Cell cycle heritability and localization phase transition in growing populations

Abstract

The cell cycle duration is a variable cellular phenotype that underlies long-term population growth and age structures. By analyzing the stationary solutions of a branching process with heritable cell division times, we demonstrate existence of a phase transition, which can be continuous or first-order, by which a non-zero fraction of the population becomes localized at a minimal division time. Just below the transition, we demonstrate coexistence of localized and delocalized age-structure phases, and power law decay of correlation functions. Above it, we observe self-synchronization of cell cycles, collective divisions, and slow ‘aging’ of population growth rates.

The duration of a cell-cycle, or inter-division time (IDT), is a fluctuating quantity in cellular populations, and its statistical properties are thought to result from biological mechanisms that regulate cell growth and division [1–5]. In most observations on mammalian cells, and in a subset of bacterial experiments, positive mother-daughter correlations of IDTs have been measured (see Table S1 in [6]). In general, heritability of a trait (i.e. positive parent-offspring correlation) enables selection to act on the trait distribution in a population to increase the long-term population growth rate. In this Letter, we take up the question of how selection and cell cycle heritability interact to determine long-term population dynamics, a fundamental step toward understanding how evolution has shaped cell cycle control mechanisms. Our work makes connections between dynamics of age-structured populations [7] and error-threshold phenomena of evolutionary theory [8, 9], and is applicable in experimental analyses of cellular population dynamics.

Model and stationary solutions.

We model a proliferating population by a branching process in which K (τ, τ′) is the transition probability density from τ′ to τ, where τ′ and τ are the parent and offspring IDTs, respectively. We focus on the analytically tractable Lebowitz-Rubinow model [10], which uses the transition kernel

$$K\left(\tau ,{\tau }^{\prime }\right)=h\delta \left(\tau -{\tau }^{\prime }\right)+(1-h)k(\tau ),$$ (1)

where δ (·) is the Dirac delta function showing accurate inheritance of the parent’s IDT, h represents the heritability of IDTs in the model (0 ≤ h < 1), and k (τ) is a probability density function on the interval [τ₀, ∞), with 0 < τ₀ < ∞. A cell produces $\widehat{z}$ offsprings at every division, where $\widehat{z}$ is independently drawn from a fixed probability distribution and its average is denoted by z > 0. The dynamics of cell divisions in the population are governed by

$$n_{\text{div}}(\tau ;t+\tau )=z{\int }_0^{\infty }K\left(\tau ,{\tau }^{\prime }\right)n_{\text{div}}\left({\tau }^{\prime };t\right)d{\tau }^{\prime },$$ (2)

where n_div (τ; t) dτdt is the expected number of dividing cells (cells at the termination of cell-cycle) between times t and t + dt with IDT between τ and τ + dτ [11]. The expected number of divisions occurring between t and t + dt is given by $N_{\text{div}}(t)dt:=dt{\int }_0^{\infty }{n}_{\text{div}}(\tau ;t)d\tau$, and the time-dependent population growth rate is defined by ${\Lambda }_t:=\raisebox{1ex}{$(z-1)N_{\text{div}}(t)$}\!\left/ \!\raisebox{-1ex}{$N(t)$}\right.$, where N (t) is the expected population size at time t (see [6] for detailed derivations).

The steady-state, exponentially growing solution of (2) is characterized by the time-independent growth rate Λ and the probability density p_div (τ) of IDTs of dividing cells, which can be found by substituting the form n_div (τ; t) = N_div (t) p_div (τ), where N_div (t) is proportional to e^Λt, yielding

$$p_{\text{div}}(\tau )=z{e}^{-\Lambda \tau }{\int }_0^{\infty }K\left(\tau ,{\tau }^{\prime }\right)p_{\text{div}}\left({\tau }^{\prime }\right)d{\tau }^{\prime }.$$ (3)

Using (1) and solving the above equation one obtains

$$p_{\text{div}}(\tau )=q_{h,\Lambda }(\tau )$$ (4)

where

$$q_{h,\Lambda }(\tau ):=\frac{(1-h)k(\tau )z{e}^{-\Lambda \tau }}{1-hz{e}^{-\Lambda \tau }},$$ (5)

which is a valid solution if there exists Λ such that q_h,Λ(τ) ≥ 0 for all τ ∈ [τ₀, ∞), and which normalizes p_div(τ), i.e. Λ is the unique real solution of Q(h, Λ) = 1, where

$$Q(h,\Lambda ):={\int }_0^{\infty }{q}_{h,\Lambda }(\tau )d\tau .$$ (6)

If z = 1, for example, then Λ = 0 and p_div (τ) = k (τ) for any 0 ≤ h < 1; this can be realized as an isolated single cell, where offspring at each division event are removed.

