Institute for Systems Theory and Automatic Control, University of Stuttgart, Germany

Institute of Cell Biology and Immunology, University of Stuttgart, Germany

Abstract

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.

Background

The main goals of research in systems biology are the development of quantitative models of intracellular pathways and the development of tools to support the modeling process. Thereby, most of the available methods and models consider only a single "typical cell" whereas most experimental data used to calibrate the models are obtained using cell population experiments, e.g. western blotting. This yields problems in particular if the studied population shows a large cell-to-cell variability. In such situations inferring a single cell model from cell population data can lead to biologically meaningless results. In order to understand the dynamical behavior of heterogeneous cell populations, it is crucial to develop cell population models, describing the whole population and not only a single individual

This has already been realized by several authors, and it has been shown that stochasticity in biochemical reactions and unequal partitioning of cell material at cell division can lead to complex population dynamics

For the purpose of this paper, heterogeneity in populations is modeled by differences in parameter values and initial conditions of the model describing the single cell dynamics

In the following, the problem of estimating the probability density of the parameters is studied. Therefore, we employ population snapshot data (PSD), which provide single cell measurements at every time instance but which do not provide single cell time series data. A typical experimental setup which provides PSD is flow cytometric analysis. In general, PSD are a common data type in the experimental analysis of biological systems.

So far, there are not many methods available for the estimation of parameter distributions. In pharmacokinetic studies mixed effect models ^{4}) and the amount of information per individual very limited (often only one data point), these methods are computationally too demanding. Furthermore, as in this study we are particularly interested in intracellular signal transduction, also methods which purely focus on the population balance

In this paper a novel Bayesian approach

To illustrate the properties of the proposed methods, a mathematical model of the tumor necrosis factor (TNF) pathway

Methods

Problem statement

Cell population model

For the purpose of this work we consider intracellular biochemical reaction networks which are modeled by systems of ordinary differential equations. This modeling framework allows to describe metabolic networks as well as signal transduction pathways, as long as spatial effects and stochasticity of the biochemical reactions can be neglected. Mathematically, the dynamic behavior of each single cell is determined by an ordinary differential equation in state space form

with state variables ^{(i) }can be kinetic constants, e.g. reaction rates or binding affinities.

Employing the definition of the single cell dynamics (1), a cell population model is given by the collection of

The heterogeneity within this population is modeled by differences in parameter values among individual cells. The parameters are distributed according to the probability density function

Note that interactions among individual cells influencing the analyzed pathway are not allowed. This is a restriction but indeed fulfilled in many

Measurement data and noise

In this paper we consider experimental setups where measurement data are obtained in the form of population snapshots, e.g. via flow cytometry. Population snapshots are taken at different times _{j}_{j }

The cells in the individual snapshots are referenced through index sets: snapshot _{0 }= 0. Thereby,

Let the data point for the cell with index

where ^{(i) }is the time at which the measurement was taken, and ^{(i) }= _{j}_{j}

Population snapshot data of heterogeneous cell population

**Population snapshot data of heterogeneous cell population**. The single cell measurement (·) is denoted by _{j }

In the parameter estimation, only the union of all snapshots is considered, and the parameter density function Θ is fitted to all snapshots simultaneously. To this end, we introduce the collection of all data, denoted as

in which

We emphasize that experimental setups are considered in which cells are not tracked over time. These setups are very common in studies on the population scale. Classical examples for measurement techniques yielding such data are flow cytometric analysis and cytometric fluorescence microscopy. These measurement techniques allow to determine protein concentrations within single cells. As the population is well mixed when the measurement is performed and no cell is measured more than once, the individual single cell measurements _{1 }≠ _{2}, holds if both cells are measured during one snapshot

Like most other experiments also the considered single cell experiments are subject to noise. We consider the noise model

in which ^{(i) }is the actual output from (1). The multiplicative noise is denoted by η^{× }∈ ℝ^{m }^{+ }∈ ℝ^{m}^{× }and ^{+ }are in the following assumed to be vectors of log-normally distributed random variables with probability density functions

for all

From (8) the conditional probability of measuring ^{(i) }can be determined. As the different output errors are assumed to be independent the conditional probability density is

with

which is illustrated in Figure

Line integral (10)

**Line integral (10)**. Set (^{× }and ^{+ }which yield for a given output

Problem formulation

As mentioned previously, when studying heterogeneous populations the density of the parameters Θ is in general unknown but necessary to gain an in-depth understanding of the population dynamics. Therefore, we are concerned with the problems:

**Problem 1 **_{pop},

**Problem 2 **_{pop},

Unfortunately, the number of cells considered in a standard lab experiment is on the order of 10^{4 }to 10^{7}. Thus, simulating the population model (2) is computationally expensive. Furthermore, it is hard from a theoretical point of view to deal with ensemble models such as (2). Density-based descriptions of the population dynamics are far more appealing for solving Problem 1 and 2. Therefore, in the next section a density description of the population is introduced.

