#### Abstract

To quantitatively study the effect of delay on selection dynamics in long-term sphere culture of cancer stem cells (CSCs), a selection dynamic model with time delay is proposed. Theoretical results show that the ubiquitous time delay in cell proliferation may be one of the important factors to induce fluctuation, and numerical simulations indicate that the proposed selection dynamical model with time delay can provide a better fitting effect for the experiment of a long-term sphere culture of CSCs. Thus, it is valuable to consider the delay effect in the future study on the dynamics of nongenetic heterogeneity of clonal cell populations.

#### 1. Introduction

In the past years research on cancer stem cells (CSCs) has become a focus of cancer research, because CSCs have self-renewing and multidirectional differentiation capability and may result in tumors [1–6]. Recently, cell state dynamics due to non-genetic heterogeneity of clonal cell populations also has received more and more attention [7–10].

In order to expand CSCs, sphere culture is performed by experimental cell biologists [11, 12]. However, whether long-term sphere culture can maintain a high ratio of CSCs is unclear. For this question, it is interesting that [13, 14] obtain a similar quantitative result through different mathematical model; that is, the ratio of CSCs will towards an apparent equilibrium state in a long-term sphere culture. Concretely, [13] proposed a kinetic model using ordinary differential equations that considered the symmetric and asymmetric division of CSCs, as well as the proliferation and transformation of differentiated cancer cells (DCCs). And [14] puts forward a Markov model in which cells transition stochastically between states. However, the time delay due to the maturation of individual cells has been ignored in [13, 14].

In the present paper, based on the kinetic model in [13], we further explore the effect of time delay on selection dynamics in long-term sphere culture. The results show that the ubiquitous time delay in cell proliferation may be one of the important factors to induce fluctuation, and the proposed selection dynamical model with time delay can provide a better fitting effect for the experiment of a long-term sphere culture of CSCs in [13]. The organization of the paper is as follows. In Section 2, we formulate the selection dynamical model with time delay in long-term sphere culture of CSCs. Section 3 first gives the analytic analysis on our proposed model and then presents numerical simulations to compare the effect of time delay. Finally, some predictive conclusions with biological implications are given in Section 4.

#### 2. Model Description

Let and denote the population sizes of CSCs and DCCs at time in long-term sphere culture, respectively. Our previous work [13] proposed the following mathematical model to describe the interactive growth of the CSCs and DCCs in a long-term sphere culture: Here the constants are called the net birth rate or intrinsic growth rate of population , respectively. denotes the conversion rate from CSCs to DCCs in the process of CSCs proliferation, and denotes the conversion rate from DCCs to CSCs in the process of DCCs proliferation.

Note that time delay may play an important role in many biological models. As shown in [15], the maturation of individual cells may need a period of time ; that is, the number of these cells at time may depend on the population at a previous time . Under the assumption of equal discrete retarded cell proliferation, model (1) can be modified to Here is the time delay due to maturation time.

For (2), inspired by [16], we give the following average fitness of the population: Thus, the selection dynamics in long-term sphere culture of CSCs can be written as For (4), let . We have Therefore, as ; that is, the total population size remains constant. Hence and in (4) can be understood as the frequency of CSCs and DCCs, respectively. Furthermore, since can be replaced by , system (4) describes only a single differential equation; that is, in which , and . Note that (6) is the final selection dynamical model with time delay in long-term sphere culture of CSCs.

#### 3. Results

##### 3.1. Dynamic Analysis

The objective of this subsection is to analyze the dynamical behavior of (6). In order to explore the effect of the delay, we split this into two cases.

###### 3.1.1. Case of

In this case, we focus on the dynamic analysis if the delay is nonexistent; that is, in (6). We start by studying the existence of nonnegative equilibria in the interval . Let . Clearly, the discriminant of the quadratic equation is Hence there are two different real roots for if . Furthermore, since we know that there is a unique positive equilibrium for (6) if (see Figures 1(a) and 1(b)). When (see Figure 1(c)), it is clear that there is only one positive equilibrium

**(a)**

**(b)**

**(c)**

**(d)**

The combination of the above results and the phase diagram (see Figure 1(d)) of system (6) yields the following result.

