Applications of stochastic semigroups to cell cycle models

We consider a generational and continuous-time two-phase model of the cell cycle. The first model is given by a stochastic operator, and the second by a piecewise deterministic Markov process. In the second case we also introduce a stochastic semigroup which describes the evolution of densities of the process. We study long-time behaviour of these models. In particular we prove theorems on asymptotic stability and sweeping. We also show the relations between both models.


Introduction
The modeling of the cell cycle has a long history [27]. The core of the theory was formulated in the late sixties [16,30,38]. The important role in these models is played by maturity of cells. A lot of new models appear in the eighties and we can divide them into two groups. The first group contains discrete-time models (generational models) which describe the relation between the initial maturity of mother and daughter cells [13,36,37]. The second group is formed by continuous-time models characterizing the time evolution of distribution of cell maturity [6,19,29] or cell size [8,11]. The longtime behaviour of continuous-time models was studied in [4,20,22,28,33].
Mathematical modelling of cell cycle is still important and topical and new interesting models appear [1,2,7,9,17].
In this paper we consider two-phase models of the cell cycle. The cell cycle is a series of events that take place in a cell leading to its replication.
Usually the cell cycle is divided into four phases [3,12,21]. The first one is the growth phase G 1 with synthesis of various enzymes. The duration of the phase G 1 is highly variable even for cells from one species. The DNA synthesis takes place in the second phase S. In the third phase G 2 significant protein synthesis occurs, which is required during the process of mitosis. The last phase M consists of nuclear division and cytoplasmic division. Some models of cell cycle contains also a G 0 phase, where the cell has left cycle and has stopped dividing. From a mathematical point of view we can simplify the model by considering only two phases [5,35,37]. The first phase is the growth phase G 1 and it is also called the resting phase.
The second phase called the proliferating phase consists of the phases S, G 2 , and M . The duration of the first phase is random variable and of the second phase is almost constant. A cell can move from the resting phase to the proliferating phase with some rate, which depends on the maturity of a cell. Each cell is characterized by its age and maturity. The maturity can be size, volume or contents of genetic material.
We investigate discrete and continuous-time models characterized by the same parameters. The discrete model is slightly extended version of that of Tyrcha [37]. In our model the growth of maturity in both phases is described by different functions. We include the derivation of this model to have the paper self-contained. A cell can move from the resting phase to the proliferating phase with the rate which depends on its maturity. Then it spends a fixed time τ in the second phase and divides into two cells which have the same maturity. The maturity of a daughter cell is determined by the maturity of the mother cell at the moment of division. The mathematical model is given by a stochastic operator P which describes the relation between densities of maturity of new born cells in consecutive generations.
The continuous-time model is given by a piecewise deterministic Markov process (PDMP), which describes consecutive descendants of a single cell.
PDMPs are nowadays widely used in modeling of biological phenomena [26,34]. Some PDMPs were applied to describe statistical dynamics of recurrent biological events and could be used in cell cycle models [15,20]. The main problem with application of PDMPs to a two-phase model is to construct a stochastic process which has the Markov property. In the first phase the state of a cell depends on its maturity, but the second phase has a constant length, so the state of the cell depends on time of visit this phase.
The novelty of our model is that it consists a system of three differential equations which describes age, maturity, and phase of a cell. We consider two different jumps. The first jump is a stochastic one, when a cell enters the proliferating phase. The second one is deterministic at the moment of division. After the division of a cell, we consider time evolution of its daughter cell, etc. Since we include into the model age, maturity and phase of a cell, our process satisfies the Markov property. The evolution of densities of the PDMP corresponding to our model leads directly to a continuous-time stochastic semigroup {P (t)} t≥0 . The densities of the PDMP satisfy a system of partial differential equations with boundary conditions similar to that in [19]. It is interesting that the density of maturity satisfies a first order partial differential equation in which there is a temporal retardation as well as a nonlocal dependence in the maturation variable (35). This observation suggests that even rather complicated transport equations can be introduced by means of simple PDMPs and one can find numerical solutions of these equations by Monte Carlo methods.
