\section*{Conclusion}

We described a completely automated loop modeling algorithm that
consists of an optimization of a defined segment of protein structure
in a fixed environment, guided by a pseudo energy function. The method
was tested on a statistically meaningful number of loops of known
structure, both in the native and near-native environments. The
evaluation indicated that loops of 8 residues predicted in the 
native environment have 90\% chance to be modeled with useful 
accuracy (\rmsl\ less than 2\Ang). Even 12-residue loops are modeled 
with useful accuracy in 30\% of the cases. When the \RMSD\ distortion 
of the mainchain stem atoms is 2.5\Ang, the average loop prediction 
error increased by 180, 25 and 3\% for 4, 8 and 12-residue loops, 
respectively. It is not anymore too optimistic to expect useful models 
for loops as long as 12 residues, if the environment of the loop is at 
least approximately correct. It is possible to estimate whether or 
not a given loop prediction is correct, based on the structural variability
of the independently derived lowest energy loop conformations.

The method is flexible and can model any subset of atoms.
It is technically applicable to modeling of
loops with bound ligands and several interacting loops, although it
has not been evaluated in such contexts yet. Moreover, the method can
incorporate additional structural information, such as restraints
implied by disulfide bonds and metal binding sites.

The CASP3 meeting revealed that practical applications of loop
modeling are severely limited by the need to model loops in an
approximately correct environment. This is necessary because the
environment has to be fixed during loop modeling; if it were not, the
number of optimized degrees of freedom would increase several fold,
making the modeling problem generally too hard for the current
methods. If there are alignment errors at the stem residues or at the
other environment residues, loop modeling is not likely to result in
an accurate model. This means that loop modeling is most useful for
target sequences that share more than 30\% sequence identity with the
template structures. Another limitation exposed by our experience at CASP3
is the need for a method that determines the regions in the target
sequence to be modeled as loops. While the insertions have to fall
into this category, there are generally additional regions that are
aligned with templates but could benefit from an accurate ``loop''
modeling. These regions cannot be reliably identified at the present.

It was shown that the accuracy of the predictions is limited primarily
by the accuracy of the energy function, not the thoroughness of the
optimizer. The improvements of the energy function that are most
likely to result in better predictions include explicit modeling
of the \cis-peptide states, a more accurate representation of 
the $\Phi, \Psi$ angles, as well as more accurate non-bonded terms. 
Better $\Phi, \Psi$ restraints might be obtained by expressing 
them with 2D cubic splines instead of bivariate Gaussian functions,
and by explicitly taking into account the dependence of the mainchain
conformation of a residue on the preceding and subsequent residues in
the sequence. The non-bonded terms would probably benefit from a 
description of hydrogen bonds and solvent that is more physical than
the current statistical potential. For example, an all-hydrogen atom 
and solvent representations, such as the generalized Born model
\cite{RappFries99,DomBro99}, may be necessary. To test some of these 
suggestions, we plan to evaluate the latest generation of molecular 
mechanics force fields \cite{CHARMM98,AMBER95} for loop modeling.
We will begin by a detailed analysis of the correlations of the 
prediction accuracy with the properties and interactions of the 
predicted and native loops.

The differences in length and conformation of loop regions in a family
of related proteins are frequently responsible for the specificity of
ligand binding. Thus, accurate modeling of loops is essential for
structure-based prediction of function from sequence. For example, a
comparative model can sometimes be used with computational ligand docking 
to find a putative ligand or resolve preferences within a limited set of 
ligands \cite{RinSunMcK93,XuSanSal96}. A serious complication is that the
ligand may induce conformational changes in loops with which it interacts. 
Thus, it is not always sufficient to be able to model loops on their own, 
as addressed in this paper. Instead, the protein and the ligand should
ideally be modeled simultaneously. Nevertheless, induced fit is frequently 
small. As a result, modeling of ligand binding loops in the {\em apo} state 
can still be useful for studying functional differences within a family of 
proteins, as well as for the ``\abinitio'' prediction of protein function 
by recognition of ligand binding sites 
\cite{JonTho97,FetGodSko98,russell98b,kleywegt99,wei99,kasuya99}.
The current method presents a  significant improvement in the 
modeling of loops in protein structures.
