Mathematical modeling is a powerful tool for understanding and predicting complex dynamical systems, ranging from gene regulatory networks to population-level dynamics. However, model predictions are highly sensitive to initial conditions, which are often unknown. In infectious disease models, for instance, the initial number of exposed individuals (E) at the time the model simulation starts is frequently unknown. This initial condition has often been estimated using an unrealistic, history-independent assumption for simplicity: the chance that an exposed individual becomes infectious is the same regardless of the timing of their exposure (i.e., exposure history). Here, we show that this history-independent method can yield serious bias in the estimation of the initial condition. To address this, we developed a history-dependent initial condition estimation method derived from a master equation expressing the time-varying likelihood of becoming infectious during a latent period. Our method consistently outperformed the history-independent method across various scenarios, including those with measurement errors and abrupt shifts in epidemics, for example, due to vaccination. In particular, our method reduced estimation error by 55% compared to the previous method in real-world COVID-19 data from Seoul, Republic of Korea, which includes likely infection dates, allowing us to obtain the true initial condition. This advancement of initial condition estimation enhances the precision of epidemic modeling, ultimately supporting more effective public health policies. We also provide a user-friendly package, Hist-D, to facilitate the use of this history-dependent initial condition estimation method.

MOTIVATION: Cell function is regulated by gene regulatory networks (GRNs) defined by protein-mediated interaction between constituent genes. Despite advances in experimental techniques, we can still measure only a fraction of the processes that govern GRN dynamics. To infer the properties of GRNs using partial observation, unobserved sequential processes can be replaced with distributed time delays, yielding non-Markovian models. Inference methods based on the resulting model suffer from the curse of dimensionality. RESULTS: We develop a simulation-based Bayesian MCMC method employing an approximate likelihood for the efficient and accurate inference of GRN parameters when only some of their products are observed. We illustrate our approach using a two-step activation model: an activation signal leads to the accumulation of an unobserved regulatory protein, which triggers the expression of observed fluorescent proteins. With prior information about observed fluorescent protein synthesis, our method successfully infers the dynamics of the unobserved regulatory protein. We can estimate the delay and kinetic parameters characterizing target regulation including transcription, translation, and target searching of an unobserved protein from experimental measurements of the products of its target gene. Our method is scalable and can be used to analyze non-Markovian models with hidden components. AVAILABILITY AND IMPLEMENTATION: Our code is implemented in R and is freely available with a simple example data at https://github.com/Mathbiomed/SimMCMC.

MOTIVATION: Simultaneous recordings of gene network dynamics across large populations have revealed that cell characteristics vary considerably even in clonal lines. Inferring the variability of parameters that determine gene dynamics is key to understanding cellular behavior. However, this is complicated by the fact that the outcomes and effects of many reactions are not observable directly. Unobserved reactions can be replaced with time delays to reduce model dimensionality and simplify inference. However, the resulting models are non-Markovian, and require the development of new inference techniques. RESULTS: We propose a non-Markovian, hierarchical Bayesian inference framework for quantifying the variability of cellular processes within and across cells in a population. We illustrate our approach using a delayed birth-death process. In general, a distributed delay model, rather than a popular fixed delay model, is needed for inference, even if only mean reaction delays are of interest. Using in silico and experimental data we show that the proposed hierarchical framework is robust and leads to improved estimates compared to its non-hierarchical counterpart. We apply our method to data obtained using time-lapse microscopy and infer the parameters that describe the dynamics of protein production at the single cell and population level. The mean delays in protein production are larger than previously reported, have a coefficient of variation of around 0.2 across the population, and are not strongly correlated with protein production or growth rates. AVAILABILITY AND IMPLEMENTATION: Accompanying code in Python is available at https://github.com/mvcortez/Bayesian-Inference. CONTACT: [email protected] or [email protected] or [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Advances in experimental and imaging techniques have allowed for unprecedented insights into the dynamical processes within individual cells. However, many facets of intracellular dynamics remain hidden, or can be measured only indirectly. This makes it challenging to reconstruct the regulatory networks that govern the biochemical processes underlying various cell functions. Current estimation techniques for inferring reaction rates frequently rely on marginalization over unobserved processes and states. Even in simple systems this approach can be computationally challenging, and can lead to large uncertainties and lack of robustness in parameter estimates. Therefore we will require alternative approaches to efficiently uncover the interactions in complex biochemical networks. RESULTS: We propose a Bayesian inference framework based on replacing uninteresting or unobserved reactions with time delays. Although the resulting models are non-Markovian, recent results on stochastic systems with random delays allow us to rigorously obtain expressions for the likelihoods of model parameters. In turn, this allows us to extend MCMC methods to efficiently estimate reaction rates, and delay distribution parameters, from single-cell assays. We illustrate the advantages, and potential pitfalls, of the approach using a birth-death model with both synthetic and experimental data, and show that we can robustly infer model parameters using a relatively small number of measurements. We demonstrate how to do so even when only the relative molecule count within the cell is measured, as in the case of fluorescence microscopy. AVAILABILITY AND IMPLEMENTATION: Accompanying code in R is available at https://github.com/cbskust/DDE_BD. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.