On Observability and Reconstruction of Promoter Activity Statistics from Reporter Protein Mean and Variance Profiles

Abstract

Reporter protein systems are widely used in biology for the indirect quantitative monitoring of gene expression activity over time. At the level of population averages, the relationship between the observed reporter concentration profile and gene promoter activity is established, and effective methods have been introduced to reconstruct this information from the data. At single-cell level, the relationship between population distribution time profiles and the statistics of promoter activation is still not fully investigated, and adequate reconstruction methods are lacking.

This paper develops new results for the reconstruction of promoter activity statistics from mean and variance profiles of a reporter protein. Based on stochastic modelling of gene expression dynamics, it discusses the observability of mean and autocovariance function of an arbitrary random binary promoter activity process. Mathematical relationships developed are explicit and nonparametric, i.e. free of a priori assumptions on the laws governing the promoter process, thus allowing for the decoupled analysis of the switching dynamics in a subsequent step. The results of this work constitute the essential tools for the development of promoter statistics and regulatory mechanism inference algorithms.

Appendices

A Definitions and Proofs

Matrix definitions. $A_{MP}$ $A_{MP,\times }$ $A_{MP,F}$ are given by

$$\begin{aligned} \begin{bmatrix} - d_M&0&0&0&0\\ k_P&- d_P&0&0&0\\ d_M&0&- 2\, d_M&0&0\\ k_P&d_P&0&- 2\, d_P&2\, k_P\\ 0&0&k_P&0&- d_M - d_P \end{bmatrix}, \quad \begin{bmatrix} 0&0\\ 0&0\\ 2\, k_M&0\\ 0&0\\ 0&k_M \end{bmatrix},\quad \begin{bmatrix} k_M&0\\ 0&0\\ k_M&0\\ 0&0\\ 0&0 \end{bmatrix}, \end{aligned}$$

in the same order, while

$$\begin{aligned} A_\otimes&= \begin{bmatrix} - d_M&0\\ k_P&- d_P \end{bmatrix},&A_{\times ,F}&= \begin{bmatrix} 0&k_M\\ 0&0 \end{bmatrix},&A_F&=\begin{bmatrix} - \alpha&0\\ \alpha -2\lambda _+&- 2\alpha \end{bmatrix}. \end{aligned}$$

Proof of Proposition 1 . Process F is a homogeneous continuous-time binary Markov chain. Letting $p(t)=\begin{bmatrix}\text {Prob}\{F(t)=0\}&\text {Prob}\{F(t)=1\} \end{bmatrix}^T$, for any t and $\tau$ it holds that

$$\begin{aligned} p(t)&=e^{Q(t-\tau )}p(\tau ),&Q&=\begin{bmatrix} -\lambda _+&\lambda _- \\ \lambda _+&-\lambda _- \end{bmatrix}. \end{aligned}$$

Mean $\mu _F=\text {Prob}\{F(t)=1\}$. Using the fact that $\dot{p}=Q p$, the differential equation for $\mu _F$, the second element of p, is $\dot{\mu }_F=\lambda _+(1-\mu _F)-\lambda _-\mu _F=-\alpha \mu _F+\lambda _+$. The solution of this equation relative to $\mu _F(0)$ yields the expression in the statement. Covariance $\rho _F(t,\tau )=\text {Prob}\{F(t)=1,F(\tau )=1\}-\mu _F(t)\mu _F(\tau )$. By Bayes’law, $\text {Prob}\{F(t)=1,F(\tau )=1\}=\text {Prob}\{F(t)={1|F(\tau )=1\}\cdot \text {Prob}\{F(\tau )=1\}}$. Second factor is equal to $\mu _F(\tau )$, while the first factor is given by the entry of row 2 and column 1 of $e^{Q(t-\tau )}$. Computing the matrix exponential thus yields the result. Stationary versions of $\mu _F$ and $\rho _F$ are found simply by taking the limit of $\mu _F(t)$ as $t\rightarrow +\infty$ and replacing the result for $\mu _F(\tau )$ and $\mu _F(t)$ in the expression of $\rho _F(t,\tau )$.

Proof of Proposition 2 . Starting from the second relation in (10),

$$\begin{aligned} \mathscr {M}_P(t)= & {} \mathbb {E}\big [\mathbb {E}[P^2|F]\big ]=\mathbb {E}\Big [\mathbb {E}\big [\big ((P-\mathbb {E}[P|F])+\mathbb {E}[P|F]\big )^2|F\big ]\Big ]\\= & {} \mathbb {E}\Big [\mathbb {E}\big [(P-\mathbb {E}[P|F])^2|F\big ]\Big ]+\mathbb {E}\Big [\mathbb {E}\big [\mathbb {E}[P|F]^2|F\big ]\Big ] \\+ & {} 2\cdot \mathbb {E}\Big [\mathbb {E}\big [(P-\mathbb {E}[P|F])\cdot \mathbb {E}[P|F]|F\big ]\Big ] \\= & {} \mathbb {E}\Big [\mathbb {E}\big [(P-\mathbb {E}[P|F])^2|F\big ]\Big ]+\mathbb {E}\Big [\mathbb {E}[P|F]^2\Big ]\\+ & {} 2\cdot \mathbb {E}\Big [\mathbb {E}\big [P-\mathbb {E}[P|F]|F\big ]\cdot \mathbb {E}[P|F]\Big ], \end{aligned}$$

where the last row vanishes since $\mathbb {E}\big [P-\mathbb {E}[P|F]|F\big ]=0$. Then, using the definitions of $\mu _P^F$ and $\sigma _{PP}^F$, the chain of equalities continues with

$$\begin{aligned} =\mathbb {E}\big [\sigma _{PP}^F(t)\big ]+\mathbb {E}\big [\big (\mu _P^F(t)\big )^2\big ] =\mathbb {E}\big [L_2^tF\big ]+\mathbb {E}\big [(L_1^tF)^2]= L_2^t\mu _F+\mathbb {E}[(L_1^tF)^2]. \end{aligned}$$

