii and bond angles are derived from simple rules.
Dreiding is useful for the main group elements (the B, C, N, O, and F columns) most prevalent in biology. A generic FF that also treats transition metals is the universal force field (UFF; Rappe 1992), which also allows the force constants to depend on the specific atoms involved. Such generic FF are most useful for systems with unusual arrangements of atoms or for new molecules for which there is no experimental or QM data.
Unfortunately, for applications where the exact molecular structure plays a critical role, the generic FFs may not be sufficiently accurate. Hence, there is a need for a generic FF which incorporates just enough specificity for accurate simulations of bio-molecules while allowing the flexibility to model all other organic molecules.
At the other extreme are highly specialized FF developed to predict accurately not just the geometry and energy but also the vibrational spectra of the molecule. This requires complex cross-terms coupling different bonds and angles. A general procedure for developing such spectroscopic FF from QM is the Biased Hessian method, which has been used for many systems (Dasgupta 1996). Usually the spectroscopic quality FF are useful only for very limited classes of molecules.
In between these two extremes are FF developed for a specific class of molecules.
The most commonly used FF for proteins and DNA are AMBER (Weiner 1984, Cornell
1995) and CHARMM (Mackerell 1995), which are parameterized to describe the naturally occurring amino acids and nucleic acids. However, unusual ligands such as drug molecules, cofactors, substrates, or their modifications are difficult to incorporate as are non-natural amino acids or bases. Useful here are FF developed for organic systems
6

[OPLS (Pranata 1991) and MM3 (and Dreiding)] which can do non-natural amino acids or bases along with most molecules which bind to bio systems.
2.1.2 Effect of Solvents
The role of solvents (particularly water) is critical in biological simulations, since the secondary and tertiary structures of proteins are determined by the nature of the solvent. Several levels of simulation have been used.
The earliest studies ignored the solvent entirely, usually replacing the effect of solvent polarization by using a dielectric constant larger than 1 (often distance dependent). At this level of approximation it is important to include counter ions to represent the effects of solvent on charged groups. Such simulations were useful in understanding gross properties of systems.
The most accurate MD treatments include an explicit description of the water using a FF for water adjusted to describe the bulk properties of water (Jorgensen 1983;
Levitt 1997; Rahman 1971; Berendsen 1981, 1987). Although accurate, such calculations may require very long time scales in order to allow the hydrogen bonding network in the water to equilibrate as the biomolecule undergoes dynamic motions. In addition the number solvent atoms for an accurate treatment of solvent effects may be 10 times that of the biomolecule.
An excellent compromise for attaining most of the accuracy of explicit water while eliminating the atoms and time scale of the solvent is the dielectric continuum model. Here the electrostatic field of the protein is allowed to polarize the (continuum) solvent, which then acts back on the protein, leading to the Poisson-Boltzmann equation
(Honig 1993). Recent developments have led to computationally efficient techniques that
(Tannor 1994) can now account accurately for effects of solvation on the forces
(geometry) and free energy.
2.2 Molecular Dynamics Methods
2.2.1 The fundamental equations
Solving the classical Newton’s equations of motion for the N atoms in a molecular system described by the FF, leads to a trajectory describing the 3N coordinates and 3N velocities as a function of time. For many applications it is this dynamical trajectory which contains the valuable information about the system. Thus, MD methods
7

have been useful for the exploration of structure-activity relationships in biological molecules.
However, some applications require the ensemble of conformations near equilibrium, which can then be used to calculate very accurate properties. Usually we consider that the collection of conformations sampled in the dynamics can be used as the ensemble for calculating properties. This assumes that the barriers between different relevant structures are sufficiently small that they can be sampled in the time scale of the simulations.
The steps in MD simulations are:
1.
Calculation of potential energy and its derivative (the force) f i
on every i th atom in a molecule as described in section (2.1).
2.
Using the forces f i calculated from the FF, Newton’s equation of motion is solved for
(1.13)
 m i
..
x i f i
Here m i
is the mass of atom i.
.
3.
The velocities of each atom are calculated by integrating the acceleration and then the velocities are integrated to determine the new atomic positions.
This integration is usually performed using Verlet leapfrog algorithm. The acceleration from equation (1.1) is given by
..
x i
 f i
/ m i
 
