Replicative senescence as a consequence of stochastic processes

The aging of multicellular organisms is a complex problem. It has been the subject of extensive research and have led to various theories. One of the remarkable features of aging is that is accompanied by the decrease in the rate of cell replication in tissue regeneration processes (replicative senescence). In the present work, we will show, if sporadic cell alterations occur in a homogenous set of stem cells, they lengthen the cell cycle and these are fixed with greater probability than non-altered stem cells. This effect is due to the inherent characteristics of the renewal dynamics and as time goes by it leads to a quiescence state for stem cells due to the recurrent fixation of such altered cell.

The effect of aging in multicellular organisms in homeostatic regime is characterized 2 by the decline of cell renewal processes in all organs and tissues [1][2][3][4][5]. This diminution 3 is closely related to the functions of stem cells (SC) responsible for cell renewal and it 4 is more evident in the tissues of which a high rate of cell renewal, such as the skin 5 epithelium or intestinal epithelium, is necessary. Although some variations in the 6 behavior of SC are known with age, and studied from different the point of view, as 7 genomic [6][7][8], epigenomic [9,10] and proteomic [11][12][13], although the scope and 8 consequences of these changes have not been well established. Consider a tissue or 9 part of it, whose cell renewal processes is due to small set of SC, (as in the intestinal 10 crypts). It has been established that the population dynamics of SC in cell renewal 11 process is neutral, from the monoclonal character of the crypts [14,15]. This was 12 profusely studied in the framework of the dynamics of birth and death processes in a 13 population of distinguishable but equally fit individuals, which leads to the survival of 14 the descendants of only one of the original individual. This is a purely stochastic 15 phenomenon and constitutes a spontaneous break of symmetry. Within the cell 16 renewal process in the crypts a similar phenomenon occurs, because the symmetrical 17 mitoses originating two cells involved in the differentiation process (DD division) 18 means eliminating a SC, while symmetrical divisions originating SC (SS division) 19 equals the birth of one of them. We show that any disturbance (from any source) 20 sporadically affecting any of SC, as to extend their cell cycle length, leads to 21 replicative senescence and is a consequence of the dynamics of cell renewal. Cell 22 renewal in homeostatic regime is determined by a delicate balance between the 23 occurrence of symmetric cell divisions of each type. So, to preserve SC population 24 finite and avoid its extinction, the occurrence of these events must be regulated. To 25 study this dynamic, two alternative points of view are usually adopted. In the 26 homeostatic regime, the SC number will be considered fixed and the regulation is 27 taken as "perfect", that is at each symmetrical DD division it will be followed again by 28 a symmetric SS division.This leads to the well-known Moran model [16] (widely used 29 to study the effect of genetic drift in a small population group) and more recently, in 30 processes of cell renewal, the fixation of neutral somatic mutations in the framework of 31 cancer development [17]. The other point of view is to assume certain regulation for 32 the occurrence of possible events for SC population and their daughter cells, which give 33 rise to the complete lineage of the considered tissue. For a homogeneous population of 34 each cellular phenotype is known as compartmental models. These models are 35 described by a system of ODEs for the populations of each compartment [18,19]. The model can be raised a priori, since the equilibrium population and its fluctuations will 46 depend on whether or not the alteration was fixed. In what follows, we will name the 47 alteration as a mutation, but it is not necessarily related to somatic mutation. 48 Therefore, in the present context we use mutation in the sense of any altered cell.

49
However there is an exception to the latter that is when the mutation was fixed but 50 the equilibrium point or the fluctuations around it do not changed. This is the kind of 51 mutation that will be studied in this work. This one will change the length of the cell 52 cycle but not the balance or the regulation between the different possible events.

53
First we introduce a differential equation associated to the compartment of the SC 54 and introduce the family of mutation that we will study. Then, we will calculate the 55 fixation probabilities of these mutations by studying the associated stochastic process   Let us consider a compartment corresponding to SCs whose populations obeys the 68 following differential equation: where s is the total number of SC present at time t in the compartment. The rate One wonders, what will happen if a different SC emerges (for example due a 75 somatic mutation or any other alteration). Supose the desapperance (birth) rate of the 76 mutant is β 1 f (s) (β 2 g(s)).

77
Clearly Eq (1) cease to have meaning because SC population in not longer 78 homogenous. To study the resulting dynamics, we must consider it as a stochastic 79 process of four possible events:

84
The description of population dynamics through an equation such as Eq (1), will 85 be recover if WSC or MSC are extinguish. If the latter, Eq (1) should be replaced by: and now, the equilibrium population should be defined by: Let consider mutations where β 1 = β 2 = α. Therefore, Eq (3) is reduced to Eq (1)

88
(it is worth to notice that there is not change in the equilibrium population) and 89 Eq (3) is rewritten as: By defining t ′ = αt this equation is identical to Eq (1). This means that MSCs 91 have different length cell cycle, τ m , than those of WSCs, τ w , such that: Given α < 1, the fixation of such mutation slows cell renewal processes, what is 93 own aging of tissues.

94
In the following section we disscus the probabilities of fixing this kind of mutations. 95

96
To analyze how mutations are fixed let consider a function V (s) as a potential: it is enough to have a deep minimum at s = s e to avoid extinctions or exponential 98 growth due to fluctuations.

