The severe simplifications inherent to the protein model described above enforce a close inspection concerning its relation to real proteins as well as a number of caveats with respect to the interpretation of the MD-studies described below. In particular we want to check in what sense our study of the low-frequency configurational dynamics of our simplified model can serve as a tool to characterize the corresponding dynamics of real proteins. To that aim we will first discuss some of the properties determining the shape of the energy and free energy landscape, respectively. Subsequently, we will check, whether the model fulfills our expectations.
The neglect of any internal structure of the 100 residues which make up the protein model, represents the most obvious simplification. Since the residues are represented by van der Waals spheres, most short-range, residue-specific interactions, such as H-bonds, which contribute to the formation of rigid secondary structure elements in proteins, are absent in our model. As a consequence, the model structure is expected to exhibit larger flexibility as compared to proteins. An additional increase of flexibility should arise from the absence of side-groups and a corresponding lack of site-specific sterical restraints. Thus, free energy barriers for conformational transitions in our model are expected to be lower than in real proteins. The neglect of angle torsion barriers should have similar consequences.
Secondly, the residue masses have been chosen smaller than those of amino acids by about one order of magnitude. However, this difference does not affect the quality of the dynamics: according to Newton's laws, scaling of masses merely corresponds to a shift of time scales --- in the present case by a factor of three to four. As a result, also conformational motions of the model will occur at a correspondingly faster time scale as compared to proteins.
We expect that as a result of both effects, the reduction of free energy barriers as well as the time scaling of the dynamics of the simplified model, the number of conformational transitions, which occur in the course of our simulations is large enough as to permit their statistical analysis.
Finally, our simple effective solvent model does not include stochastic and frictional forces, which are exerted by solvent molecules onto protein surfaces. Comparisons of MD-simulations of protein models in vacuo with simulations in solvent have shown, that such solvent-induced forces reduce the inertial character of vibrational modes in proteins and decorrelate these motions[57]. Accordingly, our in vacuo simulations should exhibit a much slower decay of the short-time displacement autocorrelation functions of surface residues than the one determined in more realistic protein-solvent models, i.e., the simplifications should enhance memory effects at fast time scales.
The folded structure of our model (Fig. 
(g))
seems to be built up from structural motifs comparable to
secondary structure elements found in proteins, such as a `helix'
at the bottom of the model as well as
`loops' at the left and right sides and at the top. The combination of these
motifs resembles typical tertiary structures of small globular proteins
(compare, e.g., the structure of Crambin, as shown in Ref. Ornstein90).
This structural similarity has not explicitly been put into our model,
but, instead, results from the particular choice of
chain-chain interactions. Analogously, we did expect that also
realistic dynamical properties should appear as a consequence of the
model design.
To confirm that expectation, we computed various properties, the combination of which is known to be characteristic to protein dynamics, and compared them with simulations of more realistic protein models or experiments. In our analysis, we will proceed from short time scales (femtoseconds) to longer ones (nanoseconds).
The short-range interactions, which determine the high-frequency dynamics of the protein model, have been chosen in close correspondence to those of hydrocarbons, as defined by the CHARMm force field[23]. Hence, the protein model should exhibit reasonably realistic dynamical properties at a time scale below some 100 femtoseconds.
Figure: Normalized velocity autocorrelation function
derived from an average over a
one nanosecond trajectory using the velocity vector 
of one particular atom of the protein model. The inset shows the corresponding
spectrum, computed using a Fourier transform of the velocity autocorrelation
function. This spectrum is comparable to that of more realistic
protein models: The two sharp peaks originate from fast
bond-stretch vibrations, whereas the broad bands in the low-frequency region of
the spectrum are predominantly caused by the stochastic character
of inter-atomic van der Waals contacts.
As an example, Figure shows the normalized velocity
autocorrelation function
, derived from an average over a one-nanosecond
trajectory using the velocity vector 
of an arbitrarily
chosen residue of the protein model. The inset
shows the corresponding spectrum, which has been derived in
a way similar to that employed in Ref. Nadler87 in order to enable a
comparison with
the results of MD-simulations of a detailed model of RNAse-A presented therein.
Similar features are sharp peaks in the
range 150 to
, originating from fast
bond stretch vibrations, as well as broad bands in the region
below 
, arising from bond angle vibrations and,
particularly, the noisy character of van der Waals collisions.
The dynamics of more realistic models
differs from that of the simplified protein model in that
it typically gives rise to a larger number of peaks in the high frequency
spectrum, due to a heterogeneity of bond stretch frequencies, which is
absent in our model.
Figure: Distance (in Å ) between residues # 12 and # 36
(cf. Fig. (g)) during an
MD-simulation of 100 nanoseconds length.
The series of plots shows the distance fluctuations at
time scales increasing by a factor of ten at each magnification step;
the bold lines in the bottom two pictures represent smoothed data.
Note that the depicted simulation has not been included within the
analysis carried out in the following Section.
These high frequency modes represent the first layer of
a hierarchy of time scales in proteins,
within each of which specific dynamical processes can be
observed[45,63,64]. These
range from bond stretching modes
(
30 fs), bond angle- and dihedral vibrations (few 100 fs),
collective motions involving groups of atoms (some 10 ps), to
conformational transitions occurring within a wide range of time scales above
100 ps. Inspection of dynamical details at
different resolutions in time has shown, that such hierarchy actually is
reproduced by our simulations, which cover time scales differing by more
than six orders of magnitude. As an example, Figure
shows the time development of the distance between two arbitrarily
chosen residues at decreasing time resolutions.
Proceeding within Fig. from top to bottom,
one observes characteristic fluctuations, which occur at time scales
increasing at each step by a factor of ten and
originate from the four different
dynamical processes enumerated above.
A characteristic feature of protein dynamics is the existence
of a variety of conformational substates, which are interconnected by
conformational transitions.
The model, too, exhibits such conformational transitions, which
reveal themselves as sudden structural rearrangements.
In particular, the rapid atomic distance changes apparent in the
bottom three pictures of Fig.
are due to such conformational transitions.
From these observations we infer, that, despite its simplifications, our protein model exhibits a set of structural and dynamical properties, which are characteristic for proteins. In particular, the model should be rather well-suited for a simulation study of the long-time conformational dynamics of proteins.