Estimation of biochemical network parameter distributions in cell populations

Populations of heterogeneous cells play an important role in many biological systems. In this paper we consider systems where each cell can be modelled by an ordinary differential equation. To account for heterogeneity, parameter values are different among individual cells, subject to a distribution function which is part of the model specification. Experimental data for heterogeneous cell populations can be obtained from flow cytometric fluorescence microscopy. We present a heuristic approach to use such data for estimation of the parameter distribution in the population. The approach is based on generating simulation data for samples in parameter space. By convex optimisation, a suitable probability density function for these samples is computed. To evaluate the proposed approach, we consider artificial data from a simple model of the tumor necrosis factor (TNF) signalling pathway. Its main characteristic is a bimodality in the TNF response: a certain percentage of cells undergoes apoptosis upon stimulation, while the remaining part stays alive. We show how our modelling approach allows to identify the reasons that underly the differential response.

💡 Research Summary

The paper addresses a fundamental challenge in systems biology: how to model and infer the heterogeneity that naturally exists among cells within a population. Traditional approaches often collapse a population into a single set of ordinary differential equations (ODEs) with averaged parameters, thereby discarding the variability that can drive dramatically different phenotypic outcomes such as cell death versus survival. The authors propose a data‑driven framework that treats each cell as an instance of the same ODE system but with its own parameter vector drawn from an unknown probability distribution. The central goal is to reconstruct this distribution directly from experimental measurements obtained by flow cytometric fluorescence microscopy, which provides high‑throughput single‑cell readouts of molecular reporters at one or several time points.

Methodological pipeline

1. Parameter space sampling – The authors generate a finite set of candidate parameter vectors using uniform grid sampling or Latin hypercube designs. Each sampled vector, denoted (\theta^{(j)}), represents a hypothetical subpopulation of cells.
2. Forward simulation – For every (\theta^{(j)}) the ODE model is numerically integrated, producing simulated trajectories of observable quantities (e.g., fluorescence intensity of a reporter protein). The simulated observables are then discretized into histograms that mimic the experimental measurement process.
3. Convex optimization of weights – The experimental histogram (H_{\text{exp}}) is approximated as a convex combination of the simulated histograms:
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