99
November 24, 2018 3 3/14 therefore, considering in Eq (3) we propose: which fulfill the conditions given above. For all results shown in this paper we take 101 a = 1 / day, b = 100 1/ day, s 0 =10 and therefore s e = 10, with these parameters the 102 cell cycle is τ = 1 day for the homogenous starting system under homeostasis 103 conditions. 104 We start from the state in which there are present (s e − 1) WSCs and only one 105 MSC, this is labeled as (s e − 1, 1). Therefore the events mentioned in the previous 106 section are: where s (s ′ ) is the number of WSCs (MSCs). The probabilities of occurrence of these 112 event are given in Table 1.
113 Table 1. Probabilities of the events that define the stochastic process mentioned in the text.
Event Probability We determine the probabilities of fixation by two ways:  Which is striking about these results is that the probabilities of fixing of MSC 124 increases with respect to the probability of fixing each WSC. So, in cell renewal 125 process from symmetrical divisions, the fixing of those SC with longer cell cycle will be 126 favored.

127
In next section we will propose a simplification that allow to determine the 128 dependence of fixation probabilities on α. The curves correspond to the probability of fixation of mutant and each of the WSC starting from a state ((s e − 1), 1). The red and green curves were obtained from the stochastic process in two different ways. Those which are green correspond to the average of 10 6 realizations of the process, while the red ones to the exact calculation of the fixed probability vector of the stochastic matrix. The blue curves correspond to considering the stochastic process as a Moran and leads to two algebraic expressions for such probabilities.
The associated probabilities are shown in Table 2.
142 Table 2. Probabilities of the events that define the stochastic process mentioned in the text.
Event Probability The resulting stochastic matrix has two absorbent states: (s e , 0) and (0, s e )(all 143 WSCs or MSC). Starting from the initial state (s e − s ′ , s ′ ), the probability of ending at 144 (0, s e ) for α > 0 results (see see section Methods): and the probability of ending at (0, s e ) is its complementary: The latter correspond to a not fixed mutation but any of (s e − 1) WSCs, while the 147 probability of one of them is fixed is: So, after k fixed mutations the cell cycle length will be α k = 1/τ (we assume as a 176 unit the cell cycle length of the original population).

177
The probability that of N mutations k are fixed is: where P α is the probability of fixation of one mutation. Thus, the cell cycle length 179 after N mutation is: and taking into account the first of Eqs (12): It is possible generalize these results for more than one α. If there are two different 182 mutations characterizes by α and α ′ , and assuming that the mutation associated with 183 α occurs with probability p + and p − for α ′ , (p + + p − = 1), we obtain: We will now assume that α increases the cell cycle length to the same extend that 185 α ′ decreases it, this means α + α ′ = 2. Therefore we have:  The graph illustrates the effects in the cell cycle τ after 200 mutations assuming that there may be two kind of mutations. One of them increases the duration of τ (α < 1) and decreases it with probability p + , while the other decreases it (α ′ > 1) with probability p − and α + α ′ = 2 (see Eq 17). The blue one corresponds to p + = 1, p − = 0, for the mutation which increases the cell cycle. The curvesred, green and black to p + = 2/3, p − = 1/3, p + = 1/2, p − = 1/2 and p + = 0.45, p − = 0.55.

196
The results shown above were found by numerical calculation and analytical 197 derivations.

198
Numerical calculations 199 These were carried out in two ways. 200 1. To simulate the stochastic process by using Gillespie´s Algorithm [20]. 201 2. Detrmination of the fixed probability vector associated to stochastic matrix.

202
The stochastic matrix is:  respectively. Their elements are: The matrix T acting on the probability vector p: where: . . .
November 24, 2018 8 8/14 P M (M−n,n) is the probability that the system has (M − n) WSCs and n MSCs.

208
The stochastic process changes the total number of SCs. In fact, each event of 209 the process creates or destroys a cell. On the other hand, the matrix T connects 210 even states (M even) with odd (odd M ) or odd with even ones. Taking 211 advantage of this fact we can reduce T. Rearranging p as: We obtain: where: and: November 24, 2018 9 9/14 Therefore T 2 has one representation on the even space (BA) and other on the 216 odd space (AB).
To perform calculations with these expressions, it must be taken into account 218 that the matrix T (like A and B) has infinite dimensions. So, they must be 219 truncated. This truncation is not always possible because we must guaranteed 220 that after it the matrices remain stochastic, then it is necessary that:

241
Regarding simplified model, we will show how obtain the probabilities of fixation of 242 MSC starting from a given state (N − n, n) (that is (N − n) WSC, n MSC). The 243 stochastic matrix T acts on column vectors like: where P N −n.n is the probability that the system is in the state (N − n, n).

245
The matrix T is tridiagonal and its elements are: T can be write as:

249
Defining the row vectors v and u as 250 v = t 12 0 · · · 0 (34) And due to We obtain: The last row of the latter (0, u(I − R) −1 , 1) contains the probabilities of fixing 255 MSC. In fact,due to the vector u it is not necessary to calculate the whole (I − R) −1 .

256
We only need to know the last row.

257
In this way we obtain: Whereas cell renewal follows a neutral dynamic from a small set of SC, we have shown 260 that fixation SC with longer cell cycle is favored and therefore, as time goes by, such 261 renewal is becoming slower. This does not mean that cellular senescence is due to this 262 effect (it is due purely to the characteristics of renewal dynamics). However, and