Cfse Case Study

Abstract

Motivation:In vitro and in vivo cell proliferation is often studied using the dye carboxyfluorescein succinimidyl ester (CFSE). The CFSE time-series data provide information about the proliferation history of populations of cells. While the experimental procedures are well established and widely used, the analysis of CFSE time-series data is still challenging. Many available analysis tools do not account for cell age and employ optimization methods that are inefficient (or even unreliable).

Results: We present a new model-based analysis method for CFSE time-series data. This method uses a flexible description of proliferating cell populations, namely, a division-, age- and label-structured population model. Efficient maximum likelihood and Bayesian estimation algorithms are introduced to infer the model parameters and their uncertainties. These methods exploit the forward sensitivity equations of the underlying partial differential equation model for efficient and accurate gradient calculation, thereby improving computational efficiency and reliability compared with alternative approaches and accelerating uncertainty analysis. The performance of the method is assessed by studying a dataset for immune cell proliferation. This revealed the importance of different factors on the proliferation rates of individual cells. Among others, the predominate effect of cell age on the division rate is found, which was not revealed by available computational methods.

Availability and implementation: The MATLAB source code implementing the models and algorithms is available from http://janhasenauer.github.io/ShAPE-DALSP/.

Contact: jan.hasenauer@helmholtz-muenchen.de

Supplementary information: Supplementary data are available at Bioinformatics online.

1 Introduction

Proliferation is essential in many biological processes, ranging from development to wound healing, immune response and stem cell renewal. Due to this eminent role, a key challenge in cell biology is to quantify proliferation dynamics. Nowadays, there are different experimental methods available to assess cell proliferation, e.g. time-lapse microscopy (Schroeder, 2011) and proliferation assays based upon carboxyfluorescein succinimidyl ester (CFSE) labeling (Lyons and Parish, 1994). While time-lapse microscopy is highly informative, it is difficult to apply in vivo. CFSE-based proliferation assays can be employed in vivo and in vitro, but the data analysis is more intricate and no single-cell time courses are available.

To study proliferation using CFSE, cells are incubated with carboxyfluorescein diacetate succinimidyl ester (CFDA-SE). CFDA-SE can diffuse across the cell membrane and once in the cytoplasm is converted to CFSE, which binds covalently to intracellular proteins. CFSE is a fluorescent dye and its concentration is reduced by protein degradation as well as cell division. During cell division, CFSE is distributed approximately equally among daughter cells (Lyons and Parish, 1994); hence, proliferation results in a progressive dilution of the dye (Fig. 1A), which can be recorded using flow cytometry. As the individual cells do not, in general, divide with the same rates, an initially unimodal distribution (Fig. 1B) becomes multi-modal as time progresses (Fig. 1C). The modes are related to different numbers of cell divisions (Hawkins et al., 2007a); however, often the cells with different numbers of divisions are not strictly separated. For this reason, the calculation of the number of cells with a certain division number using peak detection and deconvolution methods (Hawkins et al., 2007a; Luzyanina et al., 2007a) is error-prone. Furthermore, even knowledge about the number of cells with a particular division number does not enable the comparison of different hypotheses regarding, e.g. the proliferation rates.

Fig. 1.

Illustration of proliferation assay showing (A) one labeled cell and its descendants, (B) the frequency of different CFSE concentration in the cell population at t = 0 and (C) the frequency of different CFSE concentration in the cell population at t = T. Shades from dark to light gray correspond to high and low CFSE concentration, respectively

Fig. 1.

