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Received 30 Jun 2016 | Accepted 11 Jan 2017 | Published 23 Feb 2017 DOI: 10.1038/ncomms14545 OPEN Hierarchical tissue organization as a general mechanism to limit the accumulation of somatic mutations

Imre Dere´nyi1 & Gergely J. Szo¨llosi+ 2

How can tissues generate large numbers of cells, yet keep the divisional load (the number of divisions along cell lineages) low in order to curtail the accumulation of somatic mutations and reduce the risk of cancer? To answer the question we consider a general model of hierarchically organized self-renewing tissues and show that the lifetime divisional load of such a tissue is independent of the details of the cell differentiation processes, and depends only on two structural and two dynamical parameters. Our results demonstrate that a strict analytical relationship exists between two seemingly disparate characteristics of self- renewing tissues: divisional load and tissue organization. Most remarkably, we find that a sufficient number of progressively slower dividing cell types can be almost as efficient in minimizing the divisional load, as non-renewing tissues. We argue that one of the main functions of tissue-specific stem cells and differentiation hierarchies is the prevention of cancer.

1 Department of Biological Physics, ELTE-MTA ‘Lendulet’ Biophysics Research Group, Eo¨tvo¨s University, Pa´zma´ny P. stny. 1A, Budapest H-1117, Hungary. 2 Department of Biological Physics, ELTE-MTA ‘Lendulet’ Evolutionary Genomics Research Group, Eo¨tvo¨s University, Pa´zma´ny P. stny. 1A, Budapest H-1117, Hungary. Correspondence and requests for materials should be addressed to I.D. (email: [email protected]) or to G.J.S. (email: [email protected]).

