Department of Chemical and Biomolecular Engineering, Rice University, Houston, TX 77005, USA

Abstract

Background

The

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.

Background

Since the introduction of the operon concept by Jacob et al.

Most of these models, however, pertain to the single cell behavior [see, for example the models reviewed in

Furthermore, van Hoek and Hogeweg

Thus, there remain several open questions regarding the emergence of population behavior from the complex interplay between reaction dynamics and stochasticity at the single cell level. Investigating this connection and ultimately understanding cell population dynamics is significant for two reasons. First, typical biology experiments involve cell populations rather than single cells and, thus, phenotypic distributions obtained for instance from flow cytometry pertain to the cell population rather than the lifetime of a single cell. Second, there have been mathematical modeling studies for simple genetic networks, suggesting that the behavior of the single cell is very different from the behavior of the cell population

In a previously published article, Stamatakis and Mantzaris

The first framework assumes stochastic reaction occurrence and takes into account stochastic DNA duplication, division and partitioning. McAdams and Arkin

In order to better understand how stochastic effects shape cell population distributions, we develop here a second framework that assumes deterministic reaction dynamics and stochastic DNA duplication, cell division and partitioning. Throughout this work, the single cell models are derived from the reaction network developed in ref.

Comparisons of the predictions of the two frameworks reveal that stochasticity can create or destroy bimodality, and may enhance phenotypic heterogeneity by creating heavy tailed distributions, a phenomenon that has not being shown before and can be investigated experimentally. The insights provided from the new study also allow us to construct simpler and less computationally intensive models that can closely approximate the dynamics of heterogeneous cell populations. Specifically, for the case of deterministic reaction dynamics, we use the continuum model formulation which assumes that all cells in the population occupy a continuous and expanding biotic phase

Methods

Figure

Definition of cell chain and cell population

**Definition of cell chain and cell population**. A cell chain essentially stores information about the history of a single cell in time. On the other hand, a cell population consists of all the viable offspring observed at time

We are now ready to present the frameworks that will be used for the simulating cell populations in this study. Both frameworks treat the occurrence of division and DNA duplication events as stochastic processes. Their difference lies in the treatment of reactions and the partitioning of species between the daughter cells: the first framework treats these as stochastic, whereas the second one as deterministic processes. We will refer to the former as the "population model with stochastic reaction dynamics", and the latter as the "population model with deterministic reaction dynamics".

Population Model with Stochastic Reaction Dynamics

In an earlier study **z **= (**X**, **X **is a vector with n entries for species copy numbers and

This CPME describes the evolution of the probability of finding at time **X**
_{1}, _{1}),..., (**X**
_{i}
_{i}
**X**
_{ν}, _{ν}). If we denote this probability by _{ν}[(**X**
_{1}, _{1}),..., (**X**
_{i}
_{i}
**X**
_{ν}, _{ν});

where the terms _{R}
_{S}
_{G }
_{D }

**Supplementary text that includes**: (1) detailed descriptions of the population level modeling frameworks for deterministic and stochastic reaction dynamics; (2) a discussion of asymmetric volume partitioning; (3) the derivations of the partitioning probabilities for population models with deterministic reaction dynamics (4) the structured continuum model simulated in Figure

Click here for file

Population Model with Deterministic Reaction Dynamics

The extension of our approach to the case where reactions are deterministic is straightforward and is presented in detail in Section S1 of Additional file

where the terms _{S}
_{G }

Structured Continuum Model: a Lumped Description of Cell Population Dynamics

Since the simulation of detailed population models can be computationally expensive, one often resorts to simpler models. In this study, we will use and evaluate the structured continuum model formulation attributed to Fredrickson

where [_{i}

The Artificial

To simulate the artificial

Here, our intention has been to model an artificial

Table _{R }

Reactions and Propensity Functions for the Stochastic

**Reaction**

**Propensity Function**
^{
1, 2, 3
}

(1-1)

(1-2)

(1-3)

(1-4)

(1-5)

(1-6)

(1-7)

(1-8)

(1-9)

(1-10)

(1-11)

(1-12)

(1-13)

(1-14)

(1-15)

(1-16)

(1-17)

(1-18)

(1-19)

(1-20)

(1-21)

(1-22)

(1-23)

(1-24)

(1-25)

^{1 }Variables without brackets denote number of molecules of the corresponding species.

