Given a model category $\mathcal{M}$, Goerss and Hopkins constructed a subcategory (see Structured Ring Spectra, p. 160) $\mathbf{E}$ of $\mathcal{M}$ such that:
If $X\in\mathbf{E}$ and $Y$ is weakly equivalent to $X$, then $Y\in\mathbf{E}$.
A morphism $f$ in $\mathcal{M}$ is in $\mathbf{E}$ if and only if it is a weak equivalence.
If $\mathbf{E}$ is the category whose objects are $\mathcal{M}$ and morphisms are the weak equivalences, then $\mathbf{E}$ is written $\mathbf{E}(\mathcal{M})$. My question is as follows: is it possible to recover the model category $\mathcal{M}$ from $\mathbf{E}(\mathcal{M})$? Is there any nontrivial model structure on $\mathcal{M}$ that is useful/interesting?
A long comment: When you have a model category $\mathcal{M}$ in particular you have an enriched $Ho(sSet)$-module structure on $Ho(\mathcal{M})$ (in the sense of Hovey) i.e. the homotopy category $Ho(\mathcal{M})$ is tensored and cotensored over $Ho(sSet)$ in a compatible way. Let me use the following analogy: suppose that $R$ is a ring and $M$ is an abelian group and imagine you have an action of some elements of $R$ on some elements of $M$ and you are asking the question wether this action is a restriction of honest $R$-module structure on $M$. So your question (if I understand it) should be formulated as follows:
Suppose that we have a (co)complete category $\mathcal{M}$ and a subcategory $ \mathbf{E}(\mathcal{M})$ such that the category $\mathcal{M}[\mathbf{E}(\mathcal{M})^{-1}] $ exists , does it come from a model structure on $\mathcal{M}$ such that $Ho(\mathcal{M})\simeq \mathcal{M}[\mathbf{E}(\mathcal{M})^{-1}]$?
Resume:
1) In order to have a model structure on $\mathcal{M}$ whit a subcategory $ \mathbf{E}(\mathcal{M})$ of weak equivalences such that $\mathcal{M}[\mathbf{E}(\mathcal{M})^{-1}] $ exists a necessary condition is to have an enriched $Ho(sSet)$-module structure on $\mathcal{M}[\mathbf{E}(\mathcal{M})^{-1}]$.
2) The Homotopy category $Ho(\mathcal{M})$ depends on the class of weak equivalences but the additional structure of enriched $Ho(sSet)$-module structure on $Ho(\mathcal{M})$ is given by the class of cofibrations (fibrations).