Estimation methods for heterogeneous cell population models in systems biology

An integrative phenotype-structured partial differential equation model for the population dynamics of epithelial-mesenchymal transition

Phenotypic heterogeneity along the epithelial-mesenchymal (E-M) axis contributes to cancer metastasis and drug resistance. Recent experimental efforts have collated detailed time-course data on the emergence and dynamics of E-M heterogeneity in a cell population. However, it remains unclear how different intra- and inter-cellular processes shape the dynamics of E-M heterogeneity. Here, using Cell Population Balance model, we capture the dynamics of cell density along E-M phenotypic axis resulting from interplay between-(a) intracellular regulatory interaction among biomolecules, (b) cell division and death and (c) stochastic cell-state transition. We find that while the existence of E-M heterogeneity depends on intracellular regulation, heterogeneity gets enhanced with stochastic cell-state transitions and diminished by growth rate differences. Further, resource competition among E-M cells can lead to both bi-phasic growth of the total population and/or bi-stability in the phenotypic composition. Overall, our model highlights complex interplay between cellular processes shaping dynamic patterns of E-M heterogeneity.

A local polynomial moment approximation for compartmentalized biochemical systems

Compartmentalized biochemical reactions are a ubiquitous building block of biological systems. The interplay between chemical and compartmental dynamics can drive rich and complex dynamical behaviors that are difficult to analyze mathematically - especially in the presence of stochasticity. We have recently proposed an effective moment equation approach to study the statistical properties of compartmentalized biochemical systems. So far, however, this approach is limited to polynomial rate laws and moreover, it relies on suitable moment closure approximations, which can be difficult to find in practice. In this work we propose a systematic method to derive closed moment dynamics for compartmentalized biochemical systems. We show that for the considered class of systems, the moment equations involve expectations over functions that factorize into two parts, one depending on the molecular content of the compartments and one depending on the compartment number distribution. Our method exploits this structure and approximates each function with suitable polynomial expansions, leading to a closed system of moment equations. We demonstrate the method using three systems inspired by cell populations and organelle networks and study its accuracy across different dynamical regimes.

Population balance modelling captures host cell protein dynamics in CHO cell cultures

Monoclonal antibodies (mAbs) have been extensively studied for their wide therapeutic and research applications. Increases in mAb titre has been achieved mainly by cell culture media/feed improvement and cell line engineering to increase cell density and specific mAb productivity. However, this improvement has shifted the bottleneck to downstream purification steps. The higher accumulation of the main cell-derived impurities, host cell proteins (HCPs), in the supernatant can negatively affect product integrity and immunogenicity in addition to increasing the cost of capture and polishing steps. Mathematical modelling of bioprocess dynamics is a valuable tool to improve industrial production at fast rate and low cost. Herein, a single stage volume-based population balance model (PBM) has been built to capture Chinese hamster ovary (CHO) cell behaviour in fed-batch bioreactors. Using cell volume as the internal variable, the model captures the dynamics of mAb and HCP accumulation extracellularly under physiological and mild hypothermic culture conditions. Model-based analysis and orthogonal measurements of lactate dehydrogenase activity and double-stranded DNA concentration in the supernatant show that a significant proportion of HCPs found in the extracellular matrix is secreted by viable cells. The PBM then served as a platform for generating operating strategies that optimise antibody titre and increase cost-efficiency while minimising impurity levels.

Computer-aided design of reversible hybridization chain reaction (CAD-HCR) enables multiplexed single-cell spatial proteomics imaging

In situ spatial proteomics analysis of a single cell has not been achieved yet, mainly because of insufficient throughput and sensitivity of current techniques. Recent progress on immuno-nucleic acid amplification technology presents tremendous opportunities to address this issue. Here, we report an innovative hybridization chain reaction (HCR) technique that involves computer-aided design (CAD) and reversible assembly. CAD enables highly multiplexed HCR with a sequence database that can work in parallel, while reversible assembly enables the switching of HCR between a working state and a resting state. Thus, CAD-HCR has been successfully adopted for single-cell spatial proteomics analysis. The fluorescence signal of CAD-HCR is comparable with conventional immunofluorescence, and it is positively correlated with the abundance of target proteins, which is beneficial for the visualization of proteins. The method developed here expands the toolbox of single-cell analysis and proteomics studies, as well as the performance and application of HCR.