E tot
/ m i
(1.14) where

E tot is the gradient of the potential energy. Thus, for a timestep

, the acceleration can be written as
..
x n

.
x n

1
2

.
x n

1
2

(1.15) leading to
.
x n

1
2

.
x n

1
2

 m i

E n
(1.16)
8

Integrating (1.16) then leads to the coordinates at the next time step, x n

1
 x n
 
.
x n

1
2
(1.17)
Equations (1.16) and (1.17) are the fundamental equations for dynamics (Verlet velocity leapfrog). To start the calculation we usually have an initial set of coordinates, but not the initial velocities. Thus, we assign the velocities randomly in such a way as to describe the Maxwell-Boltzmann statistics. The first segment of MD is used to equilibrate the system, removing any bias from the initial conditions.
The time step of integration,

, is determined by the highest vibrational frequency of the system, usually 1 to 2 fs for good accuracy.
The trajectory of the molecular systems may require time scales ranging from picoseconds to hundreds of nanoseconds, depending on the application and the size of the system. Thus, a computationally efficient MD algorithm must allow
1.
fast and accurate calculation for atomic forces and
2.
use the longest possible time steps compatible with accurate Verlet integration so as to simulate the longest time scale molecular motions.
2.2.2 NPT and NVT dynamics
Newton’s equations of motion (1.13) describe a closed system. Thus the total energy (kinetic energy plus potential energy) of the system does not change; the system is adiabatic (it is said to be conservative). Assuming that the volume is held constant this simulation generates the microcanonical ensemble of statistical mechanics (denoted NVE for constant number of molecules, volume, and energy). However, most systems are best described in terms of a temperature bath and a pressure bath, leading to a Gibbs ensemble
(Allen 1987). For the trajectory to lead to a Gibbs ensemble one must allow the internal temperature of the molecule under study to fluctuate in the same way that it would in contact with a temperature bath and must allow the volume to fluctuate in the same way as it would in contact with a pressure bath.
Several methods (Woodcock 1971; Nose 1984; Hoover 1985; Vaidehi 1996) are used to control the temperature of a MD simulation. The most rigorous method (Nose
1984) introduces into the equations of motion a new dynamical degree of freedom

, associated with energy transfer to the temperature bath (friction). If the volume is kept
9

fixed the Nose dynamics generates a Helmholtz canonical ensemble, giving rise to the correct partition function for an NVT system. These partition functions can be used to calculate macroscopic properties of the system. For most biochemical problems
[calculating binding energy and other molecular properties to be used in deriving quantitative structure function relationships (QSAR) useful for predicting enzyme activity] we recommend using NVT canonical dynamics.
Some interesting biological applications involve the structural deformations of proteins under external stress (Floriano 1998). In this case we use periodic boundary conditions (described below), placing the molecule and solvent in a periodic cell that can be acted upon by external stresses. To examine the structural deformations of proteins under pressure or under external stress requires that the MD allow the internal pressure of a molecule to fluctuate in the same way as for a system in a constant pressure environment (Parrinello 1981). The modified Newton equations lead to NPT dynamics
(constant number of particles, pressure, and temperature)
2.2.3 Constrained internal Coordinates
The short time scale of 1fs for MD is required to describe the very rapid oscillations involved in bond stretch and angle bend motions. However, for proteins and nucleic acids it is the low frequency motions involved in conformational changes that are of most interest. Several algorithms (Ryckaert 1977; van Gunsteren 1990; Abagayan
1989; Jain 1993; Rice 1994) have been developed for fixing the bonds and angles in order to focus on the conformational motions. Such constrained dynamics algorithms lead to coupled equations of motion
M (

)

..