Density-based modeling of cell populations

To simplify the inference problems on Θ the population description is changed from an ensemble model (2) to a density model

in which ^{(i)}(

To compute the cell population response ϒ(^{(i)}(_{0}(^{(i)}). This yields ^{(i)}(

in which

Illustration of kernel density estimation

**Illustration of kernel density estimation**. The kernel density estimate (^{(i)}(

are used to conserve the property that all variables are positive. The positive definite matrix

Approaches similar to the one we use to approximate ϒ(

Estimation of the parameter density

In the previous section an approach to determine the output density ϒ within the cell population for a given parameter density Θ is presented. Based on this an approach for estimating Θ from the available data

Bayes' theorem for heterogeneous cell populations

For learning the parameter density from the data Bayes' theorem

is used, in which

in which _{1 }≠ _{2}, it is not necessary to distinguish between the cases that (1) the two cells are measured at the same instance

As this equation cannot be solved explicitly the integral has to be approximated numerically. This could be realized using importance sampling

in which ^{(i) }, ^{(i) }for a cell having parameters

Based on (15) and (17), the calculation of the posterior probability for a given probability density of the parameters Θ is possible. Unfortunately, the inference problem nevertheless cannot be solved directly, as Θ is element of a function space, and hence further steps are necessary.

Parameterization of parameter density

In order to avoid the infinite dimensional inference problem the parameter density is parameterized. Θ is modeled by a finite weighted sum of multivariate ansatz functions Λ_{j}

The ansatz functions _{φ }_{φ }_{j }_{φ }

Note that the method presented in the following is independent of the choice of ansatz functions. Nevertheless, a clever choice of the ansatz functions may improve the approximation of the true parameter density dramatically. In this work, the ansatz functions are chosen to be multivariate Gaussians.

Given a parameterization of Θ_{φ}

in which ϒ (_{j}_{j}_{φ}_{j}

Reformulation of posterior probability

Having parameterized Θ_{φ }_{j}

in which _{j}

This in general high-dimensional integral is approximated employing Monte Carlo integration, yielding

in which ^{(k) }is drawn from Λ_{j}^{(k) }~ Λ_{j}_{c }^{(k)}}_{k}_{j }_{c }

Given these precomputed

in which

where the prior probability,

enforces the satisfaction of the constraint of Θ_{φ }

in which

Computation of maximum posterior probability estimate

Given the simplified unnormalized posterior probabilities _{φ }

This optimal parameter density

in which the two constraints ensure that the obtained density is positive and has integral one. Note that as Λ_{j }_{j }

This minimization problem can for concave _{φ}

Uncertainty of parameter density

In the previous section a method is presented which allows the computation of the maximum posterior probability estimate

Sampling of posterior probability density

In order to analyze the uncertainty of the estimate, a sample of the posterior probability density

The main point of concern when using MCMC sampling for the problem at hand is that the prior probability and the posterior probability respectively are only non-zero on a (_{φ }- 1) -dimensional subset of the density parameter space (28). This is due to the fact that the sum over the elements of _{φ }

This problem can be overcome by performing the sampling in the (_{φ }- 1)-dimensional space,

1. Draw proposals _{φ }- 1)-dimensional reduced proposal density _{r}

2. Determine

In this work, the reduced proposal density is chosen to be a multivariate normal distribution,

This two-step proposal generation procedure is in the following denoted by ^{k+1}~^{k+1}|^{k}). The proposed density parameter vector ^{k+1 }is accepted with probability

The distinction of the two cases is hereby crucial to ensure that only probability densities

By combining update and acceptance step one obtains an algorithm which draws a sample of weighting vectors _{φ}. The pseudo code for the routine is given in Algorithm 1. In particular, the facts that

• the conditional probabilities

• every evaluation of the acceptance probability _{a }

ensure hereby an efficient sampling. Without this reformulation the integral defining the conditional probability

**Algorithm 1 **Sampling of posteriori distribution

**Require: **data _{φ}), model ^{0}.

Calculation of conditional probabilities

Initialize the Markov Chain with ^{0}.

**for **_{φ }**do**

Given ^{i }^{k+1 }from proposal density ^{k+1}|^{k}

Calculate posterior probability

Generate uniformly distributed random number

**if **_{a}^{k+1}|^{k}**then**

Accept proposed parameter vector ^{k+1}.