Two-Stage Approaches to Accounting for Patient Heterogeneity in Machine Learning Risk Prediction Models in Oncology

PURPOSE

Machine learning models developed from electronic health records data have been increasingly used to predict risk of mortality for general oncology patients. But these models may have suboptimal performance because of patient heterogeneity. The objective of this work is to develop a new modeling approach to predicting short-term mortality that accounts for heterogeneity across multiple subgroups in the presence of a large number of electronic health record predictors.

METHODS

We proposed a two-stage approach to addressing heterogeneity among oncology patients of different cancer types for predicting their risk of mortality. Structured data were extracted from the University of Pennsylvania Health System for 20,723 patients of 11 cancer types, where 1,340 (6.5%) patients were deceased. We first modeled the overall risk for all patients without differentiating cancer types, as is done in the current practice. We then developed cancer type-specific models using the overall risk score as a predictor along with preselected type-specific predictors. The overall and type-specific models were compared with respect to discrimination using the area under the precision-recall curve (AUPRC) and calibration using the calibration slope. We also proposed metrics that characterize the degree of risk heterogeneity by comparing risk predictors in the overall and type-specific models.

RESULTS

The two-stage modeling resulted in improved calibration and discrimination across all 11 cancer types. The improvement in AUPRC was significant for hematologic malignancies including leukemia, lymphoma, and myeloma. For instance, the AUPRC increased from 0.358 to 0.519 (∆ = 0.161; 95% CI, 0.102 to 0.224) and from 0.299 to 0.354 (∆ = 0.055; 95% CI, 0.009 to 0.107) for leukemia and lymphoma, respectively. For all 11 cancer types, the two-stage approach generated well-calibrated risks. A high degree of heterogeneity between type-specific and overall risk predictors was observed for most cancer types.

CONCLUSION

Our two-stage modeling approach that accounts for cancer type-specific risk heterogeneity has improved calibration and discrimination than a model agnostic to cancer types.

Uncertainty propagation for deterministic models of biochemical networks using moment equations and the extended Kalman filter

Differential equation models of biochemical networks are frequently associated with a large degree of uncertainty in parameters and/or initial conditions. However, estimating the impact of this uncertainty on model predictions via Monte Carlo simulation is computationally demanding. A more efficient approach could be to track a system of low-order statistical moments of the state. Unfortunately, when the underlying model is nonlinear, the system of moment equations is infinite-dimensional and cannot be solved without a moment closure approximation which may introduce bias in the moment dynamics. Here, we present a new method to study the time evolution of the desired moments for nonlinear systems with polynomial rate laws. Our approach is based on solving a system of low-order moment equations by substituting the higher-order moments with Monte Carlo-based estimates from a small number of simulations, and using an extended Kalman filter to counteract Monte Carlo noise. Our algorithm provides more accurate and robust results compared to traditional Monte Carlo and moment closure techniques, and we expect that it will be widely useful for the quantification of uncertainty in biochemical model predictions.

Pancreatic Ppy-expressing γ-cells display mixed phenotypic traits and the adaptive plasticity to engage insulin production

The cellular identity of pancreatic polypeptide (Ppy)-expressing γ-cells, one of the rarest pancreatic islet cell-type, remains elusive. Within islets, glucagon and somatostatin, released respectively from α- and δ-cells, modulate the secretion of insulin by β-cells. Dysregulation of insulin production raises blood glucose levels, leading to diabetes onset. Here, we present the genetic signature of human and mouse γ-cells. Using different approaches, we identified a set of genes and pathways defining their functional identity. We found that the γ-cell population is heterogeneous, with subsets of cells producing another hormone in addition to Ppy. These bihormonal cells share identity markers typical of the other islet cell-types. In mice, Ppy gene inactivation or conditional γ-cell ablation did not alter glycemia nor body weight. Interestingly, upon β-cell injury induction, γ-cells exhibited gene expression changes and some of them engaged insulin production, like α- and δ-cells. In conclusion, we provide a comprehensive characterization of γ-cells and highlight their plasticity and therapeutic potential.

