A mathematical and computational approach for integrating the major sources of cell population heterogeneity

Comparison of Queueing Data-Structures for Kinetic Monte Carlo Simulations of Heterogeneous Catalysts

On-lattice kinetic Monte Carlo (KMC) is a computational method used to simulate (among others) physicochemical processes on catalytic surfaces. The KMC algorithm propagates the system through discrete configurations by selecting (with the use of random numbers) the next elementary process to be simulated, e.g., adsorption, desorption, diffusion, or reaction. An implementation of such a selection procedure is the first-reaction method in which all realizable elementary processes are identified and assigned a random occurrence time based on their rate constant. The next event to be executed will then be the one with the minimum interarrival time. Thus, a fast and efficient algorithm for selecting the most imminent process and performing all of the necessary updates on the list of realizable processes post execution is of great importance. In the current work, we implement five data-structures to handle the elementary process queue during a KMC run: an unsorted list, a binary heap, a pairing heap, a one-way skip list, and finally, a novel two-way skip list with a mapping array specialized for KMC simulations. We also investigate the effect of compiler optimizations on the performance of these data-structures on three benchmark models, capturing CO oxidation, a simplified water gas shift mechanism, and a temperature-programmed desorption run. Excluding the least efficient and impractical for large-problems unsorted list, we observe a 3× speedup of the binary or pairing heaps (most efficient) compared to the one-way skip list (least efficient). Compiler optimizations deliver a speedup of up to 1.8×. These benchmarks provide valuable insight into the importance of, often-overlooked, implementation-related aspects of KMC simulations, such as the queueing data-structures. Our results could be particularly useful in guiding the choice of data-structures and algorithms that would minimize the computational cost of large-scale simulations.

Modeling cell population dynamics

 Quantitative modeling is quickly becoming an integral part of biology, due to the ability of mathematical models and computer simulations to generate insights and predict the behavior of living systems. Single-cell models can be incapable or misleading for inferring population dynamics, as they do not consider the interactions between cells via metabolites or physical contact, nor do they consider competition for limited resources such as nutrients or space. Here we examine methods that are commonly used to model and simulate cell populations. First, we cover simple models where analytic solutions are available, and then move on to more complex scenarios where computational methods are required. Overall, we present a summary of mathematical models used to describe cell population dynamics, which may aid future model development and highlights the importance of population modeling in biology.

Modeling cell population dynamics

 Quantitative modeling is quickly becoming an integral part of biology, due to the ability of mathematical models and computer simulations to generate insights and predict the behavior of living systems. Single-cell models can be incapable or misleading for inferring population dynamics, as they do not consider the interactions between cells via metabolites or physical contact, nor do they consider competition for limited resources such as nutrients or space. Here we examine methods that are commonly used to model and simulate cell populations. First, we cover simple models where analytic solutions are available, and then move on to more complex scenarios where computational methods are required. Overall, we present a summary of mathematical models used to describe cell population dynamics, which may aid future model development and highlights the importance of population modeling in biology.

Heterogeneity in Pure Microbial Systems: Experimental Measurements and Modeling

Cellular heterogeneity influences bioprocess performance in ways that until date are not completely elucidated. In order to account for this phenomenon in the design and operation of bioprocesses, reliable analytical and mathematical descriptions are required. We present an overview of the single cell analysis, and the mathematical modeling frameworks that have potential to be used in bioprocess control and optimization, in particular for microbial processes. In order to be suitable for bioprocess monitoring, experimental methods need to be high throughput and to require relatively short processing time. One such method used successfully under dynamic conditions is flow cytometry. Population balance and individual based models are suitable modeling options, the latter one having in particular a good potential to integrate the various data collected through experimentation. This will be highly beneficial for appropriate process design and scale up as a more rigorous approach may prevent a priori unwanted performance losses. It will also help progressing synthetic biology applications to industrial scale.