Illustration of proliferation assay showing (A) one labeled cell and its descendants, (B) the frequency of different CFSE concentration in the cell population at t = 0 and (C) the frequency of different CFSE concentration in the cell population at t = T. Shades from dark to light gray correspond to high and low CFSE concentration, respectively

The need for quantitative data analysis and hypothesis testing inspired the development of a multitude of model-based approaches. Nowadays, age-structured population (ASP) models (Bernard et al., 2003; Hawkins et al., 2007b), division-structured population (DSP) models (De Boer et al., 2006), label-structured population (LSP) models (Banks et al., 2010; Luzyanina et al., 2007b), age- and division-structured population (ADSP) models (Hawkins et al., 2007b), and division- and label-structured population (DLSP) models (Hasenauer et al., 2012; Schittler et al., 2011) are used to study CFSE data. These classes of population balance models account for up to two properties of individual cells:

  • number of divisions i (→ DSP and DLSP model)

  • CFSE concentration x (→ LSP and DLSP model)

  • cell’s age a (→ ASP and ADSP model)

and are mostly written as systems of ordinary differential equations (ODEs) or partial differential equations (PDEs). Population balance models which account for the cell age—the time passed since cell division—, i.e. ASP and ADSP, can also be formulated as sets of nested integrals (De Boer and Perelson, 2013). This alternative formulation is employed for the Smith–Martin model (Smith and Martin, 1973) and the cyton model (Hawkins et al., 2007b). For the Smith–Martin model and cyton model (with progression fraction equal to zero), equivalent PDE formulations are available (Bernard et al., 2003; De Boer and Perelson, 2013). For a comprehensive introduction to structured population models, we refer to the review by De Boer and Perelson (2013).

As the population models available in the literature did not capture the complexity of the process, we recently introduced the division-, age- and label-structured population (DALSP) model (Metzger et al., 2012). The DALSP model, a system of coupled PDEs, provides a flexible description of proliferation dynamics (Metzger et al., 2012), but it has never been used to analyse CFSE data. Accordingly, no parameter estimation, model selection and uncertainty analysis methods have been available for this type of model until now.

In this manuscript, we introduce methods to infer the parameters of DALSP models from CFSE distribution time-series data and to assess the dependency of the proliferation rates on factors such as cell age. For this we introduce a statistical model linking predictions of the DALSP model to measure CFSE data and formulate the corresponding inverse problem. As the optimization problem is nonlinear, we compare different optimization procedures. Furthermore, we implement the first identifiability and uncertainty analysis methods for DALSP models. Combined with model selection methods, these methods facilitate an in-depth analysis of CFSE time-series data, which we illustrate for a published dataset.

2 Methods

To analyse CFSE data, we combine mechanistic and statistical models of the biological process and the measurement process. In the following we present the different ingredients as well as the inference methods. A more detailed description is provided in the Supplementary Information.

Notation: We denote the set of non-negative real numbers by and the set of natural numbers with zero by . The units used in the following equations are number of cells (cells), unit of concentration (UC), unit of fluorescence intensity (UFI) and unit of time (UT). For simplicity, we assume that age and time are measured in the same units, generalizations are straight forward.

2.1 DALSP model

The state variable of the DALSP model is the joint number density of age, label concentration and cell division number, (in cells/UC/UT). Its dynamics are governed by a system of coupled 2D PDEs,

(1)

with initial conditions (ICs) and boundary conditions (BCs)

(3)

The i-th PDE describes the dynamics of cells with i divisions. The factorization of the initial condition, , in initial age distribution (in cells/UT) and initial label density (in 1/UC), is biologically plausible as the labeling efficiency should not depend on the cell age. This factorization will allow for an efficient numerical solution algorithm.

In (1)–(3), and denote the rates (in 1/UT) of cell division and cell death for cells with i divisions. The rate of cellular label degradation is denoted by (in 1/UT), with rate constant k(t) (in 1/UT/UC). Accordingly, the following terms contribute to the temporal change of the density :

  • , change of label x with rate ,

  • , increase of cell age a, and

  • , loss of cells from the ith subpopulation due to cell division and due to cell death,

and possess units cells/UC/UT2. The loss of cells from the -th subpopulation due to cell division results in the birth of cells in the ith subpopulation with age a = 0, defining the BCs (3). The BCs are obtained by integrating over the age. As the cells double (factor 2) and the label distribution is rescaled due to the halving of the label concentration (factor 2), this integral is multiplied by a factor 4.