Derivation of stationary distributions of biochemical reaction networks via structure transformation

Long-term behaviors of biochemical reaction networks (BRNs) are described by steady states in deterministic models and stationary distributions in stochastic models. Unlike deterministic steady states, stationary distributions capturing inherent fluctuations of reactions are extremely difficult to derive analytically due to the curse of dimensionality. Here, we develop a method to derive analytic stationary distributions from deterministic steady states by transforming BRNs to have a special dynamic property, called complex balancing. Specifically, we merge nodes and edges of BRNs to match in- and out-flows of each node. This allows us to derive the stationary distributions of a large class of BRNs, including autophosphorylation networks of EGFR, PAK1, and Aurora B kinase and a genetic toggle switch. This reveals the unique properties of their stochastic dynamics such as robustness, sensitivity, and multi-modality. Importantly, we provide a user-friendly computational package, CASTANET, that automatically derives symbolic expressions of the stationary distributions of BRNs to understand their long-term stochasticity.

A Monte Carlo method to estimate cell population heterogeneity from cell snapshot data

Variation is characteristic of all living systems. Laboratory techniques such as flow cytometry can probe individual cells, and, after decades of experimentation, it is clear that even members of genetically identical cell populations can exhibit differences. To understand whether variation is biologically meaningful, it is essential to discern its source. Mathematical models of biological systems are tools that can be used to investigate causes of cell-to-cell variation. From mathematical analysis and simulation of these models, biological hypotheses can be posed and investigated, then parameter inference can determine which of these is compatible with experimental data. Data from laboratory experiments often consist of "snapshots" representing distributions of cellular properties at different points in time, rather than individual cell trajectories. These data are not straightforward to fit using hierarchical Bayesian methods, which require the number of cell population clusters to be chosen a priori. Nor are they amenable to standard nonlinear mixed effect methods, since a single observation per cell is typically too few to estimate parameter variability. Here, we introduce a computational sampling method named "Contour Monte Carlo" (CMC) for estimating mathematical model parameters from snapshot distributions, which is straightforward to implement and does not require that cells be assigned to predefined categories. The CMC algorithm fits to snapshot probability distributions rather than raw data, which means its computational burden does not, like existing approaches, increase with the number of cells observed. Our method is appropriate for underdetermined systems, where there are fewer distinct types of observations than parameters to be determined, and where observed variation is mostly due to variability in cellular processes rather than experimental measurement error. This may be the case for many systems due to continued improvements in resolution of laboratory techniques. In this paper, we apply our method to quantify cellular variation for three biological systems of interest and provide Julia code enabling others to use this method.

Stochastic reaction networks in dynamic compartment populations

Many biochemical processes in living systems take place in compartmentalized environments, where individual compartments can interact with each other and undergo dynamic remodeling. Studying such processes through mathematical models poses formidable challenges because the underlying dynamics involve a large number of states, which evolve stochastically with time. Here we propose a mathematical framework to study stochastic biochemical networks in compartmentalized environments. We develop a generic population model, which tracks individual compartments and their molecular composition. We then show how the time evolution of this system can be studied effectively through a small number of differential equations, which track the statistics of the population. Our approach is versatile and renders an important class of biological systems computationally accessible.

Compartmentalization of biochemical processes underlies all biological systems, from the organelle to the tissue scale. Theoretical models to study the interplay between noisy reaction dynamics and compartmentalization are sparse, and typically very challenging to analyze computationally. Recent studies have made progress toward addressing this problem in the context of specific biological systems, but a general and sufficiently effective approach remains lacking. In this work, we propose a mathematical framework based on counting processes that allows us to study dynamic compartment populations with arbitrary interactions and internal biochemistry. We derive an efficient description of the dynamics in terms of differential equations which capture the statistics of the population. We demonstrate the relevance of our approach by analyzing models inspired by different biological processes, including subcellular compartmentalization and tissue homeostasis.

