  In this article we investigate a system of geometric evolution equations
describing a curvature driven motion of a family of 3D curves in the normal and
binormal directions. Evolving curves may be subject of mutual interactions
having both local or nonlocal character where the entire curve may influence
evolution of other curves. Such an evolution and interaction can be found in
applications. We explore the direct Lagrangian approach for treating the
geometric flow of such interacting curves. Using the abstract theory of
nonlinear analytic semi-flows, we are able to prove local existence, uniqueness
and continuation of classical H\"older smooth solutions to the governing system
of nonlinear parabolic equations. Using the finite volume method, we construct
an efficient numerical scheme solving the governing system of nonlinear
parabolic equations. Additionally, a nontrivial tangential velocity is
considered allowing for redistribution of discretization nodes. We also present
several computational studies of the flow combining the normal and binormal
velocity and considering nonlocal interactions.