We study long-time behaviour of the discrete-time semigroup {P n } n∈N and the semigroup {P (t)} t≥0 . We are specially interested in asymptotic stability and sweeping [14]. We recall that a stochastic semigroup is sweeping for each density f . We prove that both semigroups satisfy the Foguel alternative, i.e. they are asymptotically stable or sweeping from compact sets. This result is based on a decomposition theorem of a stochastic semigroup into asymptotically stable and sweeping components [24] (see also [25] for substochastic semigroups). We give some sufficient conditions for asymptotic stability and sweeping of the continuous-time stochastic semigroup. We also present an example such that the operator P is asymptotically stable but the semigroup {P (t)} t≥0 is sweeping from compact sets and explain this unexpected phenomenon. It should be noted that stochastic semigroups are widely applied to study asymptotic properties of biological models (see [32,34] and references cited therein).
The organization of the paper is as follows. Section 2 contains the definitions and results concerning asymptotic stability, sweeping and the Foguel alternative for stochastic semigroups. Biological and mathematical description of the cell cycle is presented in Section 3. In Section 4 we investigate the discrete-time model and we prove that the stochastic operator P related to this model satisfies the Foguel alternative (Theorem 4). We also recall some sufficient conditions for asymptotic properties of P . In Section 5 we introduce a continuous-time model as a PDMP and we show that the stochastic semigroup {P (t)} t≥0 corresponding to this process satisfies the Foguel alternative. In Section 6 we show the relations between discrete-time and continuous-time models, which allow us to formulate some conditions for asymptotic stability and sweeping of {P (t)} t≥0 . Finally, we compare asymptotic properties of both models.

Asymptotic properties of stochastic operators and semigroups
Let a triple (X, Σ, µ) be a σ-finite measure space. Denote by D the subset of the space L 1 = L 1 (X, Σ, µ) which contains all densities {P (t)} t≥0 of linear operators on L 1 is called a stochastic semigroup if it is a strongly continuous semigroup and all operators P (t) are stochastic. Now, we introduce some notions which characterize the asymptotic behaviour of iterates of stochastic operators P n , n = 0, 1, 2, . . . , and stochastic semigroups {P (t)} t≥0 . The iterates of stochastic operators form a discrete-time semigroup and we can use notation P (t) = P t for their powers and we formulate most of definitions and results for both types of semigroups without distinguishing them. A stochastic semigroup {P (t)} t≥0 is asymptotically stable if there exists a density f * such that Our aim is to find such conditions that a stochastic semigroup {P (t)} t≥0 is asymptotically stable or sweeping from all compact sets called the Foguel alternative [14]. We also want to find simple sufficient conditions for asymptotic stability and sweeping for operators and semigroups related to cell cycle models.
We assume additionally that X is a separable metric space and Σ = B(X) is the σ-algebra of Borel subsets of X. We will consider a stochastic semigroup {P (t)} t≥0 such that for each t ≥ 0 we have where q(t, ·, ·) : X × X → [0, ∞) is a measurable function and the following condition holds: (K) for every y 0 ∈ X there exist an ε > 0, a t > 0, and a measurable function where B(y 0 , ε) = {y ∈ X : ρ(y, y 0 ) < ε}.
We define condition (K) for a stochastic operator P in the same way remembering the notation P (t) = P t . Condition (K) is satisfied if, for example, for every point y ∈ X there exist a t > 0 and an x ∈ X such that the kernel q(t, ·, ·) is continuous in a neighbourhood of (x, y) and q(t, x, y) > 0. Now, we formulate the Foguel alternative for some class of stochastic semigroups. We need an auxiliary definition. We say that a stochastic semigroup {P (t)} t≥0 overlaps supports if for every f, g ∈ D there exists t > 0 such that The support of any measurable function f is defined up to a set of measure zero by the formula then for every f ∈ L 1 (X, Σ, µ) and for every compact set F we have In particular, if {P (t)} t≥0 has an invariant density f * with the support A and X \ A is a subset of a compact set, then {P (t)} t≥0 is asymptotically stable.