In the absence of IDT heritability (h = 0), one recovers the well-known result p_div (τ) = ze^−Λτ k (τ) where Λ is the unique real root of the integral equation $z{\int }_0^{\infty }{e}^{-\Lambda \tau }k(\tau )d\tau =1[3,12]$. For h > 0, one additionally must have 1 − hze^−Λτ > 0 for all τ in the support of k(τ) to ensure q_h,Λ (τ) ≥ 0. If k(τ) > 0 for all τ > τ₀, we find

$$\Lambda \ge {\omega }_0(h):=\underset{\tau >{\tau }_0}{\text{sup}}\frac{\text{ln}(hz)}{\tau }=\text{max}\left(0,\frac{\text{ln}(hz)}{{\tau }_0}\right);$$ (7)

(see [6] for k(τ) with bounded support). Since Q(h, Λ) is a monotonically decreasing function of Λ tending to zero as Λ → ∞, Q(h, Λ) = 1 has a unique root Λ = Λ(h) provided that Q(h, ω₀ (h)) ≥ 1. This condition holds for h < h_c, where h_c is a heritability threshold defined by

$$Q\left(h_c,{\omega }_0\left(h_c\right)\right)=1,$$ (8)

such that for h > h_c, Q(h, ω₀ (h)) < 1 and Q(h, Λ) = 1 does not admit a real root Λ. For h > h_c, the solution p_div (τ) given in (4) is incomplete, as there is missing probability 1 − Q. For z > 1, the full solution is

$$p_{\text{div}}(\tau )=q_{h,\Lambda }(\tau )+(1-Q(h,\Lambda ))\delta \left(\tau -{\tau }_0\right),$$ (9)

and substitution into (3) yields $\Lambda (h)={\tau }_0^{-1}\text{ln}(hz)$ for the steady-state growth rate when h > h_c [6].

Further analysis of (8) shows that a heritability threshold h_c < 1 exists if and only if ${\int }_0^{\infty }d\tau k(\tau )/\left(\tau -{\tau }_0\right)$ converges [6]; e.g. if k(τ) ~ (τ − τ₀)^γ near τ₀ for some γ > 0. The steady state population growth rate Λ(h) qualitatively changes as h crosses the threshold: it depends on k (τ) for h < h_c, and becomes independent of it for h > h_c. The expected number of offspring having the same IDT as their parent is hz, and only parents with τ′ = τ₀ can generate offspring with τ = τ₀. Thus, the fraction of the population with IDT τ₀ grows with rate ${\tau }_0^{-1}\text{ln}(hz)$. If we add a small fraction of τ₀ cells to a population, they go extinct if hz < 1, while if hz > 1, they can constitute a giant cluster in the population’s genealogy. The subpopulation localized at τ = τ₀ will be outcompeted by the rest of the population for z⁻¹ < h < h_c, with Λ(h) determined by (6); or it will dominate the population for h > h_c, and thus dictate its growth rate to be $\Lambda (h)={\tau }_0^{-1}\text{ln}(hz)$. In Fig. 1A, we show a range of examples of k(τ) which admit a threshold h_c. Increasing h from 0 to 1, Λ(h) increases monotonically with a shallow slope, while past h_c, the slope of the growth rate changes markedly.

FIG. 1.

Stationary solutions and their dependence on the heritability parameter h. Solid curves show results using z = 2 for (A) population growth rate, Λ(h); (B) reciprocal mean block size, $\overline{m}{(h)}^{-1}$; and (C) reciprocal ancestral mean IDT, $\overline{\tau }{(h)}^{-1}$. Dotted line in (B,C) indicates the result for isolated single cells (i.e. z = 1). The IDT distribution k (τ) (shown at inset of A) is chosen to be a gamma distribution, shifted by τ₀ = 0.2, with shape parameters α = 1.5 (blue), 4 (orange), and 20 (green); $k(\tau )=\Gamma {(\alpha )}^{-1}{\theta }^{-\alpha }{\left(\tau -{\tau }_0\right)}^{\alpha -1}{e}^{-\left(\tau -{\tau }_0\right)/\theta }$ for τ ≥ τ₀. For the mean IDT to be 1, the scale parameter θ is chosen as τ₀ +αθ = 1. h_c = 0.68, 0.60 and 0.58 respectively for α = 1.5, 4, and 20.