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n each multicellular organism a single cell proliferates to example, the location of the transit-amplifying cells along produce and maintain tissues comprised of large populations intestinal crypts). As a result, information on the details of Iof differentiated cell types. The number of cell divisions in the differentiation hierarchies is incomplete18–20. lineage leading to a given somatic cell governs the pace at which Nonetheless, in a recent paper, Tomasetti and Vogelstein21 mutations accumulate1. The resulting somatic mutational gathered available information from the literature and load determines the rate at which unwanted evolutionary investigated the determinants of cancer risk among tumours of processes, such as cancer development, proceed2–4. To produce different tissues. Examining cancers of 31 different tissues they N differentiated cells from a single precursor cell the theoretical found that the lifetime risk of cancers of different types is strongly minimum number of cell divisions required along the longest correlated with the total number of divisions of the normal lineage is log2(N). To achieve this theoretical minimum, cells self-replicating cells. Their conclusion that the majority of 21 must divide strictly along a perfect binary tree of height log2(N) cancer risk is attributable to bad luck arguably results from a (Fig. 1a). In multicellular organisms such differentiation typically misinterpretation of the correlation between the logarithms of takes place early in development. It is responsible for producing two quantities22,23. However, regardless of the interpretation of the cells of non-renewing tissues (for example, primary oocytes in the correlation, the data display a striking tendency: the the female germ line1,5) and the initial population of stem cells in dependence of cancer incidence on the number of stem cell self-renewing tissues (for example, hematopoietic stem cells6–8 or divisions is sub-linear, that is, a 100 fold increase in the number the spermatogonia of the male germ line1,5). of divisions only results in a 10 fold increase in incidence. This In self-renewing tissues, which require a continuous supply of indicates that tissues with a larger number of stem cell divisions cells, divisions along a perfect binary tree are unfeasible. Strictly (typically larger ones with rapid turnover, for example, the colon) following a perfect binary tree throughout the lifetime of the are relatively less prone to develop cancer. This is analogous organism would require extraordinarily elaborate scheduling of to the roughly constant cancer incidence across animals with individual cell divisions to ensure tissue homoeostasis9, and vastly different sizes and life-spans (Peto’s paradox), which would be singularly prone to errors (for example, the loss of any implies that large animals (for example, elephants) possess single cell would lead to the loss of an entire branch of the binary mechanisms to mitigate their risk relative to smaller ones tree). Instead, to compensate for the continuous loss of cells, (for example, mice)24–26. mechanisms have evolved to replenish the cell pool throughout What are the tissue-specific mechanisms that explain the the organism’s lifetime10. In most multicellular organisms differential propensity to develop cancer? It is clear that stem cells hierarchically organized tissue structures are utilized. At the that sustain hierarchies of progressively differentiated cells are root of the hierarchy are a few tissue-specific stem cells defined well positioned to provide a safe harbour for genomic informa- by two properties: self-replication and the potential for diffe- tion. Qualitative arguments suggesting that hierarchically orga- rentiation11,12. During cell proliferation cells can differentiate and nized tissues may be optimal in reducing the accumulation become increasingly specialized toward performing specific of somatic mutations go back several decades27. As mutations functions within the hierarchy, while at the same time losing provide the fuel for somatic evolution (including not only the their stem cell-like properties (Fig. 1b). A classic example is development of cancer, but also tissue degeneration, aging, germ the hematopoietic system13,14, but other tissues such as skin15 line deterioration and so on) it is becoming widely accepted that or colon16,17 are also known to be hierarchically organized. tissues have evolved to minimize the accumulation of somatic Identifying each level of the hierarchy, however, can be difficult, mutations during the lifetime of an individual27. The potential of especially if the cells at different levels are only distinguished hierarchical tissues to limit somatic mutational load simply by by their environment, such as their position in the tissue (for reducing the number of cell divisions along cell lineages, however, has not been explored in a mathematically rigorous way. Here, we discuss this most fundamental mechanism by which hierarchical tissue organization can curtail the accumulation of somatic abmutations. We derive simple and general analytical properties of N cells at the same time N cells over lifetime the divisional load of a tissue, which is defined as the number of divisions its constituent cells have undergone along the longest Mature cells cell lineages, and is expected to be proportional to the mutational load of the tissue. Models conceptually similar to ours have a long history7,28–33, going back to Loeffler and Wichman’s work on modelling hema- Progenitor 28 cells topoietic stem cell proliferation , and several qualitative argu- ments have been made suggesting why hierarchically organized tissues may be optimal in minimizing somatic evolution. Past 29 cell divisions In a seminal contribution Nowak et al. showed that Stem cells tissue architecture can contribute to the protection against the accumulation of somatic mutations. They demonstrated that Non-renewable Self-sustaining the rate of somatic evolution will be reduced in any tissue differentiation hierarchical differentiation where geometric arrangement or cellular differentiation induce Figure 1 | Differentiation in non-renewing versus self-renewing tissues. structural asymmetries such that mutations that do not occur in (a) To produce N mature cells from a single precursor with a minimum stem cells tend to be washed out of the cell population, slowing number of cell divisions, log2(N), strict division along a perfect binary tree is down the rate of fixation of mutations. Here, we begin where necessary. In multicellular organisms such ‘non-renewable’ differentiation Nowak et al.29 left off: aside of structural asymmetry, we consider typically takes place early in development. (b) However, in self-renewing a second and equally important aspect of differentiation, the tissues, where homoeostasis requires a continuous supply of cells, a small dynamical asymmetry of tissues, that is, the uneven distribution population of self-replicating tissue-specific stem cells sustain a hierarchy of divisional rates across the differentiation hierarchy. of progressively differentiated and larger populations of cell types, with cells More recently a series of studies have investigated the of each type being continuously present in the tissue. dynamics of mutations in hierarchical tissues with dynamical