**else**

Restore previous parameter vector, ^{k+1 }= ^{k}

**end if**

**end for**

**end**

Bayesian confidence intervals

The sample

In this work an approach is presented which employs the percentile method _{φ}

For the problem of estimating parameter densities, the 100_{φ}_{φ}

Consequently, the 100(1-_{φ}

As the sample

Predictions of output density

As the parameter density is not known precisely, also the model predictions show uncertainties. To evaluate the reliability of the population model and its predictive power, these prediction uncertainties have to be quantified. Therefore, the Bayesian confidence interval of the prediction around the output density with the highest a posteriori probability density,

is determined.

The 100(1-

in which the 100_{φ}

Computing _{j}_{j}

This sample can be used to approximate the prediction confidence interval _{φ }

To sum up, in this section a method for the estimation of parameter distributions in heterogeneous cell populations from population data has been presented. It has been shown that the optimal value as well as the Bayesian confidence intervals can be computed efficiently employing a parameterization of the parameter density. Also a method to determine prediction uncertainties has been presented. This allows an in-depth analysis of the reliability of the model. A summary of the procedure is shown in Figure

Illustration of the analysis procedure

**Illustration of the analysis procedure**. The main steps as well as their order is shown.

Results and discussion

To illustrate the properties of the proposed methods, artificial measurement data of a cell population responding to a tumor necrosis factor (TNF) stimulus will be analyzed. For illustration purposes, we consider a case where only one parameter is distributed in a first step. In a second step, we show that the method is also applicable in the case of multi-parametric heterogeneity.

In multicellular organisms, the removal of infected, malfunctioning, or no longer needed cells is an important issue. Therefore, multicellular organisms developed different mechanisms to externally enforce cell death. Thereby the signaling molecule TNF is one of the key players.

TNF can bind to specific death receptors in the cell membrane and is able to induce programmed cell death, also called apoptosis, via the activation of the caspase cascade. On the other hand, it promotes cell survival via the inflammatory response, specifically activation of the NF-

Model of TNF signaling pathway

The model considered in this study has been introduced in

Graphical representation of the TNF signal transduction model

**Graphical representation of the TNF signal transduction model**. Normal arrows indicate activation while arrows with flat hats indicate inhibition.

The state variables _{i}_{j}_{i}_{j}_{i}

and

The parameters _{j }_{j }

Nominal parameter values for the TNF signaling model (41).

**
i
**

**1**

**2**

**3**

**4**

**5**

_{i}

0.6

0.2

0.2

0.5

_{i}

0.4

0.7

0.3

0.5

0.4

It is known from experiments that the cellular response to a TNF stimulus is highly heterogeneous within a clonal cell population. Some cells die, others survive. The reasons for this heterogeneous behavior are unclear, but of great interest for biological research in TNF signaling, e.g. the use of TNF or related molecules as anti-cancer agent.

To understand the biological process at the physiological and biochemical level it is crucial to consider this cellular heterogeneity, using for example cell population modeling. Here, we model heterogeneity at the cell level via differences in the parameters _{3 }and _{4}. The parameter _{3 }describes the inhibitory effect of NF-_{4 }models the activation of I-

Univariate parameter density

For a first evaluation of the proposed method an artificial experimental setup is considered in which the caspase 3 activity is measured at four different time instances during a TNF stimulus,

At each time instance the C3a concentration in 150 cells is determined, _{2}, with measurement noise according to (7), where ^{× }= 0, ^{× }= 0.1, ^{+ }= log(0.05), and ^{+ }= 0.3. This corresponds to an average noise level of about 15%. The generated artificial experimental data for a bimodal distribution in _{3 }are depicted in Figure

Artificial population snapshot data of C3a used to infer the parameter density within the cell population

**Artificial population snapshot data of C3a used to infer the parameter density within the cell population**. Each

As ansatz functions Λ_{j }_{φ }

where _{j }_{j }_{φ}

in which _{β }_{j }_{j }_{β }_{j}|α_{j}|β_{j}^{2}. The distribution of a sample {^{k}_{k }_{3}. Furthermore, it does not enforce a trend to unimodal or bimodal distributions Θ_{φ }_{3}). Such distribution properties shall be inferred from the data.

Visualization of 15-dimensional MCMC sample

**Visualization of 15-dimensional MCMC sample **.

Prior, conditional and posterior probability density of Θ_{φ }_{3}) in _{3 }- Θ(_{3}) - plane

**Prior, conditional and posterior probability density of Θ**.

Given the ansatz functions Λ_{j }^{(j)}) for individual parameter values ^{(j)}, and 59% for the evaluation of the conditional probability ^{2 }= 0.05), of the conditional, and of the posterior probability distribution. The sample has _{φ }^{6 }members and the generation takes only four minutes. The computation is very fast, as the proposed approach simplified the evaluation of the conditional probability to a matrix vector multiplication. Note, that all steps during the computation of the conditional probabilities and the MCMC sampling can be parallelized, yielding a tremendous speed-up for more complex models.

