  In this article we study the Gieseker-Maruyama moduli spaces
$\mathcal{B}(e,n)$ of stable rank 2 algebraic vector bundles with Chern classes
$c_1=e\in\{-1,0\},\ c_2=n\ge1$ on the projective space $\mathbb{P}^3$. We
construct two new infinite series $\Sigma_0$ and $\Sigma_1$ of irreducible
components of the spaces $\mathcal{B}(e,n)$, for $e=0$ and $e=-1$,
respectively. General bundles of these components are obtained as cohomology
sheaves of monads, the middle term of which is a rank 4 symplectic instanton
bundle in case $e=0$, respectively, twisted symplectic bundle in case $e=-1$.
We show that the series $\Sigma_0$ contains components for all big enough
values of $n$ (more precisely, at least for $n\ge146$). $\Sigma_0$ yields the
next example, after the series of instanton components, of an infinite series
of components of $\mathcal{B}(0,n)$ satisfying this property.