Localization phase transition of population age structure.

We now show that the threshold behavior identified above constitutes a phase transition in the strict sense. We map the population to a statistical mechanical ensemble, as follows. From the viewpoint of single cell lineages – i.e. the history of an individual and all of its ancestors – an age-structured population is an ensemble of trajectories: the sequence of IDTs along a lineage (…, τ_i−1, τ_i, τ_i+1, …) is analogous to a microscopic state of a large system (e.g. configuration of spins on a lattice, conformation of a polymer in space, etc.); while a single ancestral cell division τ_i specifies the state of a single component (e.g a spin, or a monomer) [6, 13].

To analyze the structure of lineages observed above and below h_c, we consider the number of generations with which the same IDT is consecutively inherited, which we denote by m and call the “block size”. The probability distribution of m over lineages, p_lin(m), is analogous to a correlation function, and its mean measures the typical correlation length. From (3) one can infer that the joint probability distribution of block size m and IDT τ is

$$p_{\text{lin }}(m,\tau ):=\left(1-hz{e}^{-\Lambda \tau }\right){\left(hz{e}^{-\Lambda \tau }\right)}^{m-1}{p}_{\text{div }}(\tau ).$$ (10)

For h < h_c, the mean block size $\overline{m}(h)$ is finite, while for h > h_c, $\overline{m}(h)$ diverges. The probability distribution of IDTs on lineages [6] is given by

$$p_{\text{lin}}(\tau )=\{\begin{array}{cc}\frac{(1-h)k(\tau )z{e}^{-\Lambda \tau }}{{\left(1-hz{e}^{-\tau \Lambda }\right)}^2\overline{m}},& 0\le h<h_c\\ \delta \left(\tau -{\tau }_0\right),& h_c<h<1\end{array}.$$ (11)

The expression indicates that the IDT distribution on lineages in the population is localized entirely at τ = τ₀ above h_c, despite the fact that the IDT distribution of isolated lineages remains k (τ).

The mean block size $\overline{m}(h)$ serves as an order parameter that diverges for h > h_c, and the continuity of the phase transition can be characterized by its behavior near h_c. If ${\text{lim}}_{h\to h_c^{-}}\overline{m}(h)=\infty$, the transition is a continuous phase transition, while if ${\text{lim}}_{h\to h_c^{-}}\overline{m}(h)<\infty$ the transition is referred to as discontinuous, or ‘first-order’. Analogously, the lineage mean IDT, $\overline{\tau }(h)$, which is given by ^∂Λ/_{∂ ln z}, exhibits the same type of phase transition. In terms of thermodynamics, the length of lineages t defines the system size and ln z can be seen as a generalized force (e.g. chemical potential). Then tΛ is the thermodynamic potential (e.g. grand potential) whose first-order derivative by ln z coincides with the expected number of divisions on lineages averaged across population, which is the extensive variable associated with ln z [6]. Examples of the h dependence of $\overline{m}(h)$ and $\overline{\tau }(h)$ are shown in Fig. 1B and C. Assuming the law of large numbers, $\overline{\tau }{(h)}^{-1}\simeq \raisebox{1ex}{$D$}\!\left/ \!\raisebox{-1ex}{$t$}\right.$ and $\overline{m}{(h)}^{-1}\simeq \raisebox{1ex}{$S$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$ hold where D and S denote respectively the number of divisions and the number of switches to different values of τ on lineage [6].

Using the form of k (τ) given in Fig. 1, where k(τ) ~ (τ − τ₀)^γ for τ near τ₀, and computing m(h) we find that the phase transition is continuous for 0 < γ ≤ 1, and becomes discontinuous for γ > 1 (see Fig. 1B). For a discontinuous transition, one expects to observe coexistence of two phases at the transition point, which is seen in numerical simulations of finite populations shown below (Fig. 2A). We can also compute the probability distribution of block sizes, which decays exponentially below the transition, and follows power law statistics, p_lin (m) ~ m^−γ−1 for large m, in the limit $h\to h_c^{-}$ as expected for correlation functions in the vicinity of a phase transition [6]. Such an appearance of long memory of IDT inheritance at the transition point is likewise observed in simulations (Fig. 2A).

FIG. 2.