Inheritance and variability of kinetic gene expression parameters in microbial cells: modeling and inference from lineage tree data

Motivation

Modern experimental technologies enable monitoring of gene expression dynamics in individual cells and quantification of its variability in isogenic microbial populations. Among the sources of this variability is the randomness that affects inheritance of gene expression factors at cell division. Known parental relationships among individually observed cells provide invaluable information for the characterization of this extrinsic source of gene expression noise. Despite this fact, most existing methods to infer stochastic gene expression models from single-cell data dedicate little attention to the reconstruction of mother–daughter inheritance dynamics.

Results

Starting from a transcription and translation model of gene expression, we propose a stochastic model for the evolution of gene expression dynamics in a population of dividing cells. Based on this model, we develop a method for the direct quantification of inheritance and variability of kinetic gene expression parameters from single-cell gene expression and lineage data. We demonstrate that our approach provides unbiased estimates of mother–daughter inheritance parameters, whereas indirect approaches using lineage information only in the post-processing of individual-cell parameters underestimate inheritance. Finally, we show on yeast osmotic shock response data that daughter cell parameters are largely determined by the mother, thus confirming the relevance of our method for the correct assessment of the onset of gene expression variability and the study of the transmission of regulatory factors.

Availability and implementation

Software code is available at https://github.com/almarguet/IdentificationWithARME. Lineage tree data is available upon request.

Supplementary information

Supplementary material is available at Bioinformatics online.

Isolating the sources of heterogeneity in nano-engineered particle-cell interactions

Nano-engineered particles have the potential to enhance therapeutic success and reduce toxicity-based treatment side effects via the targeted delivery of drugs to cells. This delivery relies on complex interactions between numerous biological, chemical and physical processes. The intertwined nature of these processes has thus far hindered attempts to understand their individual impact. Variation in experimental data, such as the number of particles inside each cell, further inhibits understanding. Here, we present a mathematical framework that is capable of examining the impact of individual processes during particle delivery. We demonstrate that variation in experimental particle uptake data can be explained by three factors: random particle motion; variation in particle-cell interactions; and variation in the maximum particle uptake per cell. Without all three factors, the experimental data cannot be explained. This work provides insight into biological mechanisms that cause heterogeneous responses to treatment, and enables precise identification of treatment-resistant cell subpopulations.

Applications of personalised signalling network models in precision oncology

As our ability to provide in-depth, patient-specific characterisation of the molecular alterations within tumours rapidly improves, it is becoming apparent that new approaches will be required to leverage the power of this data and derive the full benefit for each individual patient. Systems biology approaches are beginning to emerge within this field as a potential method of incorporating large volumes of network level data and distilling a coherent, clinically-relevant prediction of drug response. However, the initial promise of this developing field is yet to be realised. Here we argue that in order to develop these precise models of individual drug response and tailor treatment accordingly, we will need to develop mathematical models capable of capturing both the dynamic nature of drug-response signalling networks and key patient-specific information such as mutation status or expression profiles. We also review the modelling approaches commonly utilised within this field, and outline recent examples of their use in furthering the application of systems biology for a precision medicine approach to cancer treatment.

Maximum Entropy Framework for Predictive Inference of Cell Population Heterogeneity and Responses in Signaling Networks

Predictive models of signaling networks are essential for understanding cell population heterogeneity and designing rational interventions in disease. However, using computational models to predict heterogeneity of signaling dynamics is often challenging because of the extensive variability of biochemical parameters across cell populations. Here, we describe a maximum entropy-based framework for inference of heterogeneity in dynamics of signaling networks (MERIDIAN). MERIDIAN estimates the joint probability distribution over signaling network parameters that is consistent with experimentally measured cell-to-cell variability of biochemical species. We apply the developed approach to investigate the response heterogeneity in the EGFR/Akt signaling network. Our analysis demonstrates that a significant fraction of cells exhibits high phosphorylated Akt (pAkt) levels hours after EGF stimulation. Our findings also suggest that cells with high EGFR levels predominantly contribute to the subpopulation of cells with high pAkt activity. We also discuss how MERIDIAN can be extended to accommodate various experimental measurements.