Least constraint approach to the extraction of internal motions from molecular dynamics trajectories of flexible macromolecules.

Abstract

We propose a rigorous method for removing rigid-body motions from a given molecular dynamics trajectory of a flexible macromolecule. The method becomes exact in the limit of an infinitesimally small sampling step for the input trajectory. In a recent paper [G. Kneller, J. Chem. Phys. 128, 194101 (2008)], one of us showed that virtual internal atomic displacements for small time increments can be derived from Gauss' principle of least constraint, which leads to a rotational superposition problem for the atomic coordinates in two consecutive time frames of the input trajectory. Here, we demonstrate that the accumulation of these displacements in a molecular-fixed frame, which evolves in time according to the virtual rigid-body motions, leads to the desired trajectory for internal motions. The atomic coordinates in the input and output trajectory are related by a roto-translation, which guarantees that the internal energy of the molecule is left invariant. We present a convenient implementation of our method, in which the accumulation of the internal displacements is performed implicitly. Two numerical examples illustrate the difference to the classical approach for removing macromolecular rigid-body motions, which consists of aligning its configurations in the input trajectory with a fixed reference structure.

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Least constraint approach to the extraction of internal motions
from molecular dynamics trajectories of flexible macromolecules
Guillaume Chevrot1,2, Paolo Calligari4, Konrad Hinsen1,2, and Gerald R. Kneller1,2,3 ∗
1Centre de Biophys. Mol´ eculaire, CNRS; Rue Charles Sadron, 45071 Orl´ eans,France
2Synchrotron Soleil; L’Orme de Merisiers, 91192 Gif-sur-Yvette, France
3Universit´ e d’Orl´ eans; Chateau de la Source-Av. du Parc Floral, 45067 Orl´ eans, France and
4D´ epartement de Chimie, associ´ e au CNRS, Ecole Normale Sup´ erieure,
24, rue Lhomond, 75231 Paris Cedex 05, France
Abstract
We propose a rigorous method for removing rigid body motions from a given molecular dynamics
trajectory of a flexible macromolecule. The method becomes exact in the limit of an infinitesimally
small sampling step for the input trajectory. In a recent article [J. Chem. Phys. 128, 194101
(2008)], one of us showed that virtual internal atomic displacements for small time increments
can be derived from Gauss’ principle of least constraint, which leads to a rotational superposition
problem for the atomic coordinates in two consecutive time frames of the input trajectory. Here
we demonstrate that the accumulation of these displacements in a molecular-fixed frame, which
evolves in time according to the virtual rigid-body motions, leads to the desired trajectory for
internal motions. The atomic coordinates in the input and output trajectory are related by a
roto-translation, which guarantees that the internal energy of the molecule is left invariant. We
present a convenient implementation of our method, in which the accumulation of the internal
displacements is performed implicitly. Two numerical examples illustrate the difference to the
classical approach for removing macromolecular rigid body motions, which consists of aligning its
configurations in the input trajectory with a fixed reference structure.
PACS numbers:
Keywords:
∗Corresponding author. E-mail:gerald.kneller@cnrs.orleans.fr
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I.INTRODUCTION
The construction of trajectories representing the internal motions of a flexible macro-
molecule from a molecular dynamics (MD) trajectory is a recurrent task in biomolecular
simulations. Such trajectories are particularly useful for the analysis of spectroscopic ex-
periments probing the total dynamics of the molecule, but where the internal dynamics is
of particular interest. Examples are combined experimental and simulation studies of pro-
teins in solution by nuclear magnetic resonance (NMR) relaxation spectroscopy1–4and by
