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Modeling population dynamics is a fundamental problem with broad scientific applications. Motivated by real-world applications including biosystems with diverse populations, we consider a class of population dynamics modeling with two technical challenges: (i) dynamics to learn for individual particles are heterogeneous and (ii) available data to learn from are not time-series (i.e, each individual’s state trajectory over time) but cross-sectional (i.e, the whole population’s aggregated states without individuals matched over time). To address the challenges, we introduce a novel computational framework dubbed correlational Lagrangian Schrödinger bridge (CLSB) that builds on optimal transport to “bridge" cross-sectional data distributions. In contrast to prior methods regularizing all individuals’ transport “costs” and then applying them to the population homogeneously, CLSB directly regularizes population cost allowing for population heterogeneity and potentially improving model generalizability. Specifically our contributions include (1) a novel population perspective of the transport cost and a new class of population regularizers capturing the temporal variations in multivariate relations, with the tractable formulation derived, (2) three domain-informed instantiations of population regularizers on covariance, and (3) integration of population regularizers into data-driven generative models as constrained optimization and an approximate numerical solution, with further extension to conditional generative models. Empirically, we demonstrate the superiority of CLSB in single-cell sequencing data analyses (including cell differentiation and drug-conditioned cell responses) and opinion depolarization.
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Numerical Approaches to Division and Label Structured Population Models
Division and label structured population models (DLSPMs) are a class of partial differential equations (PDEs) that have been used to study intracellular dynamics in dividing cells. DLSPMs have improved the understanding of cell proliferation assays involving measurements such as fluorescent label decay, protein production, and prion aggregate amplification. One limitation in using DLSPMs is the significant computational time required for numerical approximations, especially for models with complex biologically relevant dynamics. Here we develop a novel numerical and theoretical framework involving a recursive formulation for a class of DLSPMs. We develop this framework for a population of dividing cells with an arbitrary functional form describing the intracellular dynamics. We found that, compared to previous methods, our framework is faster and more accurate. We illustrate our approach on three common models for intracellular dynamics and discuss the potential impact of our findings in the context of data-driven methods for parameter estimation.
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- Award ID(s):
- 1838314
- PAR ID:
- 10302340
- Date Published:
- Journal Name:
- Letters in biomathematics
- Volume:
- 7
- Issue:
- 1
- ISSN:
- 2373-7867
- Page Range / eLocation ID:
- 153-170
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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