C (

,

.
)

T (

) for the P degrees of freedom (torsions). Here
(1.18)
   
is the angular acceleration,

M is the P

P mass matrix (moment of inertia tensor), which depends on the internal coordinates

.

T is the vector (diverging P ) of general forces (tensor) on the atoms where

C is the Coriolis force.
10

At each time step we know

, M, T, C and we must be solve the matrix equation (1.18) to obtain the acceleration,
   
M

1
( T

C ) (1.19)
Integration of
  
gives the velocity
.

and further integration leads to the torsion or the dihedral angles

from which the coordinates can be obtained. The problem is that solving equation (1.19) requires inverting the P

P dense mass matrix M at every time step. For a system with say 10,000 atoms there might be 1500 torsional degrees of freedom, making solving (1.19) impractical for every time step (the time to invert M scales as P
3
). Recently we developed the Newton-Euler Inverse Mass Operator
(NEIMO) method (Jain 1993; Mathiowetz 1994; Vaidehi 1996), which solves (1.19) at a computational cost proportional to P. The NEIMO method considers the molecule to be a collection of rigid “clusters” connected by flexible “hinges”. Rigid cluster can be a single atom, a group of atoms (a peptide bond) or a secondary structure (a helix or even an entire domain of a protein). Such rigid clusters allow much larger the time steps.
With fine-grain all-torsion NEIMO dynamics, we normally treat double bonds, terminal single bonds, and rings (benzene) as rigid bodies. In Hierarchical or H-NEIMO dynamics (Vaidehi 1998) we allow higher levels of coarseness, keeping rigid during the dynamics various segments or parts of a domain of a protein. This allows larger time steps as illustrated in Table 1.
Table 1. Time steps for hierarchical modeling of two large proteins.
Protein system MD method Degrees freedom
1062
Time-step
(fs)
1.0
Protein A (354 atoms) from staphylococcus aureus Newtonian
NEIMO( fine grain)
NEIMO(coarse grain)
PGK enzyme (4175 atoms)
Newtonian (all atom model)
219
92
12525
10.0
20.0
1.0
11

NEIMO(fine grain-all torsions) 2210
NEIMO(coarse grain) 80
5.0
10.0
Using hierarchical NEIMO simulations on the glycolytic enzyme phosphoglycerate kinase (PGK), we were able to follow the long time-scale domain motions in PGK responsible for its function. PGK catalyzes a key phosphorylation step in the glycolytic pathway. Under physiological conditions, PGK facilitates the phosphoryl transfer from ADP to ATP. PGK consists of two major domains (1400 atoms in each domain) denoted as the C-domain and the N-domain. In most crystal structures
(McPhillips 1996; Bernstein 1997), the substrates are found bound to the opposite domains at a distance of ~ 13A o
. Thus, Blake (1997) proposed a hinge bending mechanism by which the protein brings the substrates together to react. Using H-NEIMO
(see Fig. 1.1) we found low frequency domain motions that take the open structure examined by yeast PGK (McPhillips 1996) to the closed structure examined by T. brucei
PGK (Bernstein 1997). The NEIMO dynamics suggest that PGK undergoes long timescale motions which take the substrate binding sites together, then apart, then together again. The extent of the domain motions or the rate of the domain motions depend on the nature of the substrates bound.
12

Figure 1.1.
The H-NEIMO model used for the two domain phosphoglycerate kinase
MD simulations. Here the parts of two domains that bind the substrates (shown in yellow) are kept rigid while the loops (shown in green) are allowed to have full torsional freedom. H-NEIMO MD simulations exhibit large-scale domain motions that bring the substrates close together and then apart.
2.2.4 MPSim
In order to allow long time simulations on very large systems (up to a million atoms), we developed the massively parallel simulation (MPSim) program (Lim 1997) to operate efficiently with massively parallel computers. MPSim includes important algorithm developments such as
1.
CMM (for calculating the long-range nonbond interactions),
2.
NEIMO,
3.
Poisson-Boltzmann solvent (Tannor 1994)
13