quasielastic neutron scattering.5
The common strategy to extract the internal motions from a given MD trajectory is
to align the snapshots of the protein with a common reference structure. The coordinate
frame associated with that structure can be considered as the Eckart frame of the molecule,
referring to the early work of Eckart on the theory of spectroscopic experiments on small
molecules in the gas phase6. Eckart considered only small vibrations as possible internal
motions and assumed the internal energy of the molecule to be a quadratic function of the
atomic displacements. Observing that rigid-body displacements do not alter the internal
potential energy of the molecule, he constructed conditions for the atomic displacements
corresponding to internal motions of the molecule which exclude rigid-body displacements,
and he also described a construction of a molecule-fixed frame describing the global motions
of the molecule.
Shortly later Eckart’s approach was generalized by Sayvetz to linear and “anomalous”
molecules, which were excluded in Eckart’s work.7Here “anomalous” refers to molecules
which have internal motions of large amplitudes, such as rotations of methyl groups, which
cannot be treated within the approximation of small vibrations. Motivated by the obser-
vation by Kudin and Dymarsky that the Eckart axis conditions are closely related to the
problem of an optimal rotational superposition of molecular structures,8one of us (GRK)
has recently shown that virtual atomic displacements describing the internal motions in
arbitrary macromolecules can be derived from Gauss’ principle of least constraint,9which
leads indeed to a rotational superposition problem of molecular structures.10The adjective
“virtual” indicates that the internal displacements are reconstructed from a “real” trajec-
tory including global and internal motions. At each time one considers the motion that
a virtual rigid molecule would have performed within an infinitesimal time interval, given
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the same initial atomic positions, velocities, and forces. The internal atomic displacements
are then the differences between the real displacements and the displacements due to an
infinitesimal virtual rigid-body motion. According to Gauss’ formulation of mechanics, the
internal displacements are minimized in a least square sense and fulfill automatically the
Eckart conditions.
The aim of this paper is to discuss the construction of trajectories for the internal dy-
namics of macromolecules from the time-local virtual displacements obtained from Gauss’
principle. A straight-forward proposition has been made in Ref. [10], suggesting that the
(almost) infinitesimal displacements obtained from an equidistantly sampled molecular dy-
namics trajectory could be simply accumulated in the laboratory frame. Here it is shown
that the accumulation should instead be done in a common molecular-fixed frame, which can
be considered as the Eckart frame of the molecule and which can be formally constructed by
an accumulation of the infinitesimal virtual rigid-body motions obtained from the least con-
straint principle. The theoretical part of the paper is presented in section II, starting with a
short review of the essential points of Ref. [10]. In section III, practical aspects concerning
the explicit construction of the molecule-fixed frame and the corresponding internal-motion
trajectories are discussed. Two applications are presented in section IV to illustrate the dif-
ference between the traditional method of removing global motions and the one we propose.
The paper is concluded by a short discussion of the results.
II.THEORY
A.Virtual internal displacements
The following section of the paper gives a short review of the relation between the con-
straints for internal atomic motions in flexible macromolecules formulated by Eckart6and
Gauss’ principle of least constraint,9,11–13which has been developed in Ref. [10]. According
to Gauss the motion of a mechanical system under constraints can be derived from a local