The proof of Proposition 1 is based on theorems on asymptotic decom- (i) for every j ∈ J and for every f ∈ L 1 (X, Σ, µ) we have then for every f ∈ L 1 (X, Σ, µ) and for every compact set F we have We recall that a stochastic operator S is called periodic if there exists a sequence of densities h 1 , . . . , h k such that The operator P can be restricted to the space (see the proof of Lemma 9 [24]), which means that the functions h i are periodic densities of P such that supp P n h i 1 ∩supp P n h i 2 = ∅ for each n and can be canonically embedded in the space L 1 (X, Σ, µ) and, therefore, R j can be treated as the transformation from L 1 (X, Σ, µ) to itself. In the statement of Theorem 1 we use the following definition of a projection. A linear transformation T from a vector space to itself is a projection if T 2 = T .
Then there exist an at most countable set J, a family of invariant densities {f * j } j∈J with disjoint supports {A j } j∈J , and a family {α j } j∈J of positive linear functionals defined on L 1 such that (i) for every j ∈ J and for every f ∈ L 1 we have then for every f ∈ L 1 and for every compact set F we have In particular, we have satisfies condition (K) and has no invariant densities. Then {P (t)} t≥0 is sweeping from compact sets.
Proof of Proposition 1. First, we consider the case of a stochastic operator.
If P satisfies conditions (K) and overlaps supports, then J is an empty set If J has at least two elements, then supp P n f ∩supp P n g ⊆ A 1 ∩A 2 = ∅ for n ∈ N and f, g ∈ D such that supp f ⊆ A 1 and supp g ⊆ A 2 , which contradicts the assumption that P overlaps supports. If J is a singleton, then the periodic operator S is in fact a projection on a one dimensional space because the overlaping property of P excludes the existence of two periodic densities h 1 and h 2 such that supp P n h 1 ∩ supp P n h 2 = ∅ for each n.
Thus condition (6) takes the form (4). If P has an invariant density f * with the support A and Y = X \ A is subset of a compact set, then from condition (5) it follows that lim n→∞ Y P n f (x) µ(dx) = 0. Since P is a stochastic operator, we have α(f ) = 1 for any density f , and consequently, P is asymptotically stable. The proof for continuous-time stochastic semigroups is straightforward because we have at most one invariant density and conditions (4), (11) coincide.
If a continuous-time stochastic semigroup {P (t)} t≥0 has a unique invariant density f * and f * > 0, then according to Theorem 1 condition (K) implies asymptotic stability of {P (t)} t≥0 . We can strengthen considerably this conclusion replacing condition (K) by the following one. A substochastic

From the biological background to a mathematical description
We start with a short biological description of the two phase-cell cycle models. The cell cycle is divided into the resting and proliferating phase.
The duration of the resting phase is random variable t R which depends on the maturity of a cell. The duration t P of the proliferating phase is almost constant. Therefore, we assume that t P = τ , where τ is a positive constant.
The crucial role in the model is played by a parameter m called maturity which describes the state of a cell in the cell cycle. Without loss of generality we can assume that the minimum cell maturity m min equals zero.
A cell can move from the resting phase to the proliferating phase with rate ϕ(m), i.e., a cell with age a and with maturity m enters the proliferation phase during a small time interval of length ∆t with probability ϕ(m)∆t + o(∆t).
We assume that cells age with unitary velocity and mature with a velocity

A discrete-time model
Now we consider a discrete-time model [37]. This model describes the relation between initial maturity of mother and daughter cells. We assume that a new born cell has maturity m 0 and we want to find the distribution of maturity of the daughter cell. In order to do it we first need to set down the distribution of t R .