2 NATURE COMMUNICATIONS | 8:14545 | DOI: 10.1038/ncomms14545 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14545 ARTICLE asymmetry31–33 and found that hierarchical tissue organization the accumulation of mutations in check, and that populations of can (i) suppress single32 as well as multiple mutations33 that cells currently believed to correspond to tissue-specific stem cells arise in progenitor cells and (ii) slow down the rate of somatic may in general constitute a diverse set of slower dividing evolution towards cancer31 if selection on mutations with cell types6,41. Most importantly, we find that the theoretical non-neutral phenotypic effects is also taken into account. The minimum number of cell divisions can be very closely appro- epistatic interactions between individual driver mutations are, ached: as long as a sufficient number of progressively slower however, often unclear and show large variation among cancer dividing cell types towards the root of the hierarchy are present, types. The fact that the majority of cancers arise without a optimal self-sustaining differentiation hierarchies can produce histologically discernible premalignant phase indicates strong N terminally differentiated cells during the course of an cooperation between driver mutations, suggesting that major organism’s lifetime from a single precursor with no more than histological changes may not take place until the full repertoire of log2(N) þ 2 cell divisions along any lineage. mutations is acquired34. For this reason, here we do not consider selection between cells, but rather, focus only on the pace of the Results accumulation of somatic mutations in tissues, which provide the Divisional load of cell differentiation hierarchies. To quantify fuel for somatic evolution. how many times the cells of self-renewing tissues undergo cell The uneven distribution of divisional rates considered by divisions during tissue development and maintenance, we 32,33 Werner et al. followed a power law, however, this distribution consider a minimal generic model of hierarchically organized, was taken for granted without prior justification. Their focus was self-sustaining tissue. According to the model, cells are organized instead on ‘reproductive capacity’, an attribute of a single cell into n þ 1 hierarchical levels based on their differentiation state. corresponding to the number of its descendants, which is The bottom level (level 0) corresponds to tissue-specific stem conceptually unrelated to our newly introduced ‘divisional cells, higher levels represent progressively differentiated pro- load’, which characterizes the number of cell divisions along the genitor cells, and the top level (level n) is comprised of terminally longest cell lineages of the tissue. Here we show mathematically, differentiated cells (Fig. 2a). The number of cells at level k in fully to the best of our knowledge for the first time, that the developed tissue under normal homoeostatic conditions is minimization of the divisional load in hierarchical differen- denoted by Nk. During homoeostasis cells at levels kon can tiation indeed leads to power law distributed differentiation rates. differentiate (that is, produce cells for level k þ 1) at a rate dk, and More generally, evolutionary thinking is becoming an indis- have the potential for self-replication. At the topmost k ¼ n level pensable tool to understand cancer, and even to propose of the hierarchy terminally differentiated cells can no longer 35 directions in the search for treatment strategies . Models divide and are expended at the same rate dn À 1 that they are that integrate information on tissue organization have not produced from the level below. The differentiation rates dk are only provided novel insight into cancer as an evolutionary defined as the total number of differentiated cells produced by the 27,36,37 process , but have also produced direct predictions for N cells of level k per unit time. The differentiation rate of a single 38–40 k improved treatment . The simple and intuitive relations that cell is, thus dk/Nk. we derive below have the potential to further this field of research In principle five microscopic events can occur with a cell: by providing quantitative grounds for the deep connection (i) symmetric cell division with differentiation, (ii) asymmetric between organization principles of tissues and disease prevention cell division, (iii) symmetric cell division without differentiation, and treatment. (iv) single cell differentiation and (v) cell death (Fig. 2b). Our goal According to our results, the lifetime divisional load of a is to determine the optimal tissue organization and dynamics that hierarchically organized tissue is independent of the details of the minimize the number of cell divisions that the cells undergo until cell differentiation processes. We show that in self-renewing they become terminally differentiated. For this reason cell death, tissues hierarchical organization provides a robust and nearly except for the continuous expenditure of terminally differentiated ideal mechanism to limit the divisional load of tissues and, as a cells, is disallowed as it can only increase the number of divisions. result, minimize the accumulation of somatic mutations that fuel We note, however, that cell death with a rate proportional to that somatic evolution and can lead to cancer. We argue that of cell divisions would simply result in a proportionally increased hierarchies are how the tissues of multicellular organisms keep divisional load and, thus, would have no effect on the optimum.

abDifferentiation hierarchy Events present in optimal hierarchies  Event type: Rate per cell: n n + 1 levels Terminally 1  N p : Symmetric k p differentiated cells n k 2 N k  division k n−1 Differentiation  Asymmetric k N 1 − pk : (1 − pk) n−1 division Nk Transiently   Differentiation 1 k k−1 differentiated qk : pkqk = Cell from below 2 Nk Nk cells  replenishment Symmetric 1  1 1 − q : k k pk (1 − qk) division 2 Nk Differentiation N1 rates  Events not present in optimal hierarchies 0

Stem cell N0

Single cell differentiation Cell death

Figure 2 | Hierarchical cell differentiation in self-renewing tissue. (a) A model tissue produces terminally differentiated cells through n intermediate levels of partially differentiated cells. (b) Five microscopic events can occur with a cell: (i) symmetric cell division with differentiation, (ii) asymmetric cell division, (iii) symmetric cell division without differentiation, (iv) single cell differentiation and (v) cell death. To the right of each type of event present in optimal hierarchies we give the corresponding per cell rate that is used to derive equation (2).