minimum principle. Considering N point-like particles with given positions xαat time t and
masses mα, the principle can be formulated as
ξ =1
2
N
?
α=1
mα
?xα(t + δt) − x(c)
α(t + δt)?2= Min,(1)
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where x(c)
α (t + δt) are the constrained positions at time t + δt. The latter are thus obtained
from a least squares fit to the unconstrained positions at time t + δt. The time increment
δt must be small enough to permit the approximation xα(t + δt) ≈ xα(t) + δt ˙ xα(t) +
δt2Fα(t)/(2mα), where the dot denotes a derivative with respect to time and Fα(t) is the
force acting on atom α. Suppose now that the coordinates xα(t) = X(t) + rα(t) define the
atomic positions in a “virtual rigid molecule” at time t, where rαare the relative positions
with respect to a common rotation center, X. Within the time span δt, the atomic positions
in the virtual rigid molecule will evolve to
x(c)
α(t + δt) = X(t) + rα(t) + δτ + δφ ∧ rα(t),(2)
where δτ and δφ define, respectively, a small translation and rotation of the virtual rigid
molecule. The vector δφ points into the direction of the rotation axis and its modulus is
the rotation angle. The differences between the real atomic positions, xα(t + δt), and the
virtual atomic positions, x(c)
α (t + δt), define the atomic displacements due to the internal
motions of the molecule,
δuα(t + δt) = xα(t + δt) − x(c)
α(t + δt). (3)
The virtual rigid body displacement is performed according to the minimum principle (1).
Inserting here the form (2) for the constrained positions at time t+δt leads to the necessary
conditions10
∂ξ
∂δτ=
N
?
N
?
α=1
mαδuα(t + δt) = 0, (4)
∂ξ
∂δφ=
α=1
mαrα(t) ∧ δuα(t + δt) = 0, (5)
which are the Eckart conditions for the internal atomic displacements.6The translational
Eckart condition (4) and the rotational Eckart condition (5) constitute a system of linear
equations for the components of δτ and δφ, which can be decoupled if X is taken to be the
center of mass,
X =
1
M
?
α
mαxα,(6)
where M is the total mass of the molecule. In this case one obtains
δτ = X(t + δt) − X(t) (7)
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and the rotation vector is given by
δφ = θ(t)−1L(t)δt, (8)
where L =?
αmαrα∧ ˙ rαis the angular momentum and θ is the tensor of inertia, with
components θij=?N
frame.
α=1mα(|rα|2δij−rα,irα,j). Both quantities are referred to the laboratory
B.Trajectories of internal motions
We consider a trajectory xα(nδt) ≡ xα(n), where δt is a fixed small time increment
justifying the approximations
δτ(n) ≈ δt˙X(n),
δφ(n) ≈ δtω(n),
(9)
(10)
such that the constrained positions at t = nδt are approximated by
x(c)
α(n) ≈ xα(n − 1) + δt˙X(n − 1) + δtω(n − 1) ∧ rα(n − 1).
Using that (xα(n) − xα(n − 1))/δt ≈ ˙ xα(n − 1) one obtains
δuα(n) = xα(n) − x(c)
α(n)
?
≈ δt
˙ xα(n − 1) −˙X(n − 1) − ω(n − 1) ∧ rα(n − 1)
= δt{˙ rα(n − 1) − ω(n − 1) ∧ rα(n − 1)},
?
which becomes on a continuous time scale
δuα(t + δt) ≈ δt{˙ rα(t) − ω(t) ∧ rα(t)}. (11)
One can now construct the trajectory of the internal motions for a flexible macromolecule,
by accumulating for each atom its virtual displacements in the course of time. Here it must
be taken into account that virtual displacements at different times t and t?> t are related
to virtual rigid molecules which have been defined at, t−δt and t?−δt, respectively. Before
adding these virtual displacements they should first be transformed to a common molecule-
fixed frame, observing that the molecule has undergone a global rotation within the time
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interval t?−t. This point has not been considered in Ref. [10], but it is important as will be
shown below. The trajectory representing the internal motions of atom α then has the form
?t
where D(t) is an orthogonal matrix describing the accumulated rotation of the macro-
xint
α(t) = xα(0) +
0
DT(τ) · δuα(τ)(12)
molecule and “T” denotes a transposition. The matrix D(t) fulfills the differential equation
˙D(t) = Ω(t) · D(t),(13)
where Ω is the skew-symmetric matrix containing the Cartesian coordinates of angular
velocity in the laboratory frame,
Ω =