Proof of Proposition 3 . The following chain of inequalities hold:

$$\begin{aligned} \mathbb {E}[(L_1^tF)^2]= & {} \mathbb {E}\bigg [\bigg (L_1^t\Big (\sum _ia_i\phi _i\Big )\bigg )^2\bigg ]=\mathbb {E}\bigg [\sum _{i,j}L_1^t(a_i\phi _i)L_1^t(a_j\phi _j)\bigg ] \nonumber \\= & {} \sum _{i,j}\mathbb {E}[a_ia_j](L_1^t\phi _i)(L_1^t\phi _j)=\sum _i\sigma _i^2(L_1^t\phi _i)^2, \end{aligned}$$

(21)

where the latter equality follows from the mutual uncorrelation of the $a_i$.

Proof of Proposition 4 . Expanding the last term of (17) one gets

$$\begin{aligned} \mathbb {E}[(L_1^tF)^2]&=\int d\mathscr {P}_F(f) (L^t_1 f)^2 \\&=\int d\mathscr {P}_F(f) \left( \int _0^t d\tau \,\ell _1(t,\tau )f(\tau )\right) \left( \int _0^t dv\,\ell _1(t,v)f(v)\right) \\&=\int _0^t d\tau \int _0^t dv\, \ell _1(t,\tau ) \ell _1(t,v) \left( \int d\mathscr {P}_F(f)f(\tau )f(v)\right) \\&= \int _0^t d\tau \int _0^t dv\, \ell _1(t,\tau ) \ell _1(t,v) \big (\rho _F(\tau ,v)+\mu _F(\tau )\mu _F(v)\big ) \end{aligned}$$

where the last integrand is of course the autocorrelation of F at $\tau$ and v. Therefore

$$\begin{aligned} \sigma _{PP}(t)&=\mathscr {M}_P(t)-\mu _P^2(t) \\&=L^t_2\mu _F+\int _0^t d\tau \int _0^t dv\, \ell _1(t,\tau ) \ell _1(t,v) \big (\rho _F(\tau ,v)+\mu _F(\tau )\mu _F(v)\big ) \\&\quad - \left( \int _0^t d\tau \,\ell _1(t,\tau )\mu _F(\tau )\right) \left( \int _0^t dv\,\ell _1(t,v)\mu _F(v)\right) , \end{aligned}$$

and the result follows by collecting integrals and simplifying.

B Laplace Sensitivity Method for the Analysis of Parameter Identifiability

This section reports the identifiability analysis method of [2]. Let $\mathscr {Y}_\theta (t)$ be a vector function of $t\in \mathbb {R}$ depending on parameters $\theta$. Typically $\mathscr {Y}_\theta (\cdot )$ is an observed response of a dynamical system defined in terms of $\theta$.

Definition 1

The parametric family (of functions) $\{\mathscr {Y}_\theta :~\theta \in {\varTheta }\}$, with ${\varTheta }\subseteq \mathbb {R}^N$, $N\in \mathbb {N}$, is

• (a) locally identifiable at $\theta ^*$ if a neighborhood $B_{\theta ^*}\subseteq {\varTheta }$ of $\theta ^*$ exists such that the implication holds $\forall \theta \in B_{\theta ^*}$;

• (b) locally identifiable if (a) holds for almost every (a.e.) $\theta ^*\in {\varTheta }$.

For any given $\theta$ let $Y(s,\theta )$ be the Laplace transform of $\mathscr {Y}_\theta (\cdot )$. Let $\nabla Y(s,\theta )=\frac{\partial Y}{\partial \theta }(s,\theta )=\left[ \frac{\partial Y}{\partial \theta _1}~\cdots ~\frac{\partial Y}{\partial \theta _N} \right] (s,\theta ).$

Proposition 5

If, for some $L\in \mathbb {N}$, a set of points (or $\mathbb {C}$) exists such that the matrix

has full column rank, then $\{\mathscr {Y}_\theta :~\theta \in {\varTheta }\}$ is locally identifiable at $\theta ^*$ (in the sense of Definition 1(a)).

Now assume that the elements of $Y(s,\theta )$ are ratios of polynomials in the entries of $\theta$.

Corollary 1

If, for a given set of points and a given $\theta ^*$, matrix is full column rank, then $\{\mathscr {Y}_\theta :~\theta \in {\varTheta }\}$ is locally identifiable (a.e. in the sense of Definition 1(b)).

In the present paper, the Laplace transforms that are used to discuss identifiability belong to this last class (see [2]), whence Corollary 1 applies. In practice, these conditions can be easily checked by the use of the Matlab Symbolic Math Toolbox and evaluation of the rank conditions based on a finite set of heuristically chosen points  (see again [2]).