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Similarly, we also disregard single cell differentiation, because if between the number of cell divisions and the number of cells it is rare enough (that is, its rate is smaller than the asymmetric generated are concerned. cell division rate plus twice the rate of symmetric cell division Using the recursion relation the above differential equation without differentiation) then it can be absorbed in cell divisions system simplifies to with differentiation; otherwise it would merely delegate the D_ N ¼ ðÞÀd À d d ðÞD À D ; ð3Þ replication burden down the hierarchy towards the less k k k k À 1 k À 1 k k À 1 differentiated and supposedly less frequently dividing cells, and revealing that the average number of cell divisions is independent would be sub-optimal. of both the fraction of symmetric division pk in differentiation, Two of the remaining three microscopic events involve and the fraction of differentiation qk in replenishment. differentiation. If we denote the fraction of differentiation events From any initial condition Dk(t) converges to the asymptotic that occur via symmetric cell division at level k by pk, then the solution rate of symmetric cell division at level k can be written as pkdk/2 d0 0 (the division by two accounts for the two daughter cells produced DkðtÞ¼t þ D ; ð4Þ N k by a single division), while the rate of asymmetric cell division is 0 (1 À pk)dk. Symmetric cell division with differentiation leaves which shows that the divisional load of the entire tissue grows an empty site at level k, which will be replenished either (i) by linearly according to the differentiation rate of the stem cells differentiation from the level below or (ii) by division on the same (td0/N0), and the progenitorÀÁ cells at higher levels of the hierarchy 0 level. Assuming the first case and denoting the fraction of have an additional load Dk representing the number of replenishment events that occur by differentiation from the level divisions having led to their differentiation.ÀÁ By definition, the 0 below by qk, the combined rate of the contributing processes additional load of the stem cells D0 is zero. The convergence (asymmetric cell division and symmetric cell division with involves a sum of exponentially decaying terms, among which the differentiation from the level below) can be written as qkpkdk/2. slowest one is characterized by the time scale By definition this is equal to dk À 1, the differentiation rate from Xk N level k À 1, leading to the recursion relation ttr ¼ l ; ð5Þ k d l¼1 l À 1 dk À 1 ¼ dkpkqk=2: ð1Þ which can be interpreted as the transient time needed for the cells Alternatively, if replenishment occurs by cell division on the same tr level k, that is, as a result of symmetric cell division without at level k to reach their asymptotic behaviour. tk can also be considered as the transient time required for the initial differentiation, the corresponding rate is (1 À qk)pkdk/2. To keep track of how cell divisions accumulate along cell development of the tissue up to level k. The rationale behind lineages during tissue renewal, we introduce the divisional load this is that during development the levels of the hierarchy become populated by the descendants of the stem cells roughly Dk(t) for each level separately defined as the average number of divisions that cells at level k have undergone by time t since the sequentially, and the initial population of level l takes about stem cell level was created at time zero. Nl/dl À 1 time after level l À 1 has become almost fully populated. Plugging the asymptotic form of Dk(t) into the system of Using the rates of the microscopic events (also shown in 0 0 Fig. 2b), considering that each division increases the accumulated differential equations and prescribing D0 ¼ 0, the constants Dk number of divisions of both daughter cells by one, and taking into can be determined, and expressed as account the divisional loads that the departure of cells take and Pk Pk 0 dl À dl À 1 Nl D ¼ À d0 the arrival of cells bring, the following mean-field differential k dl À 1 dl À 1 l¼1 l¼1 equation system can be formulated for the time evolution of the Pk ð6Þ d total divisional load (DkNk) of levels kon of a fully developed ¼ ðÞÀg À 1 0 ttr; l N0 k tissue: l¼1 D_ N ¼Àdk p D þ d ðÞ1 À p where we have introduced the ratios k k 2 k k k k ð2Þ dk þ 2 pk½ŠqkðÞþDk À 1 þ 1 ðÞ1 À qk ðÞDk þ 2 : dk 2 gk ¼ ¼  2 ð7Þ Because stem cells cannot be replenished from below we have dk À 1 pkqk q0 ¼ 0. The terminal level k ¼ n can be included in the system between any two subsequent differentiation rates. The asymptotic of equations by specifying pn ¼ qn ¼ 1 and formally defining solution then becomes dn ¼ 2dn À 1. d ÀÁXk The above equations are valid when each level k contains the D ðtÞ¼ 0 t À ttr þ ðÞg À 1 : ð8Þ prescribed number of cells N of a fully developed, homoeostatic k k l k N0 l¼1 tissue and, therefore, do not directly describe the initial development of the tissue from the original stem cells. This This simple formula, which describes the accumulation of the shortcoming can, however, be remedied by introducing virtual divisional load along the levels of a hierarchically organized cells that at the initial moment (t ¼ 0) fill up all k40 levels. As the tissue, is one of our main results. virtual cells gradually differentiate to higher levels of the hierarchy, they are replaced by the descendants of the stem cells. Differentiation hierarchies that minimize divisional load. The Tissue development is completed when the non-virtual descen- number of mutations that a tissue allows for its constituent cells dants of the initial stem cell population fill the terminally to accumulate can be best characterized by the expected number differentiated level for the first time, expelling all virtual cells. of mutations accumulated along the longest cell lineages. On Using this approach the initial development of the tissue is average, the longest lineage corresponds to the last terminally assumed to follow the same dynamics as the self-renewal of the differentiated cell that is produced by the tissue at the end of the fully developed tissue. Even though cell divisions in a developing lifetime of the organism. Therefore, as the single most important tissue might occur at an elevated pace, such differences in the characteristics of a hierarchically organized tissue, we define its overall pace of cell divisions (along with any temporal variation in lifetime divisional load, D, as the divisional load of its last the tissue dynamics) are irrelevant, as long as only the relation terminally differentiated cell. If the total number of terminally