0
−ωz
0
ωy
ωz
−ωx
0
−ωy
ωx




.(14)
Writing ω ∧ rα= Ω · rα, one finds from the definition of the accumulated internal displace-
ments that
?t
=??DT(τ) · rα(τ)??t
0
DT(τ) · δuα(τ) =
?t
0
dτ DT(τ) · {˙ rα(τ) − Ω(τ)rα(τ)}
?t
0−
0
dτ



˙DT(τ) · rα(τ) + DT(τ) · Ω(τ)rα(τ)
?
= DT(t) · rα(t) − rα(0).
???
=0



Here it was used that˙DT= (Ω · D)T= DT· ΩT= −DT· Ω and that D(0) = 1. Inserting
the above relation for the accumulated internal displacements into the definition (12) for
the trajectory of internal motions leads to
xint
α(t) = X(0) + DT(t) · {xα(t) − X(t)},(15)
which defines a roto-translation for the transformation xα(t) → xint
tion guarantees that the internal energy of the molecule is left invariant,
α(t). Such a transforma-
U(x1(t),...,xN(t)) = U(xint
1(t),...,xint
N(t)). (16)
The matrix D(t) is the transformation matrix from the laboratory frame to a molecule-
fixed frame, which can be considered as the Eckart frame of the molecule. Its elements define
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the projections of the moving molecule-fixed basis vectors ?i(t) (i = 1,2,3) onto the fixed
basis vectors ej(j = 1,2,3),
Dij(t) = ?T
i(t) · ej.(17)
While the the center-of-mass vector X(t) can be easily obtained from a given molecular
dynamics trajectory, the construction of the matrix D(t) is a less simple task that will be
discussed in the following section.
III.CONSTRUCTING THE ROTATION MATRIX
This section describes the approximate construction of the rotation matrix D(t) and the
corresponding trajectory of internal motions xint
α(t) from a molecular dynamics trajectory
which is sampled with a finite time step ∆t.
A.Discretization
In the general case, where Ω is time dependent, only a formal solution of the defining
equation (13) for D(t) can be given. It follows from (13) that
D(t + ∆t) = D(t) +
?t+∆t
t
dτ Ω(τ) · D(τ)(18)
which may be approximated by
D(t + ∆t) ≈ (1 + ∆tΩ(t)) · D(t)(19)
if ∆t tends to zero. Defining ∆t = t/n, the rotation matrix is thus given by the infinite
product
D(t) = lim
n→∞
n
?
k=0
?
1 +t
nΩ([k/n]t)
?
. (20)
On recognizes that D(t) = exp(Ωt) if Ω is constant, i.e. if the molecule rotates with constant
angular velocity.
For practical purposes, a discrete approximation of the matrix D(t) can be obtained from
an approximation of expression (20),
D(n) ≈
n−1
?
k=0
∆D(k),(21)
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where the increments ∆D(k) are given by
∆D(k) = exp(∆tΩ(k))(22)
and appear in the finite roto-translation
x(c)
α(k + 1) = X(k) + ∆τ(k) + ∆D(k) · rα(k)(23)
which replaces relation (2) in the minimization problem (1). If X is chosen to be the center
of mass, problem (1) takes the form (n = 0,1,2,...)
ξ =1
2
N
?
α=1
mα
?rα(n + 1) − r(c)
α(n + 1)?2= Min,(24)
where the constrained positions with respect to the center of mass at time n+1 are obtained
by a rotation from the positions at time n,
r(c)
α(n + 1) = ∆D(n) · rα(n). (25)
Here one may use that rint
α(n) = DT(n) · rα(n), which leads to the alternative expression
r(c)
α(n + 1) = D(n + 1) · rint
α(n),(26)
observing that D(n+1) = ∆D(n)·D(n). Inserting either of the expressions (25) or (26) for
r(c)
α (n + 1) into the minimization problem (24) makes the target function ξ defined in (24)
a function of the components of a rotation matrix, which must be chosen such that ξ is
minimal.
B.Superposition problem
A convenient way to solve the rotational superposition problem (24) is to express the
rotation matrix minimizing the target function ξ in terms of four real quaternion parameters.
The general form for a rotation matrix in this parametrization is14


2(−q0q2+ q1q3)
D(q)=



q2
0+ q2
1− q2
2− q2
3
2(−q0q3+ q1q2)
q2
2(q0q2+ q1q3)
2(q0q3+ q1q2)
0+ q2
2− q2
1− q2
3
2(−q0q1+ q2q3)
q2
2(q0q1+ q2q3)
0+ q2
3− q2
1− q2
2




