The details of this approach are not valid, since we require reversability to use this form for the variance.
excised text from review draft:
% Using Sanchez Approach
Since only one state in the gestation cycle is transcriptionally competent, all but one element of $\vec{r}$ in equations \ref{eq:Sanchez2} and \ref{eq:Sanchez3} is zero. The mean expression level has the simple form,
\begin{equation}
\mu_m = \frac{k_t}{\delta} \frac{1}{ \sum\limits_{i=1}^N i k_b^{N-i} k_f^{i-1} }
\end{equation}
Consequently the second moment of the mRNA distribution depends only on the $(N,N)$ element of inverted, modified transition matrix,
\begin{equation}
<m^2> = \mu_m – 1/\delta*kt^2 \frac{1}{\sum\limits_{i=1}^N i k_b^{N-i} k_f^{i-1} \left[(M^T-\delta \mathbb{I})\right]_{N,N}}
\end{equation}
Denoting $A_{N,N} = \left[(M^T-\delta \mathbb{I})\right]_{N,N}$ and denoting by $B$ submatrix generated by leaving off the last row and last column from $A$,
\begin{equation}
A^{-1}{N,N} = \frac{\det(B)}{\det{A}} = \frac{\det(B)}{\det(B) (-k_f-k_b-\delta-B{N,1}^{-1}k_f^2 – B_{N,N}^{-1} k_f k_b)}
\end{equation}
which since $B$ is tridiagonal we can evaluate by solving the recurrence relation for the inverse of tridiagonal matrices as presented by \citet{Usmani1994}. The resulting explicit expressions for the second moment, the variance, and the coefficient of variation as a function of the rate parameters is not especially compact.
%\begin{equation}
%A^{-1}{N,N} = \left( -k_f -k_b – \delta – (-1)^N k_f^N\frac{1}{theta{N-1}} – \frac{\theta_{N-2}}{\theta_{N-1}} k_f k_b \right)^{-1}
%\end{equation}
%Note that $\theta_{i}$ come from the solutions of the recurrence relation, and that $\theta_{N-1} = \det(B)$.
%\begin{equation}
%\right[(M^T – \delta \mathbb{I})^{-1}\left]{ij} =
%\begin{cases}
%(-1)^{i+j}b_i\ldots b{j-1},\theta_{i-1}\phi_{j+1}/\theta_n & \text{if } i \leq j\
%(-1)^{i+j}c_j\ldots c_{i-1},\theta_{j-1}\phi_{i+1}/\theta_n & \text{if } i \g j\
%\end{cases}
%\end{equation}
%Where $b_i = k_b$, $c_i=k_f$ and the $\theta_i$ and $\phi_i$ are given by the recurrence relation,
%\begin{equation}
%\theta_i = a_i \theta_{i-1} = b_{i-1}c_{i-1}\theta_{i-2}
%\end{equation}