^{2 }All propensity functions have units of min^{-1}.

^{3 }Avogadro's number: _{A }^{14 }nmol^{-1}.

Rate Equations for the Deterministic

(2-1)

(2-2)

(2-3)

(2-4)

(2-5)

(2-6)

(2-7)

(2-8)

(2-9)

(2-10)

Finally, Table _{s }
_{d }
_{S }
_{D }
_{G}

Parameters of the

**Symbol**

**Value**

**Units**

**Description**

_{0,E. coli}

0.4

μm

_{E. coli}

2.3

μm

Representative

_{E. coli}

5.8

μm^{2}

_{E. coli}

1.0

fL

_{T}

1

(copy number)

operator molecular content

_{sMR}

0.23

nM·min^{-1}

_{sR}

15

min^{-1}

LacI monomer translation rate constant

_{2R}

50

nM^{-1}·min^{-1}

LacI dimerization rate constant

_{-2R}

10^{-3}

min^{-1}

LacI dimer dissociation rate constant

_{r}

960

nM^{-1}·min^{-1}

association rate constant for repression

_{-r}

2.4

min^{-1}

dissociation rate constant for repression

_{dr1}

3·10^{-7}

nM^{-2}·min^{-1}

association rate constant for 1^{st }derepression mechanism

_{-dr1}

12

min^{-1}

dissociation rate constant for 1^{st }derepression mechanism

_{dr2}

3·10^{-7}

nM^{-2}·min^{-1}

association rate constant for 2^{nd }derepression mechanism

_{-dr2}

4.8·10^{3}

nM^{-1}·min^{-1}

dissociation rate constant for 2^{nd }derepression mechanism

_{s1MY}

0.5

min^{-1}

_{s0MY}

0.01

min^{-1}

leak

_{sY}

30

min^{-1}

_{p}

0.12

nM^{-1}·min^{-1}

LacY-inducer association rate constant

_{-p}

0.1

min^{-1}

LacY-inducer dissociation rate constant

_{ft}

6·10^{4}

min^{-1}

TMG facilitated transport constant

_{t}

1.55·10^{-6}

dm·min^{-1}

TMG passive diffusion permeability constant

λ_{MR}

0.462

min^{-1}

λ_{MY}

0.462

min^{-1}

λ_{R}

0.2

min^{-1}

LacI monomer degradation constant

λ_{R2}

0.2

min^{-1}

LacI dimer degradation constant

λ_{Y}

0.2

min^{-1}

LacY degradation constant

λ_{YIex}

0.2

min^{-1}

LacY-inducer degradation constant

λ_{I2R2}

0.2

min^{-1}

repressor-inducer degradation constant

0.0231

(min^{-1})

cell growth rate parameter

_{d}

25

(dim/less)

division propensity sharpness exponent

_{d,crit}

15

(fL)

critical volume for division

80

(dim/less)

beta distribution sharpness exponent

_{s}

25

(dim/less)

DNA duplication propensity sharpness exponent

_{s,crit}

10

(fL)

critical volume for DNA duplication

There is considerable uncertainty for some parameters. For instance, the equilibrium constant for repression, k_{-r}/k_{r }is in the range of 10^{-13 }to 10^{-11 }M _{-r }= 2.4 min^{-1 }and k_{r }= 960 nM^{-1}·min^{-1 }which results in k_{-r}/k_{r }= 2.5·10^{-12 }M. This value for k_{r }turns out to be much higher than the experimentally measured one of 0.6 nM^{-1}·min^{-1}, which is deduced by the 59 s time needed for a repressor to find an operator _{E.coli }
^{-15 }L. Yet, simulations of the single cell deterministic model with this value of k_{r }produce quantitatively identical bifurcation structures with the nominal parameter set, provided that the value of k_{-r }has been adjusted to keep the thermodynamic constant for repression the same. Since the timescale for repression will be different in this case, the single cell stochastic model is expected to exhibit bistability for different induction levels. Previous work