The results of the sampling are illustrated in Figure _{10 }and _{11 }are negatively correlated for _{3 }is over-parameterized with respect to the data. Thus, the number of ansatz functions could be reduced while still achieving a good fit.

To analyze the uncertainty of Θ_{φ }_{3 }are rather small, indicating that the data contain many information about these regions. Unfortunately, in particular for _{3 }> 0.6 the confidence intervals are very wide showing that the parameter density in this area cannot be inferred precisely. But, although the amount of data is limited and the uncertainty with single _{i}_{φ }

Besides the uncertainty of Θ_{φ }

Predicted measured output densities

**Predicted measured output densities **. The colored lines indicate the distribution with the highest posterior probability

It is obvious that, although the parameter distributions show large uncertainties, the predictions are rather certain. This is indicated by tight confidence intervals. Furthermore, the mean predictions _{i }_{3}. A detailed analysis indicates (not shown here) that the number of ansatz function can be decreased, still ensuring a good approximation of the distribution of _{3}.

Multivariate parameter density

In many biological systems several cellular parameters are heterogeneous and different cellular concentrations can be measured

To perform this study we considered the same experimental setup as above. The only difference is that two concentrations are measured, C3a and NF-_{2}, _{3}]^{T}^{4 }cells are depicted in Figure _{φ }_{φ }^{2 }· I_{2 }and the extrema are equidistantly distributed on a regular 2-dimensional grid which is aligned with the axes.

Artificial population snapshot data of C3a and NF-

**Artificial population snapshot data of C3a and NF- κB used to infer the parameter density within the cell population**.

Given this setup, the convergence rate is studied in terms of the integrated mean square error,

of true distribution and distribution with highest posterior probability _{ext }are chosen as in the last section such that the prior is flat. The standard deviation on the other hand is reduced step-wise from _{i }_{i }

Integrated mean square error as function of the amount of available data and the informativeness of the prior

**Integrated mean square error as function of the amount of available data and the informativeness of the prior**. The plot shows the integrated mean square error for different numbers of measured cells per time instance and different standard deviation,

From Figure ^{true}, as shown in Figure ^{4}) is informative enough to infer key features of population heterogeneity.

Estimation result for 2-dimensional parameter density

**Estimation result for 2-dimensional parameter density**. **A**. Plot of the true parameter density, Θ^{true}. **B**. Plot of the estimated parameter density, ^{4 }measured cells at each time instance and a prior with

Conclusions

In this paper a Bayesian approach for inferring the parameter density for heterogenous cell populations with single cell resolution from population data is presented. We show that the proposed model can deal with limited and noisy measurement data as well as realistic noise models. The method utilizes a parameterization of the parameter density which, in combination with a reformulation of the conditional probability, allows a computationally efficient evaluation of the posterior probability. Compared to other methods for cell populations this approach does not rely on the approximation of the measured population response using density estimators.

For sampling from the posterior probability the Metropolis-Hastings algorithm is used. Here it has been adapted to be applicable to the considered constraint problem. Using this sampling strategy a sample from the posterior probability density is determined. This sample is employed to compute Bayesian confidence intervals for the parameter distribution, as well as for the model predictions. Also summary statistics like mean parameter density and mean predicted output density can easily be determined. For concave prior distributions we could even show that the calculation of the parameter density for the highest posterior probability is a convex problem.

The properties of the proposed scheme are evaluated using artificial data of a TNF signal transduction model. It could be shown that the proposed method yields good estimation results for a realistic experimental setup. Furthermore, although the remaining uncertainties are large, the predictions showg only small uncertainty indicating sloppiness of parameters.

Concerning the choice of the prior distribution it could be shown that the regularizing effect is beneficial if only little data is available. On the other hand, if the amount of available data increases, informative but not carefully chosen priors slow down the convergence.

Authors' contributions

JH, SW, and PS developed the problem formulation. JH developed the methods and implemented the algorithms. JH, SW, NR, and FA contributed to the systems dynamics and statistics. JH, SW, and FA constructed the application example and JH applied the proposed method. MD and PS contributed to the selection of the studied biological system, the choice of the addressed biological questions, and the interpretation of the results. JH, SW, and NR wrote the article. All authors read and approved the final manuscript.

Acknowledgements

The authors acknowledge financial support from the German Federal Ministry of Education and Research (BMBF) within the FORSYS-Partner program (grant nr. 0315-280A and D), from the German Research Foundation within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart, and from Center Systems Biology (CSB) at the University of Stuttgart.