compatible with massively parallel high performance computers (SGI-Origin-2000, IBM-
SP2, Cray T3D/T3E, HP/Convex-Exemplar, and Intel Paragon). MPSim has been applied to understand the action of drugs on the human rhinovirus (Vaidehi 1996), summarized in Section 3.2. It is also being used for material science problems (Miklis
1997).
2.2.5 Periodic Boundary Conditions
In order to control the pressure on a system and to include explicit solvents without introducing complications of free surfaces, it is convenient to place the molecule in a large box (much larger than the molecule) and then reproduce it periodically to fill space. Also some systems such as DNA are particularly usefully described as periodic and repeating in one direction. The computational box containing the molecular system is surrounded by an infinite number of copies of itself (see Fig. 1.2).
14

Figure 1.2.
Molecules in a periodic system. A.
Two-dimensional view of a tripeptide in a periodic box. B.
The simulation unit cell is the outlined box at the center. It is surrounded by infinite array of equivalent boxes so that there is no free surface.
15

Molecules are allowed to move from box to box but the number in the unit cell is always constant.
Because periodic systems involve an infinite number of atoms, some care must be taken in calculating the long-range forces. Otherwise, singularities or wild oscillations can occur. The most common accurate method is Ewald summation (Ewald 1921; Leeuw
1980; Heyes 1981; Allen 1987; Karasawa 1989; Chen 1997), which considers point charges smeared over a region of finite size 1/

, chosen to converge rapidly (in real space) and then Fourier transforms the difference between the smeared charges and the point charges to obtain rapid convergence (in the reciprocal space sums). Karasawa
(1989) showed how to choose the optimum

to minimize computational cost for a given level of accuracy. This leads to costs scaling as N
3/2
for systems with N atoms per unit cell. Ding (1992b) showed how to use the reduced CMM method to achieve linear scaling.
2.2.6 Monte Carlo Methods
For many systems the barriers between low lying structures may be too large for
MD to sample all structures. In such cases we often use Monte Carlo or statistical sampling techniques, using a random search algorithm such as Monte Carlo metropolis .
With a sufficiently large number of samples the occurrence of each conformation is proportional to the Boltzmann factor, leading to a canonical distribution. The steps involved in the Monte Carlo simulation procedure are:
1.
Starting from a given molecular conformation, a new conformation is generated by random displacement of one or more atoms. The random displacements should be such that in the limit of a large number of successive displacements the available conformation space is uniformly sampled.
2.
The newly generated conformation is accepted or rejected on the basis of the change in the potential energy of the current step compared to the previous step. The new conformation is accepted if the change in potential energy

V = V (present step) - V
(previous step)

0, or if

V > 0 when the Boltzmann factor is greater than a random number R .
Upon acceptance the new conformation is counted and used as a starting point for the generation of the next random displacement. If the criteria are not met, then the new conformation is rejected and the previous conformation is counted again as a starting
16

point for another random displacement. This method thus generates a Boltzmann ensemble of conformations. There is a variety (Allen 1987) of Monte Carlo methods available in literature. Such Monte Carlo methods have been used as a fast conformational search tool in protein folding (Sternberg 1996). A recent advance,
Continuous Conformation Boltzmann Biased Direct Monte Carlo (Sadanobu 1997) has been used to obtain all possible foldings of proteins with up to 100 residues (Debe 1998).
Such methods show considerable promise for solving the protein folding problem
(predicting the tertiary structure from primary sequence).
3.0 Application to Real Biological Problems
In this section we summarize some recent applications of modern methods of quantum chemistry and molecular mechanics and dynamics to interesting problems in structural biology.
3.1 Study of enzyme reaction mechanisms :
The reaction mechanisms for many enzymes have been studied using a combination of QM and MD methods (McCammon 1987, Cunnigham 1997). Examples include the hydrolysis of a peptide bond by serine proteases (Warshel 1991), and hydrolysis by the metalloenzymes staphylococcal nuclease of both DNA and RNA, etc.
We describe here a recent QM and MD study for the elucidation of the mechanism of family 18 and family 19 chitinases (Brameld 1998). This application relies heavily on a combination of QM and MM/MD methods and demonstrates the feasibility of solving difficult problems using modern computational methods.
Chitin (see Fig. 1.3) is a