Kinetic Modeling of ABCG2 Transporter Heterogeneity: A Quantitative, Single-Cell Analysis of the Side Population Assay

The side population (SP) assay, a technique used in cancer and stem cell research, assesses the activity of ABC transporters on Hoechst staining in the presence and absence of transporter inhibition, identifying SP and non-SP cell (NSP) subpopulations by differential staining intensity. The interpretation of the assay is complicated because the transporter-mediated mechanisms fail to account for cell-to-cell variability within a population or adequately control the direct role of transporter activity on staining intensity. We hypothesized that differences in dye kinetics at the single-cell level, such as ABCG2 transporter-mediated efflux and DNA binding, are responsible for the differential cell staining that demarcates SP/NSP identity. We report changes in A549 phenotype during time in culture and with TGFβ treatment that correlate with SP size. Clonal expansion of individually sorted cells re-established both SP and NSPs, indicating that SP membership is dynamic. To assess the validity of a purely kinetics-based interpretation of SP/NSP identity, we developed a computational approach that simulated cell staining within a heterogeneous cell population; this exercise allowed for the direct inference of the role of transporter activity and inhibition on cell staining. Our simulated SP assay yielded appropriate SP responses for kinetic scenarios in which high transporter activity existed in a portion of the cells and little differential staining occurred in the majority of the population. With our approach for single-cell analysis, we observed SP and NSP cells at both ends of a transporter activity continuum, demonstrating that features of transporter activity as well as DNA content are determinants of SP/NSP identity.

Effect of Intrinsic Noise on the Phenotype of Cell Populations Featuring Solution Multiplicity: An Artificial lac Operon Network Paradigm

Heterogeneity in cell populations originates from two fundamentally different sources: the uneven distribution of intracellular content during cell division, and the stochastic fluctuations of regulatory molecules existing in small amounts. Discrete stochastic models can incorporate both sources of cell heterogeneity with sufficient accuracy in the description of an isogenic cell population; however, they lack efficiency when a systems level analysis is required, due to substantial computational requirements. In this work, we study the effect of cell heterogeneity in the behaviour of isogenic cell populations carrying the genetic network of lac operon, which exhibits solution multiplicity over a wide range of extracellular conditions. For such systems, the strategy of performing solely direct temporal solutions is a prohibitive task, since a large ensemble of initial states needs to be tested in order to drive the system—through long time simulations—to possible co-existing steady state solutions. We implement a multiscale computational framework, the so-called “equation-free” methodology, which enables the performance of numerical tasks, such as the computation of coarse steady state solutions and coarse bifurcation analysis. Dynamically stable and unstable solutions are computed and the effect of intrinsic noise on the range of bistability is efficiently investigated. The results are compared with the homogeneous model, which neglects all sources of heterogeneity, with the deterministic cell population balance model, as well as with a stochastic model neglecting the heterogeneity originating from intrinsic noise effects. We show that when the effect of intrinsic source of heterogeneity is intensified, the bistability range shifts towards higher extracellular inducer concentration values.

On the impact of discreteness and abstractions on modelling noise in gene regulatory networks

In this paper, we explore the impact of different forms of model abstraction and the role of discreteness on the dynamical behaviour of a simple model of gene regulation where a transcriptional repressor negatively regulates its own expression. We first investigate the relation between a minimal set of parameters and the system dynamics in a purely discrete stochastic framework, with the twofold purpose of providing an intuitive explanation of the different behavioural patterns exhibited and of identifying the main sources of noise. Then, we explore the effect of combining hybrid approaches and quasi-steady state approximations on model behaviour (and simulation time), to understand to what extent dynamics and quantitative features such as noise intensity can be preserved.

Computational modeling of heterogeneity and function of CD4+ T cells

The immune system is composed of many different cell types and hundreds of intersecting molecular pathways and signals. This large biological complexity requires coordination between distinct pro-inflammatory and regulatory cell subsets to respond to infection while maintaining tissue homeostasis. CD4+ T cells play a central role in orchestrating immune responses and in maintaining a balance between pro- and anti- inflammatory responses. This tight balance between regulatory and effector reactions depends on the ability of CD4+ T cells to modulate distinct pathways within large molecular networks, since dysregulated CD4+ T cell responses may result in chronic inflammatory and autoimmune diseases. The CD4+ T cell differentiation process comprises an intricate interplay between cytokines, their receptors, adaptor molecules, signaling cascades and transcription factors that help delineate cell fate and function. Computational modeling can help to describe, simulate, analyze, and predict some of the behaviors in this complicated differentiation network. This review provides a comprehensive overview of existing computational immunology methods as well as novel strategies used to model immune responses with a particular focus on CD4+ T cell differentiation.

