Protein Folding Simulation and Self-Consistent Potential
Functions

Fumi Takazawa 1

takazawa@kuicr.kyoto-u.ac.jp

Masaki Sasai 2

sasai@info.human.nagoya-u.ac.jp

Minoru Kanehisa 1

kanehisa@kuicr.kyoto-u.ac.jp
12Institute for Chemical Research Kyoto University, Uji, Kyoto 611-0011, Japan
Graduate School of Human Informatics, Nagoya University, Nagoya 464-8601, Japan

1 Introduction
Protein folding is one of the most fundamental problems in biophysics. Molecular dynamics (MD)simulations aim to provide the most detailed description of the behavior of atomic models of proteinsduring their folding processes. However, current MD simulations, even with the most powerful computational techniques, can barely explore more than a few nanoseconds for an all-atom model of aprotein and its solvent environment. On the other hand, the folding experiment takes place not in therange of hundreds of microseconds and but usually in milliseconds, seconds or minutes. In order tosimulate the events in longer time ranges, potential functions are required to be simpli ed to represent
molecular structures in less detail. Statistical potentials extracted from a dataset of known structuresin native states are useful for the prediction, such as by 3D-1D (threading) methods. In 3D-1D prediction, the correct structure is recognized as a most stable state among template structures. However,such statistical potentials cannot always simulate successful protein folding. One of the reasons is thatthe statistical potentials are extracted from the various protein structures in native states, lacking theinformation of those in unfolded or misfolded states. Potential functions for folding simulation shouldhave the ability to evaluate energy of each state that appears during folding simulation. In this paper,
we present self-consistently adjusted potentials that are obtained by the following procedure. We startfrom the statistical potentials obtained from known structures in native states, generate a series ofstructures by MD simulations of protein folding, and re-adjust the potentials to better represent both
native and non-native structures.

2 Methods
2.1

Pairwise potential

We express the energy of the structure as a sum of pairwise potentials. When the residues p and q
are found at positions i and i + k in structure  of the library, a Gaussian function whose center is atpq; is summed into the potential V pq :r = ri;i+kk(l)

XX1
Vkpq
=0 pexp(l)Nk 2ck  ipq; )2(r 0 ri;i0 2c +kk!

Here p and q represent 2 of the 20amino acids, Nk is the number of pairs that appear in the library,pq; is the spatial distance between C atoms. For glycine, C is used
ck is chosen to be i 0 j , and ri;i+kinstead of C . The potentials are constructed from a library of 150 known protein structures. Twoproteins do not have more than 25% sequence identity in the library.
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2.2

Surface exposure

A contribution from surface exposure and packing is de ned as follows. The number of other C
atoms surrounding the C atom of a given amino acid residue is considered as a cost function ofsurface exposure of the residue.
2.3

Molecular dynamics simulation

We assume that the chain obeys the following Langevin equation of motion [1]:dp
@V=00 p +  (t)dt@rpdr=dtmHere p is the momentum. Mass of all atoms are considered to be equal; m = 1.  (t) is a random
force, which is assumed to be Gaussian white noise and its variance is related to temperature T by
the uctuation-dissipation relation;< i (t1 )j (t2 ) >= m T ij  (t1 0 t2 )can be estimated from Stokes relation,  = 6a, assuming  to be the viscosity of water at room
temperature.
2.4

Series of structures appearing in folding funnel

In order to adjust the potential, not only the structures in the native state but also a series of structuresin unfolded or misfolded states are needed. The series of structures of a given protein in the folding iscreated by folding simulation using a modi ed potential, which is constructed from the training data
set that additionally contain one hundred native-state structures of the protein.

3 Results and Discussion
The modi ed potential made it possible to recognize the native state with a higher probability inactual MD folding simulation.
Therefore it can be considered to contain funnel-like folding energy landscape. During the folding
simulation, the potential should be able to help the structure to escape from local minima and enhance
the probability to reach the global minimum of the native state. We added the surface-exposure-costfunction (see 2.2) to the pairwise potential (see 2.1), which did improve the accessibility to the global
minimum in our MD simulations (see 2.3).
We have applied the above procedure to several proteins with di erent folds and adjusted theparameters to better represent the folding process. We will use the adjusted potentials for folding
simulations, where we plan to use MD simulation together with the Monte Carlo simulation.

Acknowledgements
This work was supported in part by a Grant-in-Aid for Scienti c Research on Priority Areas, `GenomeScience', from the Ministry of Education, Science, Sports and Culture of Japan.

References[1] Sasai, M., Conformation, energy, and folding ability of selected amino acid sequences, Proc. Natl.
Acad. Sci. USA, 92:8438, 1995.
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