Inferring Stochastic Rates from Heterogeneous Snapshots of Particle Positions

Abstract

Many imaging techniques for biological systems—like fixation of cells coupled with fluorescence microscopy—provide sharp spatial resolution in reporting locations of individuals at a single moment in time but also destroy the dynamics they intend to capture. These snapshot observations contain no information about individual trajectories, but still encode information about movement and demographic dynamics, especially when combined with a well-motivated biophysical model. The relationship between spatially evolving populations and single-moment representations of their collective locations is well-established with partial differential equations (PDEs) and their inverse problems. However, experimental data is commonly a set of locations whose number is insufficient to approximate a continuous-in-space PDE solution. Here, motivated by popular subcellular imaging data of gene expression, we embrace the stochastic nature of the data and investigate the mathematical foundations of parametrically inferring demographic rates from snapshots of particles undergoing birth, diffusion, and death in a nuclear or cellular domain. Toward inference, we rigorously derive a connection between individual particle paths and their presentation as a Poisson spatial process. Using this framework, we investigate the properties of the resulting inverse problem and study factors that affect quality of inference. One pervasive feature of this experimental regime is the presence of cell-to-cell heterogeneity. Rather than being a hindrance, we show that cell-to-cell geometric heterogeneity can increase the quality of inference on dynamics for certain parameter regimes. Altogether, the results serve as a basis for more detailed investigations of subcellular spatial patterns of RNA molecules and other stochastically evolving populations that can only be observed for single instants in their time evolution.

Appendix A Numerical Implementation Details

1.1 A.1 Stochastic Simulation Algorithm for Synthetic Data

Algorithm 1 describes the generation procedure for synthetic data used in the inference setups of Figs. 5 and 6. Importantly, we simulate trajectories of the particle process, and do inference using the point process theory from Theorem 2.1. The simulation procedure is exact except for a small $\Delta _t$-controlled approximation error in computing whether a Brownian path has exited the domain (Smith and Grima 2019). In all simulations, we take $\Delta _t=10^{-5}$, several orders of magnitude smaller than any other timescale.

Algorithm 1

Stochastic simulation to generate particle positions for synthetic data.

1.2 A.2 Metropolis-in-Gibbs Sampling for Bayesian Inference

See Sect. 3.2 for an explanation of prior choices for $\lambda , \mu$. The choice of prior for $\lambda$ provides a (marginal) conjugate posterior that can be computed analytically, so a Gibbs-in-Metropolis MCMC approach is used and described in Algorithm 2.

Algorithm 2

MCMC Gibbs-in-Metropolis sampling of posterior distribution.

Fig. 7

Rubin-Gelman $\hat{R}$-1 demonstrating convergence of MCMC (on the setup in Fig. 5 with $M=10$) for two chains. Convergence is considered good when $\hat{R}-1<10^{-1}$, achieved in approximately $10^3$ MCMC steps

As is standard practice (Gelman et al. 1995), we do a sequence of warm-up phases to target the range of acceptance probabilities between 0.3 and 0.5, starting with $\sigma =1$ and rounds of 500 steps. After each warm-up round, if the acceptance frequency is below the target range, $\sigma \rightarrow \sigma /2$, and if above $\sigma \rightarrow 2\sigma$. Once in the range, the ending values of $\lambda ,\mu$ are used as initial values, and the full algorithm is run with this step size $\sigma$ for a much greater number of steps. The values from the chains for this burn-in phase are discarded.

To choose the number of MCMC steps, we investigate the Gelman-Rubin diagnostic $\hat{R}$ (Gelman et al. 1995) for the presumed worst-case scenario of $M=10$ snapshots in the setup of Fig. 5. For $M=10$, we initialize multiple chains and compute $\hat{R}$ as a function of the number of MCMC steps, shown in Fig. 7. Convergence is considered satisfactory for $\hat{R}<1.1$ (Gelman et al. 1995), which is achieved around $10^3$ MCMC steps. We take $10^5$ MCMC steps for all setups in the main text and do not monitor convergence further.