The finite state projection based Fisher information matrix approach to estimate information and optimize single-cell experiments

Abstract

Modern optical imaging experiments not only measure single-cell and single-molecule dynamics with high precision, but they can also perturb the cellular environment in myriad controlled and novel settings. Techniques, such as single-molecule fluorescence in-situ hybridization, microfluidics, and optogenetics, have opened the door to a large number of potential experiments, which begs the question of how to choose the best possible experiment. The Fisher information matrix (FIM) estimates how well potential experiments will constrain model parameters and can be used to design optimal experiments. Here, we introduce the finite state projection (FSP) based FIM, which uses the formalism of the chemical master equation to derive and compute the FIM. The FSP-FIM makes no assumptions about the distribution shapes of single-cell data, and it does not require precise measurements of higher order moments of such distributions. We validate the FSP-FIM against well-known Fisher information results for the simple case of constitutive gene expression. We then use numerical simulations to demonstrate the use of the FSP-FIM to optimize the timing of single-cell experiments with more complex, non-Gaussian fluctuations. We validate optimal simulated experiments determined using the FSP-FIM with Monte-Carlo approaches and contrast these to experiment designs chosen by traditional analyses that assume Gaussian fluctuations or use the central limit theorem. By systematically designing experiments to use all of the measurable fluctuations, our method enables a key step to improve co-design of experiments and quantitative models.

Optimal Design of Single-Cell Experiments within Temporally Fluctuating Environments

“…2 with respect to θ to find the maximum likelihood estimates (MLE) of the parameters, which will vary depending on each new set of experimental data. We next demonstrate how this likelihood function and the FSP model of the HOG-MAPK system can be used to design optimal smFISH experiments using the FSP-based FIM [6].…”

“…The Fisher information matrix (FIM), is a common tool in engineering and statistics to estimate parameter uncertainties prior to collecting data, and which allows one to find experimental settings that can make these uncertainties as small as possible [3, 4, 30–33]. Recently, it has been applied to biological systems to estimate kinetic rate parameters in stochastic gene expression systems [3–6, 34]. In general, the FIM for a single measurement is defined: where log p ( θ ) is the log-likelihood of observing that measurement, and the expectation is taken across over the probability distribution of states p ( θ ) assuming the specific parameter set θ .…”

“…A systematic approach to solve this problem is model-driven experiment design, in which one uses an assumed (and potentially incorrect) mathematical model of the system to estimate and optimize the value of potential experimental settings. In recent years, model-driven experiment design has gained traction for biological models of gene expression, whether in the Bayesian setting [1] or using Fisher information for deterministic models [2], and even in the stochastic, single-cell setting [3–6]. Despite the promise and active development of model-driven experiment design from the theoretical perspective, more general, yet biologically-inspired approaches are needed to make these methods suitable for the experimental community at large.…”

“…Furthermore, when a dynamic response is under investigation, the specific times at which measurements should be taken (i.e., the times after induction at which cells should be fixed and analyzed) is also unclear. In this work, we use the newly developed finite state projection based Fisher information matrix (FSP-FIM, [6]) to optimize these experimental quantities for osmotic stress response genes in yeast.…”

“…In this work, we use the chemical master equation framework [24] of stochastic chemical kinetics, which has been the workhorse of stochastic modeling of gene expression, whether through simulated sample paths of its solution via the stochastic simulation algorithm [25], moment approximations [7, 26], or finite state projections (FSP) [27]. Recently, it has come to light that for some systems it is critical to consider the full distribution of biomolecules across cellular populations when fitting CME-based models [6, 9], which can be done with guaranteed errors using the FSP approach [27, 28]. This method truncates a CME into a finite state, continuous time Markov chain, for which the set of ordinary differential equations, describes the flow of probability among all of the most likely observable states for the system.…”