APM 530 - Lecture 5

38 Pages·2010·0.31 MB·English

by  Jay Taylor

Non-bondedInteractions Overview
Computation of the Non-Bonded Potential
Recall that the non-bonded contribution to the potential function takes
the form
(cid:32) (cid:33)
(cid:88) Aij Bij (cid:88) qiqj
V (R) = − + + .
nb r6 r12 r
i<j ij ij i<j ij
Since each of these sums contains O(N2) terms, direct evaluation of the
non-bonded potential is impractical in systems containing thousands or
more atoms. In such cases, it is usually necessary to estimate V using
nb
faster, approximate methods.
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Non-bondedInteractions SphericalCutoffs
Cutoff Techniques
Cutoff techniques reduce the cost of the non-bonded computations to
order O(N) by ignoring non-bonded interactions between nuclei separated
by more than some maximum distance, b.
Typically, the interaction function is altered multiplicatively: f(r) is
replaced by f(r)S(r) where S(r) is zero for r > b.
The cutoffs can be applied using distances between atoms or between
group centers. Group-based methods treat atoms that belong to the
same residue or base similarly.
Cutoffs can be applied directly to the energy and to the force field.
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Non-bondedInteractions SphericalCutoffs
Truncation
Truncation methods abruptly set the interaction term equal to 0 when
r ≥ b:
(cid:26)
1 if r < b
S(r) =
0 if r ≥ b.
Advantage: easy to implement.
Disadvantage: because the truncated potential function is
discontinuous, optimization and MD routines run poorly.
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Non-bondedInteractions SphericalCutoffs
Switching
Switch functions smoothly reduce the energy to 0 over an interval [a,b].
One example is

1 if r < a

S(r) = 1+y(r)2(2y(r)−3)) if a ≤ r ≤ b
 0 if r ≥ b,
where y(r) = (r2−a2)/(b2−a2).
S(r) as defined above is continuously differentiable, so the
corresponding force field is continuous.
If a is close to b, then the derivative in [a,b] will be large.
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Non-bondedInteractions SphericalCutoffs
Shift Functions
Shift functions gradually reduce the energy over the entire interval [0,b].
One formulation used in CHARMM is:
(cid:26) [1−r/b]2 if r < b
S(r) =
0 if r ≥ b.
There is a tradeoff between smoothness and underestimation of the
potential:
Switch functions have larger derivatives but only underestimate the
energy over the region [a,b].
Shift functions underestimate the energy at all distances, but vary
more smoothly.
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Non-bondedInteractions SphericalCutoffs
Bookkeeping
Calculating the distance r for every pair of atoms in the molecule
ij
requires O(N2) steps.
To implement cutoff techniques in order O(N) calculations, each
atom is assigned a pairlist of atoms that are within some distance
b+c at the start of the calculation.
c should be taken large enough that only those atoms contained
within the pairlist are likely to come within distance b of the
associated nucleus over the course of the calculation.
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Non-bondedInteractions SphericalCutoffs
Cutoff Techniques in Practice
Truncation is almost never used because of the problems caused by
discontinuities.
Shift functions are used with van der Waals forces, which decay
rapidly over long distances (like r−6).
Because the Coulomb potential decays slowly (like r−1), long-range
electrostatic interactions are especially important in charged
molecules like DNA.
Many studies have reported that structural modeling using spherical
cutoffs of the electrostatic potential gives poor results.
Group-based methods perform especially poorly.
Accurate results may be attainable using atom-based shifts with
cutoffs of 12˚A (Norberg & Nilsson, 2000).
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Non-bondedInteractions Ewaldsummation
Overview
Ewald summation originated in crystallography as a method for
calculating slowly convergent electrostatic potentials.
We consider a group of point charges in an infinite periodic lattice.
Each point charge is screened by a smooth charge distribution of the
opposite sign. This leads to a rapidly converging potential (direct
sum).
The screening charges are compensated by a smooth distribution
that leads to slowly converging potential (indirect sum).
Because the compensatory distribution is smooth, it can be converted
to a rapidly converging Fourier series (Poisson summation).
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Non-bondedInteractions Ewaldsummation
Periodic Extension of the Charge Distribution
To apply Ewald summation to the electrostatic potential encountered in
MM, we imagine that the molecule sits in a cubic cell of side length L that
is used to form an infinite periodic lattice. We then seek to calculate
(cid:48) N
1 (cid:88) (cid:88) qiqj
V =
el
2 |x +n|
ij
n i,j=1
where
n = L(n ,n ,n ) denotes a cell-coordinate vector;
1 2 3
The (cid:48) over the first sum means that terms with i = j are omitted
when n = (0,0,0).
Ewald summation can also be carried out over other periodic lattices.
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Non-bondedInteractions Ewaldsummation
The Screening and Canceling Distributions
The charge distribution is decomposed as
(cid:88) (cid:88) (cid:0) (cid:1) (cid:88)
q δ = q δ −(ν +x ) + q (ν +x )
i xi i i i i i
i i i
where ν is a probability distribution with a smooth density and ν +x
i
denotes the translate of ν to x .
i
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