A new look at geometrical simulations applied to proteins

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A new look at geometrical simulations applied to
proteins

• Dan Farrell, Scott Menor, Mike Thorpe

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The original FRODA procedure
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The original FRODA procedure
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The original FRODA procedure
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There is not an explicit energy function in
FRODA. However, we will gain some mathematical
insight into FRODA if we consider the following
energy function and see how it is related to the
FRODA procedure.
Ghost-atom positions
Atom-atom separation
Atom positions
There are 6 degrees of freedom per ghost
(uniquely determining ghost-atom positions), and
3 degrees of freedom per atom.
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The FRODA fit-ghosts-to-atoms procedure is
mathematically equivalent to a steepest-descent
minimization of V in the subspace of the ghost
degrees of freedom (atom degrees of freedom are
held constant, so the repulsive energy term does
not contribute).
For one of the six ghost degrees of freedom Qa,
, where
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For the atom degrees of freedom, the gradient
components are
The FRODA fit-atoms-to-ghosts procedure is
mathematically equivalent to taking a single step
along the negative gradient in the subspace of
atom degrees of freedom, with each gradient
component scaled by 1/(NghostatomsNrepulsions).
The scaling is good for a single step. Note,
however, that this is not a minimization of the
atom degrees of freedom, but rather a single
step. In a crowded environment, the energy may
actually be higher after taking this step if new
clashes are created.
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The new procedure
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The molecule is represented by rigid units
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Initially, the rigid units overlap perfectly
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The rigid units are perturbed. Springs
interconnect the rigid units, and will help
restore the proper geometric constraints of the
molecule.
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We restore constraints by conjugate gradient
minimization of the energy function
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The explicit potential in Froda2 is
Atoms are rigidly embedded in the rigid units,
their positions completely determined by the
rigid body degrees of freedom. For atoms that
are shared by overlapping rigid units, the atoms
position is defined as the mean of the
positions of its shared copies,
If there are Nm shared atoms for atom m, there
are Nm(Nm-1)/2 springs that interconnect all
possible pairs. But this is equivalent to just
Nm springs extending from each shared copy to the
mean of the shared copies. It can be shown that
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Constraints are restored by conjugate gradient
minimization of the potential
Allowed
Disallowed
Disallowed
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Benefits of the new approach For the usual
tolerance setting of 0.125 Angstroms, speedup is
about a factor 4 (data shown later), presumably
due to the conjugate gradient minimization and
also from eliminating atom degrees of
freedom. Can achieve tighter tolerances (0.01
Angstroms) than FRODA is capable of
achieving Stabilitystructures that FRODA could
not work on without immediately getting stuck now
work fine with the new method. Having an explicit
energy function and a well-understood
minimization algorithm is mathematically clear.
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To further speed up sampling, we introduce a
momentum-like perturbation strategy After a
single round of perturbation and minimization, we
have gained useful informationwe have learned
that a good collective direction to travel from
point Q1 is along Q2-Q1.
Perturb
Minimize
Q1
Q2
Now located at Q2, we assume that the net
direction just traveled is still a good
direction, and we use the vector Q2-Q1 as the
next perturbation.
(We also add a small random perturbation on top
of the momentum-like perturbation)
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Initial state of Adenylate Kinase
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With the momentum-like perturbation strategy, any
motion that happens to be along large-amplitude
collective modes is able to persist for many
cycles. In this case, the protein closes and
opens repeatedly.
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Other large motions are also sampled that are
consistent with the given constraints, like this
over-extended open configuation.
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How fast does method X sample motion?

• Some factors
• single-step rmsd and cpu time
• correlation length

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In MD, From the equipartition theorem, the
average kinetic energy of an atom is
The mean square velocity for atoms of mass m is
Defining
,
we find that the RMSD step size achieved in an
infinitesimal time ?t obeys
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In MD, the integrated path length grows linearly
with time, with slope
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In Froda, the integrated path length also grows
linearly with time
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In Froda, we define the effective time for a
single step as
, where
For a series of steps, we integrate over the
steps to obtain
We do not necessarily equate the effective time
with real time. It serves as a measure of how
much motion has been sampled. But, the effective
time could be on par with real time for purely
diffusive processes whose dynamics are not
limited by energy-barrier hopping.
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Benchmarking Froda and Froda2 (using random
perturbation onlynot momentum) against Amber
Note Amber in vacuum with 2 fs step (SHAKE) is
about 30 times faster than the Amber GB-HCT 1 fs
used here
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With momentum perturbation added, sampling is
dramatically increased
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Compare how fast each program reaches 1 Angstrom
RMSD from the initial state (cpu time)
Froda2 with momentum -12 s Amber GB 3200
s Froda2 4000 s speedup 400
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Acknowledgments

• Stephen Wells
• Scott Menor
• Mike Thorpe