Let Φ(t) be the cumulative distribution function of t R , i.e. Φ(t) = Prob (t R ≤ t). Then From this equation we obtain and we get Since dπ 1 /ds = g 1 (π 1 (s, m 0 )) we obtain  According to (M4) lim m→∞ Q(m) = ∞, which guaranties that each cell enters the proliferating phase with probability one. From (15) it follows that (16) t,m 0 )) .
Since the random variable π 1 (t R , m 0 ) is the maturity of the cell when it enters the proliferating phase, its maturity at the moment of division is given by π 2 (τ, π 1 (t R , m 0 )). Finally the maturity of the daughter cell is given by the random variable ξ = ψ(π 1 (t R , m 0 )).
In order to find the density of the random variable ξ we determine the expectation of the random variable E(F (ξ)), where F is any bounded and continuous real function. We have Thus the random variable ξ has the density Moreover, if we assume that the distribution of the initial maturity of mother cells has a density f , then the initial maturity of the daughter cells has  lim n→∞ P n f − P n g = 0 for f, g ∈ D.
If the operator P has an invariant density f * , then we can find the stationary distribution of age and maturity in both phases. From (15) it follows that if a cell has the initial maturity m 0 , then it will not have left the resting phase before age a with probability e Q(m 0 )−Q(π 1 (a,m 0 )) and has maturity π 1 (a, m 0 ) at age a. Thus the probability that cell remains in the resting phase at age a and has maturity ≤ m at this age is given by the formula Denote byf * (a, m, i) the stationary density of the distribution of age and maturity in both phases. Then from (18) it follows that for m ≥ π 1 (a, 0) andf * (a, m, 1) = 0 for m < π 1 (a, 0), where c > 0 is a normalized constant. Integrating (19) over the age variable a gives In order to findf * (a, m, 2) we need to find the distribution of maturity at the beginning of proliferating phase. We claim that the density of this distribution is given by f * p (m) = ψ (m)f * (ψ(m)). Indeed, if ζ is a random variable having density f * , then the density of random variable λ(ζ) coincides with the density of maturity at the beginning of proliferating phase.
Substituting y = π 1 (−a, m) and then x = π 1 (a, y) we obtain 1 (a,y)) dy da Thus c = 1/(T R + τ ) assuming that T R < ∞.   Fig. 2). Our aim is to check that the stochastic semigroup {P (t)} t≥0 defined on L 1 (X, Σ, µ) corresponding to the process ξ(t) satisfies the Foguel alternative and then to give some conditions for its asymptotic stability and sweeping.
Condition (24) is not particularly restrictive because if h(π 2 (τ, m 0 )) ≥ m 0 for some m 0 > m P , then independently on the maturity of a cell, the probability that descended cells will have maturity m < m 0 goes to zero as t → ∞ and we can consider a model with the minimal maturity m 0 . If a mother cell has maturity m > m P , then any number from the interval (ψ(m), ∞) can be initial maturity of a daughter cell, any number from the interval (ψ 2 (m), ∞) can be initial maturity of a granddaughter cell if ψ(m) > m P , etc. From condition (24) it follows that for sufficiently large n we have ψ n (m) ≤ m P . Since ψ(m P ) = 0 we conclude that after a finite number of generations the initial maturity of a descended cell can be any positive number m, m > π 1 (a, 0) can be the maturity of a descended cell at age a in the resting phase and m > π 2 (a, m P ) can be the maturity of a descended cell at age a in the proliferating phase.

Condition (25) seems to be technical but if
h (π 2 (τ, m))g 2 (π 2 (τ, m))g 1 (m) = g 1 (h(π 2 (τ, m)))g 2 (m) for all m ≥ m P , then all descendants of a single cell in the same generation have the same maturity at a given time t. It means that the cell have synchronous growth and we cannot expect the model is asymptotically stable. In particular if g 1 ≡ g 2 and h(m) = m/2, then (25) reduces to 2g 2 (m) = g 2 (2m) for some m > π 2 (τ, m P ). A similar condition appear in many papers concerning size-structured models [4,8,11,33,34].

