Algebraic K-theory of an exact category $\mathcal{C}$ is a certain universal non-connective spectrum $K(\mathcal{C})$. In particular, objects of $\mathcal{C}$ give elements of $K_0(\mathcal{C})$.

There are models for the delooping of $K(\mathcal{C})$, i.e. spectra $X(\mathcal{C})$ such that $\Omega X(\mathcal{C})\cong K(\mathcal{C})$. For instance, one can take $X(\mathcal{C})$ to be the K-theory of Tate objects in $\mathcal{C}$ (see http://arxiv.org/abs/1203.0831). This, in particular, explains that Tate objects of $\mathcal{C}$ give rise to elements of $K_{-1}(\mathcal{C})$.

Are there models for the looping of K-theory? More precisely, is there an exact category $\mathcal{D}$ constructed from $\mathcal{C}$ such that $K(\mathcal{D})\cong \Omega K(\mathcal{C})$?

I am mainly interested in the case when $\mathcal{C}$ is the category of finitely-generated projective modules over a ring $R$.