,(27)
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where q2
0+q2
1+q2
2+q2
3= 1. Different variants for the solution of the rotational superposition
problem with quaternions have been discovered and described by several authors (see for
example Refs. [15–17] and the review [18]) and more details may be found in Ref. [10].
Using the method described in [17], the target function ξ given by relation [24] becomes a
quadratic form in the quaternion parameters q
ξ(q) =1
2qT· µ · q,(28)
If if the coordinate sets to be superposed are {rα(n)} and {rα(n + 1)} (see Eq. (25)),
we have q ≡ ∆q(n) and furthermore define rα≡ rα(n). For the alternative form given by
Eq. (26), we have q ≡ q(n + 1) and rα ≡ rint
r?
α(n) instead. For both cases we also define
α≡ rα(n + 1) and write the matrix µ in the form
µ =
N
?
α=1
mαµα,(29)
where the atomic contributions are given by
µα=



(rα− r?
2(r?
α)2
2(r?
α∧ rα)T
α∧ rα)(rα+ r?
α)21 − 2(r?
αrT
α+ rαr?T
α)

.

(30)
Minimization of the target function ξ(q) subject to the normalization constraint of the
quaternion parameters leads to solving an eigenvector problem for a 4 × 4 matrix,
µ · q = λq.(31)
Here the eigenvalues represent the (mass-weighted) mean square superposition error and the
normalized eigenvector corresponding to the smallest eigenvalue contains thus the quaternion
parameters describing the optimal solution. In a more general context, the mass-weighting
scheme of the atomic contributions to the total superposition error can be replaced by any
set of positive numbers.
We note that the quaternion-based superposition method also yields a measure for the
orientational distance of two molecular structures. If {rα} and {r?
coordinates of the two structures under consideration, their orientational distance can be
α} are the sets of atomic
defined as their Euclidean distance divided by the corresponding maximum distance. The
latter is given by the eigenvalue λmaxof the eigenvector problem (31) and using the definition
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of the matrix µ, the orientational distance is given by19
∆Ω=
?µ11
λmax. (32)
By construction 0 ≤ ∆Ω≤ 1.
C. Algorithmic considerations
If the choice (25) for r(c)
α (n + 1) is used in the target function (24), the accumulated
rotation matrix D(n) and the trajectory rint
α(n) are constructed as follows
rα(n)
?→ rα(n + 1) yields∆q(n)
D(n + 1)= D(∆q(n))D(n)
rint
α(n + 1) = DT(n + 1) · rα(n + 1)
(33)
setting D(0) = 1. The explicit matrix multiplication in the accumulation of D and the
corresponding accumulation of numerical errors can be avoided by minimizing the target
function (24) with the choice (26) for r(c)
α (n + 1). This leads to
rint
α(n)
?→ rα(n + 1)yieldsq(n + 1)
D(n + 1)= D(q(n + 1))
rint
α(n + 1) = DT(n + 1) · rα(n + 1)
(34)
with the starting point rint
α(0) = rα(0).
Finally, we note that approximations for the angular velocity can be computed from the
quaternion parameters ∆q obtained from the superposition fits according to scheme (33). It
follows from the expression (22) for the increment ∆D(k) and the general form (27) for a
rotation matrix expressed in quaternion parameters that (the time argument is dropped)