Now we can formulate the Foguel alternative for semigroup {P (t)} t≥0
corresponding to the process ξ(t).
{P (t)} t≥0 is asymptotically stable or sweeping from compact sets.
If {P (t)} t≥0 has no invariant density, then according to Corollary 1 it is sweeping from compact sets.

Master equation
and the boundary conditions (31) r(t, 0, m) = k (m)p(t, τ, k(m)). and does not lead directly to a stochastic semigroup. In our case we replace one mother cell by one daughter cell which has allowed us to use a piecewise deterministic Markov process in the model's description.
Let r(a, m) =f * (a, m, 1) and p(a, m) =f * (a, m, 2), wheref * is given by (19) and (21). It is not difficult to check that r(a, m) and p(a, m) are solutions of (29)-(30) with boundary conditions (31)- (32). If then an invariant density exists and the semigroup {P (t)} t≥0 is asymptotically stable. Condition (33) is equivalent to T R < ∞, where T R is given by (22). Moreover, one can check thatf * is a unique, up to a multiplicative constant, positive stationary solution of (29)- (32), which gives that if t R = ∞ then the semigroup has no stationary densities, and therefore it is sweeping from compact sets. We skip here the rigorous justification of this statement.
The second integral in (33) is finite, and therefore, in order to check if an invariant density exists it is enough to check that the first integral is finite. In order to do it we investigate the function R(t, m), which is the total number of cells in the resting stage with given maturity m at time t, i.e.
Now we are looking for a stationary solution of (35). If R(m) satisfies (35) then R is a solution of the equation It is not surprising that iff * (m, 1) is given by (20), then R(m) =f * (m, 1) is a solution of (36). Moreover, if R is a solution of (36) with R(0) = 0, then the following formula holds: and U is an isometric operator from L 1 [m P , ∞) onto L 1 [0, ∞). Therefore, P n = U −1 P n U , and, in consequence, the operators P andP have the same asymptotic properties. Observe that P has an invariant density f * if and only iff * = U −1 f * is an invariant density forP .
Let us assume that P has an invariant density f * and ϕ(m) ≥ ε > 0 for sufficiently large m. Since Rϕ is a fixed point ofP , we have Rϕ = cU −1 f * for some c > 0. Hence, ∞ m P R(m) dm < ∞, which implies that the semigroup {P (t)} t≥0 is asymptotically stable. According to Proposition 2, if lim inf m→∞ Q(λ(m)) − Q(m) > 1 and ϕ(m) ≥ ε > 0 for sufficiently large m, then the semigroup {P (t)} t≥0 is asymptotically stable. Now, we assume that P has no invariant density and ϕ is a bounded function. Then R cannot be an integrable function. Assume contrary to our claim, that R is integrable. Then Rϕ is an integrable function and the operatorP has a positive fixed point. HenceP and P have invariant densities, a contradiction. According to Proposition 2, if Q(λ(m))−Q(m) ≤ 1 for sufficiently large m and ϕ is bounded, then the semigroup {P (t)} t≥0 is sweeping.
Remark 1. It can happen that the operator P is asymptotically stable but the semigroup {P (t)} t≥0 is sweeping. Indeed, if we choose g 2 , h and τ such that λ(m) = m + 2 for m ≥ 0 and we choose g 1 and ϕ such that Q(m) = m for m ≥ 3, then lim inf m→∞ (Q(λ(m)) − Q(m)) = 2 and the operator P is asymptotically stable. Let f * be an invariant density for P . The density f * depends only on Q and λ, so we can choose g 1 and ϕ such that ϕ(m) = g 1 (m) = f * (m−2) for m ≥ 3. Then R(m) = cU −1 f * (m)/ϕ(m) = c.
Consequently, R is not integrable and the semigroup {P (t)} t≥0 is sweeping.
The explanation of this phenomenon is that in this example the rate of entering the proliferating phase is very small for large m. Then the mean length of the resting phase can be large and more and more cells have arbitrary large maturity as t → ∞.