Coarse-grained analysis of stochastically simulated cell populations with a positive feedback genetic network architecture

Among the different computational approaches modelling the dynamics of isogenic cell populations, discrete stochastic models can describe with sufficient accuracy the evolution of small size populations. However, for a systematic and efficient study of their long-time behaviour over a wide range of parameter values, the performance of solely direct temporal simulations requires significantly high computational time. In addition, when the dynamics of the cell populations exhibit non-trivial bistable behaviour, such an analysis becomes a prohibitive task, since a large ensemble of initial states need to be tested for the quest of possibly co-existing steady state solutions. In this work, we study cell populations which carry the lac operon network exhibiting solution multiplicity over a wide range of extracellular conditions (inducer concentration). By adopting ideas from the so-called "equation-free" methodology, we perform systems-level analysis, which includes numerical tasks such as the computation of coarse steady state solutions, coarse bifurcation analysis, as well as coarse stability analysis. Dynamically stable and unstable macroscopic (population level) steady state solutions are computed by means of bifurcation analysis utilising short bursts of fine-scale simulations, and the range of bistability is determined for different sizes of cell populations. The results are compared with the deterministic cell population balance model, which is valid for large populations, and we demonstrate the increased effect of stochasticity in small size populations with asymmetric partitioning mechanisms.

Gene expression noise facilitates adaptation and drug resistance independently of mutation

We show that the effect of stress on the reproductive fitness of noisy cell populations can be modeled as a first-passage time problem, and demonstrate that even relatively short-lived fluctuations in gene expression can ensure the long-term survival of a drug-resistant population. We examine how this effect contributes to the development of drug-resistant cancer cells, and demonstrate that permanent immunity can arise independently of mutations.

Deterministic and stochastic population-level simulations of an artificial lac operon genetic network

BACKGROUND

The lac operon genetic switch is considered as a paradigm of genetic regulation. This system has a positive feedback loop due to the LacY permease boosting its own production by the facilitated transport of inducer into the cell and the subsequent de-repression of the lac operon genes. Previously, we have investigated the effect of stochasticity in an artificial lac operon network at the single cell level by comparing corresponding deterministic and stochastic kinetic models.

RESULTS

This work focuses on the dynamics of cell populations by incorporating the above kinetic scheme into two Monte Carlo (MC) simulation frameworks. The first MC framework assumes stochastic reaction occurrence, accounts for stochastic DNA duplication, division and partitioning and tracks all daughter cells to obtain the statistics of the entire cell population. In order to better understand how stochastic effects shape cell population distributions, we develop a second framework that assumes deterministic reaction dynamics. By comparing the predictions of the two frameworks, we conclude that stochasticity can create or destroy bimodality, and may enhance phenotypic heterogeneity.

CONCLUSIONS

Our results show how various sources of stochasticity act in synergy with the positive feedback architecture, thereby shaping the behavior at the cell population level. Further, the insights obtained from the present study allow us to construct simpler and less computationally intensive models that can closely approximate the dynamics of heterogeneous cell populations.

Identification of models of heterogeneous cell populations from population snapshot data

Background

Most of the modeling performed in the area of systems biology aims at achieving a quantitative description of the intracellular pathways within a "typical cell". However, in many biologically important situations even clonal cell populations can show a heterogeneous response. These situations require study of cell-to-cell variability and the development of models for heterogeneous cell populations.

Results

In this paper we consider cell populations in which the dynamics of every single cell is captured by a parameter dependent differential equation. Differences among cells are modeled by differences in parameters which are subject to a probability density. A novel Bayesian approach is presented to infer this probability density from population snapshot data, such as flow cytometric analysis, which do not provide single cell time series data. The presented approach can deal with sparse and noisy measurement data. Furthermore, it is appealing from an application point of view as in contrast to other methods the uncertainty of the resulting parameter distribution can directly be assessed.

Conclusions

The proposed method is evaluated using artificial experimental data from a model of the tumor necrosis factor signaling network. We demonstrate that the methods are computationally efficient and yield good estimation result even for sparse data sets.