∆q0
∆q1
∆q2
∆q3







=







cos(ω∆t/2)
sin(ω∆t/2)nx
sin(ω∆t/2)ny
sin(ω∆t/2)nz







, (35)
where ω is the modulus of ω and nx,ny,nzare the components of the unit vector n = ω/ω.
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IV.APPLICATIONS
In the following some examples will be discussed in which the method described above is
compared with the usual approach of removing global motions from the molecular dynamics
trajectories of macromolecules, which consists of aligning all configurations with a fixed
reference structure. Two examples are considered:
(a) the dynamics of the TRP-cage miniprotein,20which consists of only 20 amino acids,
and which is believed to be the smallest protein exhibiting a stable fold (code 2JOF
of the protein data bank (PDB)21),
(b) the folding of a polypeptide chain of the same sequence length ( ”C-tail”, in the
following), taken from the C-terminal region of the eukaryotic anti-association factor
6, and believed to be very mobile and mostly unstructured.
A.TRP-cage miniprotein
We performed one MD simulation of the TRP-cage molecule at ambient conditions and
one at T = 400K and normal pressure, in order to enhance internal motions in the protein.
As initial structure we used in both cases the crystallographic coordinates deposited in
entry 2JOF of the PDB, adding the hydrogen atoms according to the known chemical bond
structure in amino acids. The protein was immersed in a solvent of 7159 water molecules,
choosing a cubic simulation box with a box length of 6 nm and periodic boundary conditions.
All simulations were performed with the Molecular Modeling Toolkit (MMTK),22using the
Amber99 force field23which includes the TIP3P force field24for the simulation of water
molecules. Coulomb interactions were treated with the method proposed by Wolf et al.,25
using a cutoff radius of 1.4 nm. Tests of the method can be found in Refs. [26,27]. For both
temperatures we performed equilibration runs of 100 ps at constant temperature and pressure
in the NpT ensemble using the Nos´ e-Andersen extended system method,28each followed by
a production run of 2 ns simulation at constant temperature in the NVT ensemble. For
the integration of the equations of motion we used a time step of 1 fs and the protein
configurations were stored every 20 fs for later analysis.
Fig. 1 displays some snapshots taken along the trajectories for internal motions which
have been obtained by aligning the molecule with the first configuration in the production
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trajectory (light gray) and by using the method proposed in this paper (dark blue). In
the following the two approaches are referred to as “fit to first” and “fit to preceding”,
respectively. It should be noted that the resulting coordinate sets differ by a global rigid-
body motion if the same time frames are compared and the internal energy is thus strictly
the same. The deviation of the global orientation increases, however, rapidly with time, and
this effect is more pronounced for the strongly heated protein at T = 400K. To quantify
this deviation we display in Fig. 2 the corresponding orientational distance which is defined
by relation (32). More important for applications are the resulting differences for time
averaged quantities, such as static averages, time-correlation functions and time-dependent
mean square displacements (MSD). The first comparison of the different methods to remove
global motions concerns the root mean square fluctuations (RMSF) of the atoms in the
TRP-cage protein. We define the total RMSF as
∆ =
?
?
?
?
N
?
α=1
wα?(rα− ?rα?)2?,(36)
where ?...? denotes an average over the MD trajectory and wα are atomic weights with
?N
weighting, i.e. wα = mα/M, with M being the total mass of the protein. The RMSF
α=1wα= 1 (N is the number of atoms in the protein). In our calculations we use mass-
values are displayed in the second column of table I. The RMSF values are almost equal at
T = 300K, but at higher temperature (T = 400K) the difference is much more pronounced.
So, we note that at T = 400K, the “fit to preceding” method leads to a greater flexibility
of the protein compared to the “fit to first” method.
The second comparison of the different methods concerns the time-dependent mean-
square displacement which is defined as:
W(t) =
N
?
α=1
wα?[rα(t) − rα(0)]2?,(37)
where wαare atomic weights with the same definition as mentioned previously. It measures
how far (on average) the system moves away from its original configuration in a given amount
of time and for confined diffusion the plateau value is related to the RMSF:
lim
t→∞W(t) = 2∆2.(38)
Fig. 3 displays the mass-weighted average MSDs for the TRP-cage molecule corresponding,
respectively, to the “fit to first” and “fit to preceding” trajectory. The comparison shows
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that practically identical results are obtained for T = 300K, whereas differences appear
for 400K. Although these differences are small in amplitude, they are not irrelevant since
the form of the MSD changes. The asymptotic values limt→∞W(t) are shown in the third
column of table I next to the values of ∆ from which they were computed. As we could
expect from the trends of the MSD curves during the first picoseconds (Fig. 3), the W(∞)