For the _{E.coli}
_{0 }
_{0 }
_{0}
_{0 }

The cell membrane area can be then calculated as a function of the volume (for given _{0}

Results and discussion

Comparison of Deterministic and Stochastic Reaction Dynamics

Deterministic Reaction Dynamics: Phenotypic Distributions and Statistics

The

Thus, Figure _{ex}
_{T}, for every cell in the population versus time. Notice that all offspring remain close to the off state of the _{T}, and the number of cells considered in the population (_{cellsmax }
^{4}, a value reached at 415 min). On the other hand, panels (c) and (d) pertain to the case where the population is initialized with a single cell having species concentrations close to the on-state of the

Simulation results from the CPME model with deterministic reaction dynamics

**Simulation results from the CPME model with deterministic reaction dynamics**. The simulations for panels (a) and (b) start with a single cell close to the off-state, while the simulations for panels (c) and (d) start with a single cell close to the on-state. [_{ex}_{cellsmax }^{4}).

The oscillations shown in Figure _{mother}

The results of Figure _{cellsmax }

Attracting states and transitions in the CPME model with deterministic reaction dynamics

**Attracting states and transitions in the CPME model with deterministic reaction dynamics**. [_{ex}_{daughter1}_{mother }

For the simulations of Figure

Effect of Stochasticity on Cell Population Behavior

Having analyzed the case where reaction dynamics follow deterministic laws, we will now investigate the case of stochastic reaction occurrence since the small copy numbers of molecules encountered in this system are expected to result in significant intrinsic stochasticity.

Stochasticity in reaction occurrence results in transcriptional and translational bursts, as shown in panel (a) of Figure _{cellsmax }

Simulation results from the CPME model with stochastic reaction dynamics

**Simulation results from the CPME model with stochastic reaction dynamics**. [_{ex}_{cellsmax }

We have just discussed the effects of stochasticity on the transient behavior of the cells as well as the cell population average. In order to characterize the cell population dynamics, however, one needs to know the entire number density function (NDF) which expresses the number of cells that exist in states (**X**, **X**,

Comparison of the NDFs computed with deterministic versus stochastic reactions

**Comparison of the NDFs computed with deterministic versus stochastic reactions**. NDFs computed with deterministic dynamics are marked as "Deter. Rxn" whereas the ones with stochastic reaction occurrence are marked as "Stoch. Rxn". For all deterministic simulations, the cell population was initiated with 20 cells and _{cellsmax }^{4}. For all stochastic simulations batches of 20 simulations were run with _{cellsmax }_{ex}_{ex}_{ex}

For low extracellular TMG concentrations, [_{ex}

For intermediate inducer concentrations [_{ex}
_{ex}
_{T}. In this case, stochasticity in reaction occurrence appears to extend the region where the NDF is bimodal and widen the upper mode of the distribution.

In all cases (Figure

Structured Continuum Model for Deterministic Reaction Dynamics

Simulating in detail the intra- and inter-cellular processes at the population level is computationally demanding. Therefore, it is natural to ask whether one can adequately predict the dynamics of the population average with the use of a lumped model that neglects heterogeneity. For this purpose, we will use the structured continuum model formulation

Simulations with the structured continuum model require estimates for the average membrane area over the volume 〈_{T}

The average ratio of area over volume 〈_{d,crit }
_{d,crit}

Therefore, the average 〈

Using the parameters of Table ^{-1}. Cell population simulations give an average equal to 5.76 μm^{-1}. Thus, using these heuristic arguments, we estimated the average ratio 〈