4 NATURE COMMUNICATIONS | 8:14545 | DOI: 10.1038/ncomms14545 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14545 ARTICLE differentiated cells produced by the tissue during the natural tissue has developed) the expected number of divisions of a stem lifetime of the organism per stem cell is denoted by N, cell, which is equivalent to the expected number of differentiation E then the lifetime of the organism can be expressed as to level 1 per stem cell is only (d0/N0)(N0N/dn À 1) ¼ 1/ln 2 1.44. tr tlife ¼ tn À 1 þ N0N=dn À 1, where the first term is the development time of the tissue up to level n À 1, and the second term is the à time necessary to generate all the N0N terminally differentiated Implications of the analytical results. Remarkably, Dopt corre- cells by level n À 1 at a rate of dn À 1. Because the last terminally sponds to less than two cell divisions in addition to the theoretical differentiated cell is the result of a cell division at level n À 1, its minimum of log2(N), achievable by a series of divisions along a expected divisional load, D, is the average divisional load of level perfect binary tree characteristic of non-renewing tissues. In other n À 1 increased by 1: words, in terms of minimizing the number of necessary cell divisions along cell lineages, a self-renewing hierarchical tissue d nXÀ 1 D ¼ D ðÞþt 1 ¼ N 0 þ ðÞþg À 1 1 can be almost as effective as a non-renewing one. Consequently, n À 1 life d l hierarchical tissue organization with a sufficient number of n À 1 l¼1 hierarchical levels provides a highly adaptable and practically nYÀ 1 nXÀ 1 1 ideal mechanism not only for ensuring self-renewability but also g : ¼ N þ ðÞþl À 1 1 ð9Þ keeping the number of cell divisions near the theoretical absolute gl l¼1 l¼1 minimum. tr Note that the complicated tn À 1 term drops out of the formula. An important result of our mathematical analysis is that it A remarkable property of D is that it depends only on two provides a simple and mathematically rigorous formula structural and two dynamical parameters of the tissue. The two (equations (13 and 14), and Fig. 3) for the lower limit of structural parameters are the total number of the terminally the lifetime divisional load of a tissue for a given number of differentiated cells produced by the tissue per stem cell, N, and hierarchical levels and a given number of terminally differentiated the number of the hierarchical levels, n. The two dynamical cells descending from a single stem cell. This lower limit can be parameters are the product and sum of the ratios of the differ- reached only with a power law distribution of the differentiation entiation rates, gk. The lifetime divisional load neither depends on rates (that is, with a uniform ratio between the differentiation most of the microscopic parameters of the cellular processes, nor rates of any two successive differentiation levels), justifying the on the number of cells at the differentiation levels. assumptions of the models by Werner et al.32,33. gà à For fixed N and n the ratios k of the differentiation rates that In the optimal scenario, where gk ¼ gopt ¼ 2, the recursion minimize the lifetime divisional load D can be easily determined relation imposes pn ¼ qn ¼ 1, thereby, all cell divisions must be by setting the derivatives of D with respect to the ratios gk to zero, symmetric and involve differentiation. This is a shared feature resulting in with non-renewable differentiation, which is the underlying nYÀ 1 1 reason, why the number of cell divisions of the optimal self- gà ¼ N : ð10Þ renewing mechanism can closely approach the theoretical k gà l¼1 l minimum. à As a salient example of self-renewing tissues, let us consider the This expression shows that gk is identical for all intermediate à human skin. Clonal patches of skin are of the order of square levels (0okon) and, therefore, can be denoted by g without a 42 subscript. This uniform ratio can then be expressed as millimetres in size , the top layer of skin, which is renewed daily, is composed of approximately a thousand cells per square gà ¼ N1=n; ð11Þ as long as the condition gÃZ2 holds, that is, when n log (N). 200 r 2 N = 3 × 108 For nZlog2(N), however, the ratio has to take the value of gà ¼ 2: ð12Þ *) 150 Plugging gà into equation (9) results in D  Dà ¼ nN1=n À 1 þ 2 ð13Þ 100 for nrlog (N) and 2  1 n À 1 n (N1/n –1) + 2 Dà ¼ N þ n ð14Þ 2 50 n–1