Dynamics and aging of growth rates in a finite population. (A,B) Time courses of reciprocal empirical minimal IDT, ${\widehat{\tau }}_0^{-1}$ (top panels, orange, solid), compared with the minimal IDT, ${\tau }_0^{-1}$ (gray dashed line); and growth rates (bottom panels, blue line) compared with predicted stationary value (gray dashed line). (A) Coexistence of delocalized and localized states near the transition (h = 0.7); (B) aging dynamics above the transition (h = 0.8). Parameters are as in Fig. 1 using α = 4, which exhibits a first-order localization transition. Population size in simulations is maintained at N = 100, and the effective heritability threshold is h_c,N = 0.73; see [6] for simulation details.

Dynamics and aging of population growth rate.

In finite sized populations, the stationary distribution (9) for h > h_c is not achievable because any parent cell with IDT τ′ > τ₀ has probability zero of generating offspring with IDT τ₀. Additionally, the distribution (9) cannot be maintained as a steady-state because cells with IDT τ₀ can be lost from the population within finite time due to coalescence. To observe dynamics in finite populations, we conducted exact stochastic simulations in which cells are randomly removed to maintain a fixed population size [6]. We sampled the initial population with size N = 100 independently from the stationary probability distribution with h = 0, which we refer to as the delocalized state, and simulated the population forward in time for a given value of h > 0.

In simulations with h < h_c, growth rates Λ_t fluctuate around the expected steady-state growth rate (see [6]). Near the heritability threshold, however, Λ_t exhibits sudden transitions between two distinct, long-lived states (Fig. 2A), which is expected for systems near a first-order phase transition. One of these states represents localization at the empirical minimum IDT, denoted by ${\widehat{\tau }}_0$, which is the minimum IDT among all the dividing cells within each time bin [6]. In this state, the growth rate fluctuates around ${\widehat{\tau }}_0^{-1}\text{ln}(hz)$ over a sufficient period during which ${\widehat{\tau }}_0$ is constant. The other phase represents delocalization, where ${\widehat{\tau }}_0$ exhibits large fluctuations. For higher values of h, the observed growth rate increases stepwise and fluctuates around ${\widehat{\tau }}_0^{-1}\text{ln}(hz)$ (Fig. 2B). In this case, a fraction of the population localized at ${\widehat{\tau }}_0$ is maintained over a significant time interval until a new value of ${\widehat{\tau }}_0$ replaces the current empirical minimum IDT. The time intervals between these replacement events become increasingly long as ${\widehat{\tau }}_0$ approaches τ₀, because IDTs that are shorter than the current minimum become increasingly rare. Plotting values of Λ_t at different simulation times as function of h, the curve possesses an inflection point h_c,N slightly greater than h_c, which we refer to as the effective transition point at fixed population size N (see [6] for further details).

Collective divisions, cell cycle synchronization, and noisy inheritance.

In addition to aging dynamics, above the transition divisions occur collectively and periodically in finite populations. The number of divisions that occur, binned in short intervals, is plotted over time (Fig. 3, left panels), along with its auto-correlation function (Fig. 3, right panels). The auto-correlation function does not oscillate below the transition, but exhibits decaying oscillations near the transition point and sustained oscillations above the transition. The fact that the period of the autocorrelation function is approximately τ₀ reflects localization at ${\widehat{\tau }}_0$ close to τ₀. We note that division rate oscillations have also been predicted to arise in cell size control models, due to negative IDT correlations [14].

FIG. 3.

Cell cycle synchronization in finite populations. Parameters are as in Fig. 2. Number of divisions B(t, δt) binned over δt = 0.1 doubling time (A, C, E) and their auto-correlations (B, D, F) are shown. Simulations were run over t_max = 10⁴ time units and the last 10 time units of the series are shown for A, C and E, where time point 0 indicates the simulation end. Auto-correlation function ACF(Δt) was computed over entire time series. The correlation coefficient was normalized to equal 1 at Δt = 0. Dotted line indicates zero correlation. (A,B) Below the transition (h = 0.6) timing of divisions is not synchronized. (C, D) Near the transition point (h = 0.7), timing of divisions is partially synchronized, observed as decaying oscillation of the auto-correlation function. (E, F) Above the transition (h > h_c,N), collective divisions are observed indicating self-synchronization of cell cycles.