To estimate the average operator concentration 〈[_{T}
_{d,crit }
_{s,crit }
_{d,crit}

Using the parameters of Table

Panel (a) of Figure

Comparison of the average stationary behavior of the CPME model that incorporates deterministic reaction dynamics with the steady state of the structured continuum model

**Comparison of the average stationary behavior of the CPME model that incorporates deterministic reaction dynamics with the steady state of the structured continuum model**. [_{ex}_{T }_{T }

The qualitative agreement between the steady states of the structured continuum model and the population averages is excellent. The agreement between the two models is remarkably good when the population-average values for _{T }

The above results pertain to the time invariant behaviors of the cell population. It is interesting, however, to also compare the dynamical behavior of the structured continuum and the cell population model.

Panels (a) and (b) of Figure _{ex}
_{cellsmax }
^{4}) has equilibrated to the stationary distribution that corresponds to [_{ex}
_{ex}

Comparison of the dynamical behavior of the CPME model that incorporates deterministic reaction dynamics with that of the structured continuum model

**Comparison of the dynamical behavior of the CPME model that incorporates deterministic reaction dynamics with that of the structured continuum model**. Parameters as in Table 3 unless otherwise noted. Panels (a, b): Transient dynamics of the population mean for the switching from the low ([_{ex}_{ex}_{T }_{s0MY }^{-1}, _{s1MY }^{-1}, λ_{MY }

The dashed and dash-dotted curves on panels (a) and (b) of Figure _{T}
_{ex}
_{ex}

The agreement between the transient behaviors predicted by the structured continuum model and the population model is excellent even in the case where the intracellular dynamics are significantly slower than the proliferation rates of the cell as shown in panels (c) and (d) of Figure

This agreement between the steady state and the transient behavior of the structured continuum and the cell population models can be explained as follows. In this system, the only coupling between the cells is due to stochastic partitioning that can generate variability but no bias on the total LacY concentration, in the sense that one does not observe consistently higher LacY_{T }concentrations in one daughter versus the other. In fact, the two newborn daughters may have different LacY_{T }contents, but they always have equal LacY_{T }concentrations, which are identical to their mother's LacY_{T }concentration just before division. Similarly, unsynchronized DNA duplication and division events also contribute to the observed variability in LacY_{T}, but they cannot consistently bias the LacY concentration.

In general, the key to understanding this effect lies in the fact that in the cell population, division results in the removal of an "old" mother cell and the addition of two "young" daughter cells. If the properties of the "old" cells are different than those of the "young" daughters, then this effect results in the properties of the "young" ones being overrepresented in the population. On the other hand, such an effect is absent in a cell chain, where each division results in the removal of a mother cell and the addition of a single daughter, and thus, the properties of both subpopulations are equally represented in the cell chain probability distribution.

Let us now suppose that the property we chose to investigate is a species concentration (intensive variable). The concentrations of mother and daughter cells are on average the same, due to the symmetric properties of the binomial and beta distributions that model division. Consequently, the cell chain and cell population distributions will be practically indistinguishable. On the other hand, if we chose an intensive property to investigate, the two distributions would be different in general, as we have shown in our previous work [Figure

These observations contradict previously published results by Mantzaris for different biological systems

More specifically, these studies incorporated into the CPB single cell models that are written for species concentrations. However, the CPB accepts single cell models written for cellular contents (amounts), which are extensive quantities, and not concentrations which are intensive

Furthermore, some of these studies used unequal partitioning to artificially generate complex behavior, such as oscillations in a reaction network with 0^{th }and 1^{st }order reactions. However, the asymmetry in

Simulation of Cell Chains for Stochastic Reaction Dynamics

Given the high computational expense of simulating the population model with stochastic reaction dynamics, we pose the question of whether there is a simpler method for obtaining good approximations for the distributions of phenotypic characteristics. In contrast to the case of deterministic reaction dynamics, we cannot use a continuum model for comparison purposes here. The main reason is that in chemical systems far from the thermodynamic limit, the size influences noise strength. In the case of stochastic reaction occurrence, therefore, growth results in dilution of molar contents and also suppression of stochastic fluctuations. Consequently, we cannot create a lumped model that accounts for growth with just the incorporation of a dilution term as was done in the deterministic case.