Lifetime divisional load ( N (1/2) + n log2 (N) for nZlog2(N). Equation (13) is a monotonically decreasing function of n, while equation (14) has a minimum at

nopt¼ log ðNÞþ1 þ log ðln 2Þlog ðNÞþ0:471 ð15Þ 0 2 2 2 0 5 10 15 20 25 30 35 40 levels. This nopt together with the ratio Number of hierarchical levels (n) à gopt ¼ 2 ð16Þ Figure 3 | The lower limit of the lifetime divisional load as a function of represent the optimal tissue-structure in the sense that it the number of hierarchical levels. The black and grey solid lines (with filled minimizes the lifetime divisional load of a self-renewing tissue, circles at integer values of n) show the lower limit of the lifetime divisional à yielding load of a tissue, D , as a function of the number of hierarchical levels, n, for nrlog2(N) and nZlog2(N), respectively. The theoretical minimum, log2(N), à Dopt¼ log2ðNÞþ1 þ log2ðln 2Þþ1=ln 2 achievable by a series of divisions along a perfect binary tree characteristic of non-renewing tissues, is displayed with a dashed line. Here we have  log ðNÞþ1:914: ð17Þ 2 assumed N ¼ 3  108 roughly corresponding to the number of cells shed by Note that under this optimal condition the divisional rate of the a few square millimetres of human skin that is sustained by a single stem stem cell level is very low: in a mature tissue (that is, after the cell.