(1,4)-linked Nacetyl-glucosamine (GlcNAc) polysaccharide, which is a major structural component of fungal cell walls and the exoskeletons of invertebrates, including insects and crustaceans. This linear polymer may be degraded through the enzymatic hydrolysis action of chitinases. Chitinases have been found in a wide-range of organisms including bacteria (Watanabe 1990; Perrakis 1994), plants (Collinge 1993), fungi (Bartinicki-Garcia 1968), insects (Kramer 1985), and crustaceans (Koga 1987). For organisms that utilize the structural properties of chitin, chitinases are critical for the normal life cycle functions of molting and cell division
(Fukamizo 1985; Kuranda 1991). Because chitin is not found in vertebrates, inhibition of
17

chitinases is a promising strategy for treatment of fungal infections and human parasitosis
(Robertus 1995).
Based on amino acid sequence, the glycosyl hydrolases have been classified into
45 families. Using this classification method, the chitinases form families 18 and 19, which are unrelated, differing both in structure and mechanism. Sequence analysis shows little homology between these classes of chitinases. Family 19 chitinases (found in plants) share the bi-lobal

+

folding motif of lysozyme, which forms a well-defined substrate binding cleft between the lobes. In contrast, family 18 chitinases share two short sequence motifs, which form the catalytic (

)8-barrel active site. Family 18 chitinases with diverse sequences have been isolated from a wide range of eukaryotes and prokaryotes. The hydrolysis site of chitin is shown in Fig. 1.3. hydrolysis site
RO
HO
OH
O
HN
O
O
HO
H
3
C
OH
O
HN
O
O
HO
H
3
C
OH
O
HN
O
O
HO
H
3
C
OH
O
HN
H
3
C
O
O
R
Figure 1.3.
An arrow marks the hydrolysis site of chitin, the

(1,4)N -acetylglucosamine (GlcNAc) polysaccharide substrate of chitinases.
Brameld investigated the hydrolysis mechanisms of the chitinases by examining the reactivity of the chitin substrate alone and in the presence of the enzyme. This was done using ab initio quantum mechanical calculations on three possible reaction intermediates, for the enzymatic hydrolysis of chitin. It was found that anchimeric assistance from the neighboring N -acetyl group of the chitin is critical in stabilizing the resulting oxazoline ion intermediate.
MD simulations of the complete enzyme with bound substrate offer further insights to the differing mechanisms of family 18 and 19 chitinases. All MD simulations were carried out using the MSC-PolyGraf program using the Dreiding FF (Mayo 1990).
Averaged charge equilibrium (QEq; Rappe 1991) charges were used for all GlcNAc residues.
A standard Coulomb potential was used without a distance dependent dielectric
18

constant and all nonbond interactions were considered explicitly. A nonbond cutoff of
9.5 Å was used during MD simulations and extended to 13.5 Å for single point energy calculations. Solvation energies were estimated using the continuum solvent model in the Delphi program (Tannor 1994). Two possible intermediates, namely the oxazoline ion and the oxocarbenium ion were proposed. The oxazoline ion intermediate forms as a result of substrate distortion (see Fig. 1.4) induced within the active site of family 18 chitinases. This substrate distortion was observed in the MD simulations and is necessary to design inhibitors for the family 18 chitinase. Yet surprisingly, the family 19 chitinases do not utilize an oxazoline ion intermediate and undergo a considerable change in enzyme conformation to stabilize the resulting oxocarbenium ion intermediate.
Figure 1.4.
The minimum energy structure for the -1–boat hexaNAG conformation. A boat geometry for GlcNAc residue -1 and the twist between residues -1 and +1 strains the linking glycosidic bond. This distortion observed in the simulations should be included when designing new inhibitors.
The oxazoline transition state serves as a target for the rational design of more potent glycosidase inhibitors specific to family 18 chitinases. Simple analogs of allosamidin which incorporate the key features of a delocalized positive charge while
19