However, we can take a different approach. We have already observed that cell population dynamics emerge from single cell behavior and that, on average, the two daughters share the same concentrations as their mother cell. Therefore, instead of tracking the NDF in a cell population, we could compute the PDF in a cell chain. Simulation of a cell chain tracks only one daughter after each division event. Thus, instead of focusing on the expected number of cells of the population that exist in state **z**, we turn our attention to the probability of finding a single cell of the cell chain at state **z**. Note, however, that we will be comparing the PDF and NDF of intensive quantities, in particular the total LacY concentration.

A comparison of the cell population NDF with the single cell PDF for [Y]_{T }shows a remarkable agreement between the two (Figure

Comparisons of cell chain probability distribution functions with cell population NDFs

**Comparisons of cell chain probability distribution functions with cell population NDFs**. Panel (a): For the population distribution, a batch of 20 simulations was run with _{cellsmax }_{ex}_{final }= 10^{5 }min and samples were taken periodically in time with Δ_{s1MY }^{-1}, _{s1MY }^{-1}) and slower translation (_{sY }^{-1}). The simulation batch consisted of 20 simulations and was sampled at t = 250 min. The cell line was tracked for 10^{5 }min of simulated time.

Conclusions

This study generalized the deterministic and stochastic single cell

We also carried out a systematic comparison of predictions obtained by a structured continuum model and a detailed cell population model with deterministic reaction dynamics. These comparisons showed that the structured continuum model gives satisfactory results for the average LacY_{T }concentration of the population, even in the case where the reaction dynamics are much slower than the proliferation dynamics. This agreement between the two models was attributed to the similar intensive properties (such as species concentrations) between mother and daughter cells.

Finally and in the case of stochastic reaction dynamics, we demonstrated that by simulating the dynamics of a cell chain we can obtain very accurate approximations of the cell population dynamics. The PDFs obtained by cell chain simulations for the LacY_{T }concentration were in excellent agreement to the cell population NDF computed with the Monte Carlo algorithm implementing the CPME of Eq. 1.

Our study shows that for cell populations in which the cells interact weakly, through division only, it is possible to accurately model and explain the population behavior in terms of the single cell dynamics. For such systems, the key parameters for describing the behavior of the population are the kinetic constants of the underlying pathway of interest, and the physiological functions that express the single cell growth rate, DNA-duplication and division propensity, as well as the partitioning mechanism. Intrinsic noise can be inherently accounted for, once the cell size has been specified and thus no additional parameters are needed for this purpose. Such a description is expected to be of great importance in bioinformatics studies focusing on population variability, since, for cells interacting though division only this variability can be explained in terms of a limited number of parameters.

Finally, deviations between the experimentally observed population dynamics and the behavior predicted by our framework may indicate the presence of more complex effects that are not accounted for in this framework. For instance, the cells could be non-isogenic, or coupled with strong interaction mechanisms. Another source of complexity is the existence of multiple compartments in the cell, which would invalidate the assumption of a single well-mixed intracellular space. Such effects would have to be incorporated to the framework in order to obtain a more accurate description of the system of interest.

Competing interests

The authors declare that they have no competing interests.

Author's Contributions

MS developed the mathematical models and performed the simulations, analyzed the results and drafted the manuscript. KZ conceived of the study, participated in its design and coordination, and refined the manuscript. Both authors read and approved the final manuscript.

Acknowledgements

The financial support of NIH-NIGMS through grant R01 GM071888 is gratefully acknowledged. This work was also supported in part by the Shared University Grid at Rice funded by NSF under Grant EIA-0216467, and a partnership between Rice University, Sun Microsystems, and Sigma Solutions, Inc.