NATURE COMMUNICATIONS | 8:14545 | DOI: 10.1038/ncomms14545 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14545 millimetre43. If we assume that a 10 mm2 patch is maintained by in n. Alternatively, if we consider for example the human a single stem cell for 80 years, this corresponds to about hematopoietic system, where B104 hematopoietic stem cells N ¼ 3 Â 108 cells. As Fig. 3 demonstrates, the DÃ versus n curve (HSCs) produce a daily supply of B3.5 Â 1011 blood cells, we can becomes very flat for large values of n, indicating that in a real calculate that over 80 years each stem cell produces a total of tissue the number of hierarchical levels can be reduced by at least NE1012 terminally differentiated cells. For this larger value of N a factor of two from the optimal value, without significantly the DÃ versus n curve reaches twice the optimal value of DÃ at compromising the number of necessary cell divisions along the n\15 after which, similarly to Fig. 3, it becomes progressively cell lineages. flatter and the reduction in divisional load diminishes as It is a question how the total number of terminally additional levels are added. This rough estimate of n\15 levels differentiated cells (N0N) produced by the tissue during the is consistent with explicit mathematical models of human natural lifetime of the organism can be best partitioned into the hematopoiesis that predict between 17 and 31 levels14. Active number of tissue-specific stem cells (N0) and the number of or short term HSCs (ST-HSCs) are estimated to differentiate terminally differentiated cells per stem cell (N). The initial about once a year, whereas a quiescent population of HSCs that generation of the stem cells along a binary tree requires log2(N0) provides cells to the active population is expected to be divisions. The production of the terminally differentiated cells in characterized by an even lower rate of differentiation. This is in a near-optimal hierarchy requires about log2(N0) divisions. Their good agreement with our prediction about the existence of a sum, which is about log2(N0), depends only on the total number heterogeneous stem cell pool, a fraction of which consists of of terminally differentiated cells, irrespective of the number of quiescent cells that only undergo a very limited number of cell stem cells. This means, that the minimization of the divisional cycles during the lifetime of the organism. Indeed, recently Busch load poses no constraint on the number of stem cells. However, et al.6 found that adult hematopoiesis in mice is largely sustained since both maintaining a larger number of differentiation levels by previously designated ST-HSCs that nearly fully self-renew, and keeping the differentiation hierarchy closer to optimum and receive rare but polyclonal HSC input. Mouse HSCs were involve more complicated regulation, we suspect that a relatively found to differentiate into ST-HSCs only about three times per large stem cell pool is beneficial, especially as a larger stem cell year. population can also be expected to be more robust against For most somatic tissues the differentiation hierarchies that stochastic extinction, population oscillation, and injury. underpin the development of most cellular compartments remain inadequately resolved, the identity of stem and progenitor cells remains uncertain, and quantitative information on their Discussion proliferation rates is limited20. However, synthesis of available In general, how closely the hierarchical organization of different information on tissue organization by Tomasetti and tissues in different organisms approaches the optimum described Vogelstein21, as detailed above, suggests that larger tissues with above depends on (i) the strength of natural selection against rapid turnover (for example, colon and blood) are relatively less unwanted somatic evolution, which is expected to be much prone to develop cancer. This phenomenon, as noted in the stronger in larger and longer lived animals and (ii) intrinsic introduction, can be interpreted as Peto’s paradox across tissues physiological constraints on the complexity of tissue organization with the implication that larger tissues with rapid turnover rates and potential lower limits on stem cell division rate. Neither the have hierarchies with more levels and stem cells that divide at a strength of selection nor the physiological constraints on tissue slower pace. Accumulating evidence from lineage-tracing organization are known at present. However, in the case of the experiments50 is also consistent with a relatively large number germ line mutation rate, which is proportional to the number of of hierarchical levels. Populations of stem cells in blood, skin, and cell divisions in lineages leading to the gametes, current evidence the colon have begun to be resolved as combinations of cells that indicates that physiological constraints are not limiting44. Across are long-lived yet constantly cycling, and emerging evidence species, differences in effective population size, which is in general indicates that both quiescent and active cell subpopulations may negatively correlated with body size and longevity45, indicate the coexist in several tissues, in separate yet adjoining locations41. effectiveness of selection relative to drift. As a result, differences Lineage-tracing techniques50 are rapidly developing, and may be in effective population size between species determine the used for directly testing the predictions of our mathematical effectiveness of selection in spreading of favourable mutations model about the highly inhomogeneous distributions of the and eliminating deleterious ones and, as such, can be used as differentiation rates in the near future. In the context of estimates indicator of the efficiency of selection46,47. This implies that, in of the number of stem cells in different tissues that underlie contrast to somatic tissues, we expect germ line differentiation Tomasetti and Vogelstein’s results, the potential existence of such hierarchies to be more optimal for smaller animals with shorter unresolved hierarchical levels suggests the possibility that the life-spans as a result of their increased effective population sizes. number of levels of the hierarchy are systematically under- For species for which information is available, the number of estimated and, correspondingly, that the number of stem cells at levels across species indeed follows an increasing trend as a the base of these hierarchies are systematically overestimated. function of the effective population size, ranging from n ¼ 5in Independent of the details of the hierarchy the dynamics of humans with relatively small effective population size of B104 how divisional load accumulates in time is described by two and correspondingly less efficient selection, n ¼ 8 in macaque phases: (i) a transient development phase during which each level with intermediate effective population size of the order of 105, of the hierarchy is filled up and (ii) a stationary phase during and n ¼ 10 in mice with the largest effective population size of which homoeostasis is maintained in mature tissue. The dynamic B5 Â 105 (refs 48,49). details and the divisional load incurred during the initial A qualitative examination of Fig. 3 suggests that a similar development phase depend on the details of the hierarchy number of levels, of the order of nE10 may be present in most (cf. equations (5 and 8)). In contrast, in the stationary phase, somatic tissues, because the DÃ versus n curve becomes further accumulation of the mutational load is determined solely progressively flatter after it reaches around twice the optimal by d0/N0 the rate at which tissue-specific stem cells differentiate at value of DÃ at n\10, and the reduction in the divisional load the bottommost level of the hierarchy. Such biphasic behaviour becomes smaller and smaller as additional levels are added to the has been observed in the accumulation of mutations both in hierarchy and other factors are expected to limit further increase somatic51 and germ line cells1,52,53. In both cases a substantial