maintaining a chair-like sugar conformation may prove to be synthetically more accessible than allosamidin. Such analogs could lead to a new generation of chitinase transition state inhibitors.
3.2 A model for drug action on Rhinovirus-1A and Rhinovirus-14
Human rhinovirus (HRV) belongs to the picornavirus family. It has over 100 serotypes, providing a challenge to drug design. The serotypes of HRV are classified into two groups. The major receptor group (including HRV-14 and HRV-16) binds to the intercellular adhesion molecule 1(ICAM-1) receptors. The minor receptor group
(including HRV-1A) binds to low density lipoprotein-type receptor molecules. The protein capsid of HRV consists of an icosahedral shell with 60 copies of the 4 viral proteins [VP1, VP2, VP3, and VP4 (see Fig. 1.5)], totaling to 480,000 atoms (
 300 Å in diameter). Fig. 1.5 shows the icosahedral shell, its elements such as the pentamers that make up the icosahedron, the asymmetric unit that makes up the pentamer and the basic four viral proteins of the asymmetric unit. A single strand RNA is enclosed in the protein shell. The sequence of events involved in the endocytosis is not clear yet but circumstantial evidence suggests that the RNA is released through the pentamer in the virus coat (Rueckert 1991).
There are several known isoxazole derived drugs for HRV-1A and HRV-14
(Couch 1990). It is known that binding of these drugs to HRV-14 prevents binding of the virus to the ICAM-1 receptors. However, for HRV-1A binding of the drugs does not block receptor attachment; rather it prevents uncoating of the virus. One speculation is that this binding leads to stiffening of the viral capsid. Based on this speculation and that the RNA is released through the pentamer channel, Vaidehi (1997) proposed the
Pentamer Channel Stiffening Model (PCSM): drug action on HRV-1A constricts or stiffens the pentamer channel sufficiently that the RNA cannot exit, thus preventing uncoating. Using MPSim (see Section 2.2.4) on the KSR-64 processor parallel computer, we showed a strong correlation (see Fig. 1.6) of drug effectiveness (Minimum
Inhibitory Concentration) with the strain energy increase calculated for various drugs.
Here we defined the strain energy as the energy required to expand the pentamer channel to 25 Å. The strain energy was calculated using MPSim with the Amber FF for the viral
20

Figure 1.5.
Structure of Rhinovirus. The complete icosahedral viral capsid (
 Å diameter) is shown it the bottom right. The four viral proteins making up the asymmetric unit of the virus are shown in the top half. The five asymmetric units forming the
21

pentamer is shown in the bottom left. Twelve such pentamers form the complete viral capsid. proteins and the Dreiding FF for the drugs. Fig. 1.6 shows that all effective drugs cause an increase in the strain energy required to open up the pentamer channel. The best drug
WIN56291 (MIC=0.1

M) shows the sharpest increase in the strain energy compared to the native HRV-1A or an ineffective drug WIN54954 (MIC=2.5

M). This suggests that the PCSM can be used to predict the effectivity of a drug prior to synthesis and testing.
Three drugs, MSC1, MSC2 and MSC3, were tested in this way.
8000
7000
6000
5000
4000
3000
WIN56291(0.1)
R61837(0.7)
WIN56278(2.25)
WIN54954(2.5)
WIN52035(4.2)
WIN54221(IA)
SUCROSE
HRV1A
MSC1
MSC2
MSC3
WIN56291(best)
R61837(good)
WIN54221(Inactive)
2000
Native-HRV-1A
1000
0
0 5 10 12 15 18 20
-1000
Van der Waals radius of He(A)
Figure 1.6.
Pentamer Stiffness for HRV-1A with various Winthrop drugs. The MIC values for the drugs are given in parenthesis. MSC1, MSC2 and MSC3 denotes new candidates for drug molecules.
3.2 Calculation of binding energy using free energy perturbation theory.
A major application of MD simulations is for drug design, where high binding energy is expected to be a necessary condition for a good drug. MD provides microscopic level information (atomic and molecular positions, velocities, etc.) about the dynamics of macromolecules which can be averaged over time (using appropriate
22

formulae from statistical mechanics) to obtain macroscopic thermodynamic properties of system such as the free energy, temperature, and heat capacity. Thus, one assumes that the macroscopic property G obs
, can be expressed as an average over the ensemble of conformation from the MD, G obs
=