The rates as well as the initial conditions are usually unknown and have to be estimated from experimental data. Therefore, rates and initial conditions are parameterized in terms of a parameter vector (see Application section). For a more detailed statement of the model we refer to the Supplementary Information, Section 1.1.1.

2.2 Modeled and measured quantities

The number density encodes the properties of the proliferating cell population. By marginalizing over all cell ages and label concentrations, the number of cells which underwent i divisions, is obtained. The subsequent summation over the division number i yields the overall number of cells, . Marginalization over label concentrations and division number and subsequent normalization with the number of cells yields the age distribution,
The label distribution, , is obtained by marginalizing over cell age and division number and normalizing with the overall cell number,
Proliferation assays provide information about the overall number of cells as well as the sum of label-induced fluorescence (Fig. 1B and C) and cellular background fluorescence. The label induced fluorescence, y (in UFI), is proportional to the label concentration, with and proportionality constant c > 0 (UFI/UC). The background fluorescence yb is a random variable, (in UFI), whose distribution (pb) depends potentially on the biological system, the measurement procedure and technical factors. The distribution of the total measured fluorescence, (in UFI), obeys the convolution integral,

(see Supplementary Information, Section 1.1.2). In the presence of outliers, can be mixed with an outlier distribution, (see Supplementary Information, Section 2.1). The measurement of the fluorescence distribution does not provide information about the absolute values of the concentration x. Any changes in c can be compensated by changes in the initial label distribution and k(t) (Hasenauer, 2013), rendering c = 1 (UFI/UC) a valid parameterization.

Commonly used measurement devices possess a finite resolution and collect interval censored samples from . The resulting binned snapshot data provides the number of cells observed in a bin j with intensity range Ij at time tk. These counts along with intervals provide a histogram. The probability of observing an individual cell at time point tk in bin j is the integral of over Ij. The overall cell count measured at time point tk is denoted by .

2.3 Numerical simulation

To compute the number density and further model properties, the DALSP model (1)–(3) is solved numerically. For this we exploit that the solution of the system of coupled 2D PDEs (1)–(3) is factorable (Metzger et al., 2012). The first factor is the solution to a system of coupled 1D PDEs describing an ADSP. This solution is computed using an efficient iterative numerical scheme. The second factor is the solution to a set of decoupled 1D PDEs describing the label distribution in cellular subpopulations with similar division number. This solution can be determined analytically. The factorization accelerates the numerical evaluation by several orders of magnitude compared to naive numerical methods. A similar decomposition approach is used to compute the sensitivities of with respect to the parameters θi, . Given and its sensitivities, the model properties and their derivatives are determined via numerical integration. The convolution integral defining the measured fluorescence intensity is evaluated using Fenton’s approximation (Fenton, 1960). For further details we refer to the Supplementary Information, Sections 1.2–1.5.

2.4 Parameter estimation and uncertainty analysis

We employed maximum likelihood and Bayesian parameter estimation to determine the unknown model parameters θ, with , from the collection of binned snapshot data

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In order to pass the exam the candidate will have to score at least 80% for the total examination. All CFSP exams are open book with all readily available published references allowed to be used during the exam. Please also note that the candidates are specifically encouraged to at least bring a copy of part 1 of 61508 and similar relevant parts of the sector-specific IEC standards relevant to their area. For example, Process Applications candidates should also bring at least part 1 of 61511, Machine Applications candidates should also bring at least 62061, Software candidates should also bring at least part 3 of 61508, and Hardware candidates should also bring at least part 2 of 61508. Other textbooks, publicly available published course materials, and workbooks are also allowed and may be useful to the candidates. Although occasional general hand notes in these reference materials may be allowed at the discretion of the Governance Board appointed exam proctor, it is expressly forbidden to bring in any personal exam-specific notes.



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