We tested the robustness of the transition properties to inaccuracy of the inheritance of IDTs by allowing small fluctuations of the offspring’s IDT when it inherits its parent’s IDT with probability h. To do so, we modified the transition kernel to be

$$K\left(\tau ,{\tau }^{\prime }\right)=h{p}_{\text{norm}}\left(\tau -{\tau }^{\prime },\sigma \right)+(1-h)k(\tau )$$ (12)

where p_norm (x, σ) is a normal distribution density function with standard deviation σ, truncated at τ₀. As a result, the population can reach equilibrium over a reasonable time scale, that is, the long-term behaviors coincide between the two distinct initial conditions. Collective divisions are weaker but still detectable through the decaying oscillations of the autocorrelation function of divisions in the population. Additionally, we analyzed the behavior of a general Gaussian kernel, truncated at τ₀, with parameters specifying the mean, variance and parent-offspring correlation of IDTs (the latter can be negative or positive). In all cases, for sufficiently high heritability, we found that as the population size increases from N = 1 to N = 1000, Λ exhibits a pronounced increase (see Fig. 4A and additional results in [6]). In contrast, for low heritability or for negative IDT correlations, Λ varies little with N. Similar behavior is observed for ${\overline{\tau }}^{-1}$, indicating that even without exact inheritance of parental IDTs, signatures of the phase transition are observed when varying the population size [6].

FIG. 4.

(A) Dependence of Λ on population size N. Simulations use Eq. 12 with σ = 0.1τ₀ and parameters as in Fig. 2. (B) Fitting the Lebowitz-Rubinow model using single cell lineage data (F3 rpsL-gfp, glucose, 37°C from [3]). Empirical (shaded) and fitted (solid) cumulative distribution of p_lin (τ). Inset: empirical p_div (τ). Fitting yields h = 0.23; for comparison, the Pearson correlation of mother-daughter IDTs is 0.26. See [6] for methods and additional data.

Discussion.

In this Letter, we analyzed how the strength of IDT heritability affects age-structured population dynamics. In a model with heritable cell cycle durations first introduced in [10], we demonstrate the existence of a localization phase transition with a heritability threshold above which the population’s distribution of IDTs localizes at the minimal IDT, corresponding to the fastest possible single-cell growth rate. We show that the ancestral mean IDT provides an order parameter of the transition, and above the heritability threshold predict the emergence of single cell lineages that maintain perfect inheritance of a minimal IDT.

A similar transition in exponentially growing populations is known in mutation-selection models of theoretical population genetics [9, 15, 16]. For example, Kingman’s house-of-cards model [16], which describes population dynamics on a specific type of fitness landscape, exhibits localization at maximal fitness below a critical mutation rate. The results can be generalized to cases in which fitness is correlated between parent and offspring using theory of positive linear operators [17].

The Lebowitz-Rubinow model is an idealization in which IDTs are either inherited precisely (with probability h) or not at all, and a key property of the model which underlies localization is the existence of a positive, minimal IDT, τ₀. In [6], using experimental measurements in E. coli we show that by fitting empirical lineage IDT distributions (using Eqs. 4 and 11), the model correctly infers h (see Fig. 4B). Additionally, the data support the existence of τ₀ > 0. Thus, the Lebowitz-Rubinow kernel captures key features of real biological data, and correctly predicts its underlying parameters.

We showed that the signature of the localization transition can be experimentally observed by measuring the population growth rate Λ at different population sizes in the range N = 1 − 1000, which is feasible in microfluidic experiments [3, 18, 19]. Varying the population size enables one to control the strength of selection in the experiments, as the size of heritable fitness differences that selection can act on efficiently scales with N⁻¹. Negative IDT correlations, as often observed in bacteria (Table S1 in [6]), correspond to a lack of heritability and do not yield transition-like behavior for increasing N [6]. Above the IDT heritability threshold, population growth rates exhibit aging dynamics similar to evolutionary dynamics in a random, unbounded fitness landscape in the low mutation rate limit [20, 21]. While aging dynamics occur for precise IDT heritability, transition-like behavior is present in less accurate IDT inheritance systems typical of biological systems [6].

Remarkably, the Lebowitz-Rubinow model does not include cell-cell interactions, which are thought to be a major mechanism for cell-cycle synchronization, yet our analysis predicts that strong but imperfect IDT heritability may be sufficient to cause self-synchronization of cell cycles in finite populations. This finding provides a starting point for the design of new types of synthetic biological oscillators that leverage population-level selective forces to establish robust cell cycle synchronization with sustained oscillations.