values are similar at T = 300K and a little higher for the “fit to preceding” method than
for the “fit to first” method at 400K.
B.C-tail
The C-tail polypeptide chain (sequence EDAQPESISGNLRDTLIETYS in the one-letter
code notation for residues types) was simulated starting from a linear configuration created
by using the AMBER929simulation package. All equilibration and production runs were
performed using the AMBER0330force field and an implicit solvent. The latter is described
by the Generalized Born (GB) solvation model, developed by Hawkins and coworkers,31,32
where mean forces are obtained from the estimation of the total solvation free energy of
the molecule into water. Initial equilibration of the linear structure was performed with
progressive temperature re-scaling from 0K to 300K with an increase of 50K every 500 ps.
The time step during this initial equilibration was varied from 0.1 to 0.5 fs, in order to reduce
the extent of force variation and thus the probability of unphysical close contacts. After
a final equilibration simulation of 700 ps, performed at constant temperature (300K) and
constant volume, the production run of 40 ns was performed with an integration time-step
of 1 fs in the NVT ensemble. Configurations of the C-tail polypeptide chain were saved
every 500 fs for later analysis.
Some snapshots of the folding path of C-tail are displayed in Fig. 4 where the config-
urations obtained by aligning the polypeptide with the ”fit to first” method (light gray)
and by using the ”fit to preceding” method (dark gray) are shown at 0.5, 5, 20 and 40 ns.
The progressively divergent configurations obtained by these two methods become well dis-
tinguishable after a few nanoseconds as also suggested by the orientational distance ∆Ω
between corresponding configurations obtained by the two methods (see Fig.5). Here the
reader should note that the slowly increasing value of ∆Ωfrom zero to a plateau value of 0.6
is strictly related to the folding evolution: as the polypeptide reaches a more or less folded
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conformation, ∆Ωstarts oscillating around an average value. The latter fact is due to the
negligible difference between the two methods when applied to folded conformations.
As for the TRP-cage simulation, we calculated the mass-weighted average RMSDs (Tab. I)
and MSDs (Fig. 6) in order to explore the difference of the two fitting methods on the internal
dynamics of C-tail. As for the TRP-cage at 400K, we note that the method proposed in
this work leads to a larger fluctuations compared to the conventional “fit to first” method,
the difference being even more striking, as is to be expected given the large orientational
differences between the two trajectories.
V. CONCLUSIONS
It has been shown that the construction of internal trajectories for flexible macromolecules
from a given molecular dynamics trajectory can be achieved in a systematic way by accumu-
lating the virtual internal displacements obtained from Gauss’ principle of least constraint
in a molecule-fixed frame. For a given time and a short time increment, these virtual dis-
placements are defined as the differences between the actual displacements of the atoms in
the molecule and those obtained from a corresponding rigid body motion within the same
time interval. The translation of the molecule-fixed frame is determined by the translational
motion of the center of mass of the molecule and its rotation is described by the accumu-
lated infinitesimal rotations of the virtual rigid bodies which are used to define the internal
atomic displacements. The coordinate transformation describing the extraction of internal
motions from the input trajectory can be written as a roto-translation of the molecule and
thus strictly preserves its internal potential energy.
The application to the dynamics of TRP-cage miniprotein around its equilibrium con-
formation at ambient conditions shows that the standard procedure for removing global
motions from the trajectory of a macromolecule, which consists of aligning its configura-
tions with a fixed reference structure, yields results that are practically identical with the
procedure proposed in this paper. When the protein is strongly heated, such that it starts to
explore non-native conformations, the analyses of the respective trajectories displays slight
differences. If a folding process is considered, where the conformation of the molecule un-
der consideration changes considerably during the simulation, the comparison with a fixed
reference structure is not appropriate for extracting internal motions and the method pro-
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posed in this article should be used instead. In this method, least-square fit alignments are
always made between neighboring steps in the trajectory, and thus between configurations
that differ only slightly. The method does not depend on the arbitrary choice of a reference
configuration and yields the exact solution in the limit ∆t → 0. Since the computational
effort is the same as for the standard method, our method can replace it safely and efficiently
in all situations to extract the internal motions of macromolecules from a given molecular
dynamics trajectory.
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