G
 ens
, and that this can be expressed as the time average of G ( r i
) , over the trajectory over a sufficiently long time interval. This is written as
G obs

G ( r i
)
 time
 t lim
  
1 t
 t
0
G ( r ) dt (1.20) where in G ( r i
) , r i
is a function of time and is generated as the trajectory of the MD simulations.
The problem is that the convergence of (1.20) is sufficiently slow that the error may be large compared to the difference in bond energies for the drugs. This problem is solved by transforming the system slowly from drug A (

=0) to drug B (

=1) and integrating the difference in G (

) along the trajectory. This is called free energy perturbation (FEP) theory. It has been demonstrated to gives a quantitative estimate of the relative free energy of binding of various drug molecules or inhibitors to its receptors.
The integration in (1.20) is replaced by finite sum over time steps. Free energy of molecular systems can be calculated (van Gunsteren 1989) using equation (1.20). The free energy difference given by
G (

)

G (
 i
)
 
RT ln
 exp(

(
V


RT
V
 i ))

 i
(1.21) provides a means of calculating the free energy difference any two states A (

) and
A (
 i
).
However, unless the states share a significant fraction of conformation space, convergence is very slow. The convergence time can be reduced by calculating the relative free energy difference between closely related states, using the thermodynamic cycle. To visualize the method (see Fig. 1.7), we consider the relative binding of two ligands L
1
and L
2
to a receptor which is a protein of DNA. The appropriate thermodynamic perturbation cycle (Zwanzig 1956) for obtaining the relative binding constant is given in Fig. 1.7.
23

Receptor + L
1
Step 1
Step 4
Complex1
Step 2
Receptor + L
2 Complex2
Step 3
Figure 1.7.
Thermodynamic cycle for the calculation of the relative free energies of binding.
The ratio of the binding constants for L
1
and L
2
can be calculated from the equation
K
L 2
K
L 1
 exp[

(

G
L 2
 
G
L 1
) / RT ]
(1.22) where R denotes the gas constant, K
1
and K
2
are the binding constants for L
1
and L
2
.
Simulation of steps 1 and 3 involve replacement of solvent molecules in the binding pocket of the protein (receptor) by the ligand and also removing the ligand from the solvent and binding it to the protein. These are very long time processes for the time scale of MD simulations. However steps 2 and 4 are possible to simulate using MD although provided the chemical composition and the structure of the ligands are very closely related as described in the example below. Hence the free energy difference for steps 1 and 2 are related to free energy difference between steps 2 and 4 by (Fig. 1.7)

G
3
 
G
1
 
G
2
 
G
4
The crucial factor in the simulations of steps 2 and 4 is in the adequacy of the sampling in the configuration space of the system. Longer MD simulations are required for accurate free energies.
This method has been extensively used in literature to study the relative binding energies of drugs and inhibitors for enzymes (van Gunsteren 1989; Marrone 1997; Plaxco
24

1994) and inhibitors. Here we describe briefly one of the recent applications (Marrone
1996) of the free energy perturbation simulations to the study of inhibition of the enzyme adenosine deaminase by 8R-coformycin and (8R)-deoxycoformycin. This study has been done by McCammon and his coworkers (Marrone 1996) at University of California, San
Diego. The inhibition of the enzyme adenosine deaminase that deaminates the base adenosine could be an effective treatment of some of the immunological disorders
(Goodchild 1982). Thus MD simulations can play a critical role in designing inhibitors for this enzyme since it gives a good model for the binding site of the inhibitors in the enzyme and also an estimate of their binding energies. The two inhibitors studied are by
Marrone and coworkers are coformycin and deoxycoformycin. These two molecules differ in the sugar moiety attached to them which is ribose in the case of coformycin and and deoxyribose for deoxycoformycin. The molecular structure of these two substrates is shown in Fig. 1.8.
OH
H
X
N
N
O
OH
N
CH
2
OH
Figure 1.8
Molecular structure of adenosine deaminase inhibitors X = H for deoxycoformycin and X=OH for coformycin.
25

