Mathematical Modeling and Optimization Methods for De Novo Protein Design - Systems Biology: Networks, Models, and Applications - page 51

known NMR structures), or simply by allowing some deviation from a
subset of distances provided by the structural template, and hence they
allow for a flexible template on the backbone.
The formulations for the folding calculations are reminiscent of
structure prediction problems in protein folding [59]. In particular, a
novel constrained global optimization problem first introduced for
structure prediction using NMR data [60], and later employed in a
generic framework for the structure prediction of proteins [61], is used.
The global minimization of a detailed atomistic energy force field E ff is
performed over the set of independent dihedral angles, f, which can be
used to describe any possible configuration of the system. The bounds
on these variables are enforced by simple box constraints. Finally, a set
of distance constraints, E l dis

, N , which are nonconvex in the
internal coordinate system, can be used to constrain the system. The
formulation is represented by the following set of equations:

, l = 1, …

min
f

E ff

subject to

dis ref

E φ

φφφ

( ) ≤ E j = 1

1

,...,
,...,

N

j j

i L

U

≤≤ i = N

i φ

i

Here, i , N f corresponds to the set of dihedral angles, f i , with

and representing lower and upper bounds on these dihedral
angles. In general, the lower and upper bounds for these variables
are set to

= 1, …

f i L f i U

E ref p and p. are reference parameters for the distance

constraints, which assume the form of a typical square-well potential
for both upper and lower distance violations. The set of constraints is
completely general, and can represent the full combination of distance
constraints or smaller subsets of the defined restraints. The force field
energy function E ff can take on a number of forms, although the current
work employs the ECEPP/3 model [62].
The folding formulation represents a general nonconvex constrained
global optimization problem, a class of problems for which several meth-
ods have been developed. In this work, the formulations are solved via
the aBB deterministic global optimization approach, a branch and bound
method applicable to the identification of the global minimum of non-
linear optimization problems with twice-differentiable functions
[59,60,63-67]. In the aBB approach, a converging sequence of upper and
lower bounds is generated. The upper bounds on the global minimum
are obtained by local minimizations of the original nonconvex problem.
The lower bounds belong to the set of solutions of the convex lower
bounding problems that are constructed by augmenting the objective and
constraint functions through the addition of separable quadratic terms.

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