There are at least a couple of definitions in the literature for an $E_2$-algebra, also known as a homotopy Gerstenhaber algebra, also known as $G_{\infty}$-algebra.

Suppose $V$ is a graded vector space. Let $S(V), \text{Lie}(v)$ denote the graded symmetric algebra and the graded free Lie algebra of $V$ respectively and $S_+(V)$ be the symmetric algebra in strictly positive degree.

$\textbf{Definition (Tamarkin)}$ A $G_{\infty}$-algebra structure on $V$ is a degree $+1$ map $$ \delta: S_+(\text{Lie}(V^*[1])[1]) \rightarrow S_+(\text{Lie}(V^*[1])[1]) $$ which behaves as a derivation of both $\cdot$ and $[\,,]$ such that $\delta^2 = 0$.

Unpacking this leads to saying that we have a collection of maps $$ m_{k_1, \dots, k_n}: V^{\otimes k_1} \otimes \dots \otimes V^{\otimes k_n} \rightarrow V$$ of degree $3-(k_1 + \dots k_n + n)$ obeying appropriate symmetry and associativity relations.

The second definition pertains to algebras over the little disk operad. Let $D_2(k)$ denote the configuration space of $k$ little disks inside a big disk and let $\text{Chains}_{\bullet}(D_2(k))$ be the singular chain complex. Letting $\mathcal{P}(k) = \text{Chains}_{\bullet}(D_2(k))$, one can define an operadic structure on this collection of vector spaces. This leads one to the

$\textbf{Definition (Getzler-Jones?)}$ An $E_2$-algebra is an algebra over the operad $\text{Chains}(D_2)$.

How can one show that these two definitions are equivalent?

Short of a full proof of equivalence, it would be nice to understand a description of the cycle in $D_2(k_1 + \dots +k_n)$ which corresponds to the map $m_{k_1, \dots, k_n}$. For example: if one were working in $H_{\bullet}(D_2)$, the homology operad, one associates to the point class in configuration space of two disks the operation $\cdot = m_2$, and to the cycle involving one little disk going around the other the bracket $[\,,] = m_{1,1}$ in the Gerstenhaber algebra. Is there an explicit description of the cycle corresponding to $m_{k_1, \dots, k_n}$?