It is clear that these two inhibitors differ only by only a small functional group and hence a well suited case for free energy perturbation calculations. Molecular dynamics and free energy simulations of coformycin and deoxycoformycin and their complexes with adenosine deaminase show a difference of -1.4 Kcals/mole in binding energy between deoxycoformycin and coformycin. Deoxycoformycin and coformycin differ by a hydroxyl group but the relative binding energy is small enough showing that this hydroxyl group is buried near a flexible region hydrophilic region of the enzyme conformation than being sequestered in a hydrophobic pocket. Thus detailed structural aspects of the transition state analogue has been derived in this study (Marrone 1996).
3.3 Quantitative Structure-Activity Relationships (QSAR)
Following the calculation of relative binding constants one of the most important steps in the process of drug discovery is the study of QSAR (Franke 1984; Fanelli 1998).
The aim is to explain the experimental data obtained for binding of various ligands to a receptor, at the molecular level, in terms of physicochemical properties of the ligands, and to predict/estimate similar biodata for new analogs. QSAR are equations that relate an observed experimental quantity for ligand binding for example, the Minimum
Inhibitory Concentration (MIC) measured routinely in drug industries, to the calculated molecular properties of various regions of a drug molecule. QSAR are widely used today in pharmaceutical industries to design new drugs or inhibitors. Fig.1.9 shows the major steps involved in the QSAR analysis. Today computational chemistry using the QM/MD simulations allow us to define and compute ad hoc shape and size descriptors on the different conformations assumed by drugs in biotest solutions. Together with the statistically sound experimental data measured on well-identified target receptor, these descriptors are essential elements for obtaining simple, consistent, comparable, and easily interpretable theoretical QSAR models based on ligand similarity-target receptor complementarily paradigm. The molecular properties calculated can be of a varied nature. For example, based on the structural formula of the drugs they can be broken into fragments and the properties of the subsystem or fragments that contribute to the drug activity can be calculated. Intermolecular interaction properties like polarizability or the
26

hydrophilic/hydrophobic nature of groups can be computed. Thus, QM/MD simulations are used extensively in pharmaceutical companies to derive QSAR (Fanelli 1998).
Drug Library
Computation of molecular properties (QM/MD)
Mathematical analysis and QSAR derivation
Suggest new drug
candidates
Interpretation and
testing
Figure 1.9 Principal steps involved in QSAR analysis
4.0 Summary
Atomistic simulations offer powerful tools to understand the dynamical behavior of biological systems. They can be used in a wide variety of problems in the heart of biological research enumerated in this chapter. The major thrust in this area is to be able to simulate large systems (hundreds of angstroms) for long time scales (microseconds).
Also, coarse grain simulations of proteins while retaining its atomic level force
27

description would be vital in understanding long time scale dynamical behavior. Another major thrust is the prediction of tertiary structure of proteins, which is not dealt with in this chapter.
Acknowledgements
The research ( by the authors) presented here was funded by National Science
Foundation (CHE-95-22179, GCAG CHE 95-2217, and SGER DBI-9708929) and DOE-
BCTR. The Materials Simulation Center is supported by grants from BP Chemical,
Beckman Institute, Seiko-Epson, Exxon, Owens-Corning, Asahi Chemical, Chevron
Petroleum Technology, Chevron Chemical Co., National Institute of Child Health and
Development, Chevron Research Technology, and Avery Dennison. Some calculations were carried out on the National Center for Supercomputing Applications (L. Smarr) at
U. Illinois.
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