Bioinformatics 2 lecture 3

1
Bioinformatics 2 -- lecture 3

• Energy and Energy minimization.
• Building a small molecule.

2
What is energy?

• A measure of how unstable a system is
• Energy only makes sense in a relative sense, i.e.
  when comparing two states.A lt--gt B
• A system is one or more molecules that do not
  interact with the outside world. Energy is
  conserved within a system.
• Energy comes in interchangeable two forms,
  potential and kinetic.

3
What is energy minimization?

• Energy minimization is a molecular simulation the
  leads the system to a lower potential energy.
• Kinetic energy is ignored.
• This is similar to the problem of finding the
  parameters that minimize a function, but there
  are generally many parameters, thus no optimal
  solution is possible.
• Energy minimization is heuristic (like almost all
  algorithms in molecular modeling!)

4
How is the energy of a molecular model calculated?

• Energy is a function of the parameters of the
  system (1) The coordinates of the atoms. (2)
  Their names. (3) Their numbers.

The names and numbers tell the program what
type of element the atoms are, how they are
supposed to be connected, and what hybridization
state they should have.
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Molecular mechanics energy
An energy function is a sum over a set of simple
functions. This sum is the so-called energy of
the system.
E f(a1,a2) f(a1,a3) f(a2,a3) f(a1,a2,a3)
etc.
Each simple energy function (f) may have 2,3 or
more atoms as parameters coordinates, names and
numbers. Each function uses stored information
about each atom name to choose constants within
each function. Together the entire set of
functions and constants is called a force field.
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Molecular mechanics

• A molecular mechanics energy function includes
  the following components (and others)
• bond lengths
• bond angles
• torsion angles
• non-bonded collisions

See Orengo p. 129
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constraint/restraint
restraint a function that approaches a minimum
as the parameters approach ideal values. For
example, the distance A-B is restrained to 3.8Å
using the restraint E(A,B) (DAB -
3.8)2 constraint a function that reduces the
number of variable parameters in the system. For
example, atoms A,B,C and D are constrained to be
in the same plane.
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Harmonic potentials and Morse potentials
Harmonic and Morse potentials are restraint
functions.
Restraint forces are applied to move the atoms to
their ideal distances/angles.
Harmonic potential
where xij is the distance between i and j, and T
is the ideal distance between i and j.
0 1 2 3 4 5
Å
9
Newton's Second Law
F ma Force equals mass times acceleration. Force
units Newton kg m/s2 Atomic scale forces
dalton Å/ps2
Forces motion along the vector joining the atoms,
apart if the sign is positive, together if
negative.
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Force is the derivative of energy with respect to
distance.
Fij uijdE/dDij Force is a vector between the
two atoms. The magnitude of the force is the
slope of the energy function at Dij the distance
between atoms i and j.
The force between 2 atoms is applied to each atom
equally (equal and opposite)
j
uij
i
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van der Waals (VDW) energy force
E
x
F
x
r r1 r2 where r1 and r2 are the VDW radii
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Typical VDW radii
Oxygen 1.52Å Nitrogen 1.55Å Carbon
1.70Å Hydrogen 1.2Å Sulfur 1.80Å
For example, an oxygen and nitrogen have zero VDW
energyat a distance of 3.07Å.
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Summing forces summing vectors
Two atoms attract or repel, causing motion along
the vector joining the atoms. (red) The force on
the center atom is the sum of the three vectors
(blue). The actual amount the atom moves depends
on its mass (m), and the time step (t).
In a dynamics simulation, each atom is moving
with a constantly-changing velocity (
acceleration x time) v a t (F/m)t New
position x v t
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ExerciseCalculate the VDW force vector
Space between black lines1.0Å rblackrbluerredr
green1.0, ?1.0 daltonÅ/fs2, t1 fs
Calculate the forces on the center atom using the
VDW equation.
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Bond angles are 1-3 distance restraints
If used in conjunction with bond restraints, then
bond angles can be restrained by restraining the
1-3 distance (1-3 refers to first and third atoms
in a covalent chain)
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Torsion angle restraints may be unnecessary.

• Torsional potentials are a function of 4 atoms in
  a chain, where the middle bond is rotatable. The
  angle-based force is converted to a Cartesian
  force and then applied.
• Energetic barriers result from 1-4 collisions,
  but these are also modeled using the VDW
  function. Why do it this way too?
• Torsion potentials have been used as a correction
  factor when the force field does not produce the
  expected result in a simulation.

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MOE tutorial building and minimizing a small
molecule
Go to Help--gtTutorials--gtGetting started... Then
click on Building a small molecule and follow
the instructions.