Proposition 1. *For system (6), when , a unique positive equilibrium always exists, and it is globally asymptotically stable.*

###### 3.1.2. Case of

In this case, we focus on the dynamic analysis if the delay is existent; that is, in (6). Clearly, the unique positive equilibrium still remains for (6) in spite of the delay. To study the stability of the equilibrium , we first translate to the origin. Let Then (6) becomes, after replacing by again,

The variational system of (11) at the origin is given by The characteristic equation of linear system (12) is given by If we let , then (13) becomes By setting in (14), we have

Solving the first algebraic equation in (15), we have Substituting (16) into the second equation of (15), we have Now, substituting (17) into (16), we have

Next, we compute . Differentiating (13) with respect to , we have thus,

Therefore, similar to [17], according to the results in [18, 19] or [15, Theorem 2.2], we can obtain the following results on (6).

Proposition 2. *Suppose . Then system (6) has a Hopf bifurcation at
*

Furthermore, according to the results in [15, 20, 21], we have the following.

Proposition 3. *Suppose in (6). Then the unique positive equilibrium is stable if and unstable if .*

##### 3.2. Numerical Simulations

For model (1), we designed a long-term sphere culture of human breast cancer MCF-7 stem cells [13]. Based on the experimental data, using an adaptive Metropolis-Hastings (M-H) algorithm to carry out an extensive Markov-chain Monte-Carlo (MCMC) simulation, we obtained the estimated parameter values as follows:

When retarded cell proliferation was considered, based on the induced selection dynamic model (6) and the experimental data in [13], using extensive MCMC simulation again, we can obtain the estimated delay (Figure 2).

**(a)**

**(b)**

Using the estimated values, we can plot the best-fit solution by fitting model (1) and (6) to the experimental data, respectively, (Figure 3). From Figure 3, we find that there is a better simulation effect in model (6) than that in model (1). In fact, the sum of squares of the deviations in (1) is , whereas in (6). Note that is far less than . These results quantitatively confirmed that the induced selection dynamic model (6) with time delay can provide a better fitting effect in long-term sphere culture of CSCs.

#### 4. Conclusions

In order to demonstrate the interesting facts about the structural heterogeneity of cancer (the stable ratio between CSCs and DCCs), many studies have been reported because it is helpful for the cancer community to elucidate the controversy about the CSC hypothesis and the clone evolution theory of cancer [10, 13, 14] and references cited therein. In the present paper, a selection dynamic model with time delay is proposed, and its dynamical behavior is studied. Based on the theoretical analysis and numerical simulations, we can conclude the following predictive conclusions.

(i) The maturation of individual cells may produce a significant effect on the dynamic behavior of the selection dynamics. When the delay is nonexistent, the frequency of CSCs will tend to a stable size because the unique positive equilibrium is globally asymptotically stable (Proposition 1). Conversely, if the delay is existent, the unique positive equilibrium may not always maintain its stability and a Hopf bifurcation may be induced (Propositions 2 and 3); that is, an oscillated phenomenon may be induced by the maturation of individual cells.

(ii) Since the induced selection dynamic model (6) with time delay can provide a better fitting effect in long-term sphere culture of CSCs (Figure 3), it is reasonable to consider the delay effect in the future study on the dynamics of non-genetic heterogeneity of clonal cell populations.

Since mathematical models can be at best approximate the behavior of real biological process, the results presented here may extend those studies on the structural heterogeneity of cancer. Note that distributed delay may be more tractable and realistic than discrete delay in the applications of biology. Hence it is a worthwhile study in future work to better understand these topics based on the idea of [15].

#### Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (nos. 30872517 and 11271369) and the Natural Science Foundation Project of CQ CSTC (2010BB5020).