This research project aims to understand the physical relevance of
a dynamical description for equilibrium and non-equilibrium phenomena
emerging in simplified models mimicking the essential features
of the protein folding problem.
The main goals can be summarized as follows:
- Molecular dynamics studies, within microcanonical (NVE)
and canonical (Nose'-Hoover dynamics) ensemble, of toy off-lattice
models in 2-d and 3-d recently investigated with Montecarlo (MC) approaches.
- Study of the role of dynamical collective variables on the
folding process. Comparison between the dynamics of "good"
and "bad" folders at different temperatures (energies).
- Characterization, in terms of the corresponding time
scales, of the three transitions tipically observed for
heteropolymers : namely, the
$\theta$-transition (associated to the transition from
coil to globular state); the folding transition
and the glassy transition
(associated to a dramatic slowing down in the collective
and single particle dynamics).
Project Description:
Protein molecules play a vital role in life. They are
heteropolymers made up of aminoacids along a backbone.
The primary structure (or the sequence of aminoacids) determines
the native folded structure (tertiary structure).
Solutions of the direct and inverse folding problem both involve an
understanding of the correlation between the aminoacid sequence and the
structure of the native state and is the goal of our proposal.
The strategies so far adopted by the scientific community
to tackle the protein folding
problem are mainly of "static" nature, i.e. conceived in the
framework of equilibrium statistical mechanics. However, the
folding process is a dynamical process both in space and time,
starting from an aminoacidic chain and ending with a globular-folded,
biologically active structure.
Nonetheless, protein folding is a very fast process, despite
the astronomically huge number of conformations in which the
protein could find itself. Would a protein reach its native
structure after a systematic testing of all the allowed
configurations, the folding time should be measured in
cosmological time units [1].
Fundamental issues
We aim to pursue a dynamical approach
to single out the relevant aspects of the problem.
In particular, we would like to obtain even partial answers
to the following questions:
what are the dynamical features characterizing the dynamics of "good"
sequences and thus of "good" proteins ? Which kind of link
should be established between the static feature of a
"good" proteic sequence and the dynamical parameters (e.g.
range and kind of interactions, internal and environmental
constraints) ?
We plan to investigate these problems by
employing a ``coarse-grained'' Molecular Dynamics (MD) approach
(both at constant energy and temperature) for simulating
the dynamics of chains of globally
coupled monomeric units bounded by local potentials.
Specifically, models will be considered to represent
heteropolymers with different structures (random versus
real sequences) in 2- and 3-d [2,3,4].
Our goal is to detect peculiar dynamical behaviours
distinguishing
the dynamics of "good" folders from the dynamics of sequences
that do not give rise to proteins.
Connections with experiments
The ab-initio design of new proteins with desired functionality (and hence
native state) on solving the inverse folding problem would unleash a host of
laboratory applications in fields including catalysis, biosensors, drugs,
hormones etc. Knowledge of the dynamical details characterizing the
folding process may greatly help efficient protein design and clarify also
the mechanisms yielding native structures through the cooperation
between intrinsic features of the chain and environmental constraints
of chemical (solvent) and thermodynamic nature.
Research Plan
The folding problem cannot be approached starting from
`` ab initio '' models taking into account all the
interactions present in the system. This is due
to the fact that while the characteristic times
associated to the microscopic dynamics are of
order $10^{-11}$ sec. (once the degrees of freedom
of solvent molecules are traced out from the
interaction hamiltonian) the folding process
occurs on time scale of order 0.1 - 1.0 sec.
Therefore, ``coarse grained'' potentials, reproducing the
main effective interactions present in the system,
should be employed in order to have a chance to
study such a transition.
In particular, we have already approached the study of a simple
generalization of a 2-d off-lattice model introduced
by Stillinger et al. [2]. This model is made up
of a chain of pointlike monomers of two different types
: polar and hydrophobic. Moreover the interaction potential
is composed of two terms : an orientational
term, associated to the angle formed between adjacent
monomers, and a long-range interaction (Lennard-Jones like),
that mimicks the effective action of the solvent.
We have also included a nearest-neighbour harmonic interaction
in order to simulate an almost rigid (stiff) bond.
The MC analysis of these models has already shown
that, analogously to real proteins, only few sequences
fold into a native structure (good folders), while the
majority of the possible sequences do not posses an
unique folded state [3].
Our first tests confirm the main results found via MC
techniques and also show that a
dynamical approach is able to reproduce
the native configuration previously identified through MC
for sequence corresponding to good folders [3].
Once the validity of the dynamical approach has been tested
and verified, we plan to investigate in detail the time
scales involved in the folding process.
We want to stress here that the problem of time scales
has been already tackled in the framework of MC simulations
[5,6], but, in
our opinion, it is extremely difficult to connect a time
expressed in number of MC steps with true
time scales. Instead, a MD approach allows one to obtain
naturally an estimate of such time scales, while detecting
all the successive stages associated to
the protein folding process.
In particular, we shall consider
a set of sequences
at different temperatures and we shall analyze the time needed
for approaching their equilibrium states.
We aim to identify the scaling law (if any) that characterizes
the equilibration time when approaching the
"folding temperature" (i.e. the maximum temperature
at which a good sequence reaches a unique native state).
A reliable estimate of the time scales can be obtained
only by averaging
over several different initial conditions. Moreover, the dynamics
of many different ``good'' and ``bad'' sequences have to be studied
and compared.
This analysis demands a considerable numerical effort
and it can be implemented fruitfully on parallel machines.
Two different indicators will be considered
for studying the dynamics of relaxation to equilibrium:
a configurational distance and an energy distance [5]
with respect to the native structure. It will be also
important the measurement of the fluctuations of such distances
and the verification of the criterion for the foldability of proteins.
recently introduced in [7].
We plan to extend the analysis also to 3-d models [4] and to more realistic potentials, where the parameter entering in the mesoscopic interactions can be extracted directly from the protein databank, similarly to what has been recently done in [8].
References
1) T.E. Creighton, "Proteins: Structures and Molecular
Properties" (W.H. Freeman & Co., New York, 1993).
2) F.H. Stillinger, T. Head-Gordon and C.L. Hirshfeld,
`` Toy Models for Protein Folding '', Phys. Rev. E 48,
1469 (1993).
3) A. Irbaeck, C. Peterson and F. Potthast,
`` Identification of Amino Acid Sequences with Good Folding Properties
in an Off-Lattice Model'', Phys. Rev. E 55, 860 (1997);
A. Irbaeck and F. Potthast, `` Studies of an off-lattice model for
protein folding: Sequence dependence and improved sampling at finite
temperature '', J. Chem. Phys. 103, 10298 (1995).
4) A. Irbaeck, C. Peterson, F. Potthast and O. Sommelius,
`` Local Interactions and Protein Folding: A 3-d Off-Lattice Approach '',
J. Chem. Phys. 107, 273 (1997).
5) G. Iori, E. Marinari and G. Parisi,
"Random self-interacting chains: a mechanism for protein
folding", J. Phys. A 24, 5349 (1991).
6) A.G. Gutin, V.I. Abkevich and E.I. Shakhnovich,
"Chain Lenght Scaling of Protein Folding Times",
Phys. Rev. Lett. 77, 5433 (1996).
7) D.K. Klimov and D. Thirumalai,
"Criterion that Determines the Foldability of Proteins",
Phys. Rev. Lett. 76, 4070 (1996).
8) C. Clementi, A. Maritan and J.R. Banavar,
"Folding, Design and Determination of Interaction Potentials
Using Off-Lattice Dynamics of Model heteropolymers",
Phys. Rev. Lett. 81, 3287 (1998).