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43. Hoath, S. B. & Leahy, D. G. The organization of human epidermis: functional Acknowledgements epidermal units and phi proportionality. J. Invest. Dermatol. 121, 1440–1446 This work was supported by the Hungarian Science Foundation (grant K101436). We (2003). would like to acknowledge the comments of anonymous reviewers on a previous version 44. Sung, W., Ackerman, M. S., Miller, S. F., Doak, T. G. & Lynch, M. Drift-barrier of the manuscript, as well as discussion with and comments from Bastien Boussau, hypothesis and mutation-rate evolution. Proc. Natl Acad. Sci. USA 109, Ma´rton Demeter, Ma´te Kiss and Da´niel Grajzel. 18488–18492 (2012). 45. Nabholz, B., Uwimana, N. & Lartillot, N. Reconstructing the phylogenetic Author contributions history of long-term effective population size and life-history traits using I.D. and G.J.S. designed the study, carried out research and wrote the paper. patterns of amino acid replacement in mitochondrial genomes of mammals and birds. Genome Biol. Evol. 5, 1273–1290 (2013). 46. Kimura, M. The Neutral Theory of Molecular Evolution (Cambridge University Additional information Press, 1983). Competing financial interests: The authors declare no competing financial interests. 47. Charlesworth, B. Fundamental concepts in genetics: effective population size and patterns of molecular evolution and variation. Nat. Rev. Genet. 10, 195–205 Reprints and permission information is available online at http://npg.nature.com/ (2009). reprintsandpermissions/ 48. Lynch, M. Evolution of the mutation rate. Trends Genet. 26, 345–352 (2010). + 49. Ramm, S. A., Scha¨rer, L., Ehmcke, J. & Wistuba, J. Sperm competition and the How to cite this article: Dere´nyi, I. & Szo¨llosi, G. J. Hierarchical tissue organization as a evolution of spermatogenesis. Mol. Hum. Reprod. 20, 1169–1179 (2014). general mechanism to limit the accumulation of somatic mutations. Nat. Commun. 8, 50. Blanpain, C. & Simons, B. D. Unravelling stem cell dynamics by lineage tracing. 14545 doi: 10.1038/ncomms14545 (2017). Nat. Rev. Mol. Cell Biol. 14, 489–502 (2013). Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in 51. Rozhok, A. I. & DeGregori, J. Toward an evolutionary model of cancer: published maps and institutional affiliations. considering the mechanisms that govern the fate of somatic mutations. Proc. Natl Acad. Sci. USA 112, 8914–8921 (2015). 52. Kong, A. et al. Rate of de novo mutations and the importance of father’s age to This work is licensed under a Creative Commons Attribution 4.0 disease risk. Nature 488, 471–475 (2012). International License. The images or other third party material in this 53. Rahbari, R. et al. Timing, rates and spectra of human germline mutation. Nat. article are included in the article’s Creative Commons license, unless indicated otherwise Genet. 48, 126–133 (2016). in the credit line; if the material is not included under the Creative Commons license, 54. Frank, S. A. & Nowak, M. A. Cell biology: developmental predisposition to users will need to obtain permission from the license holder to reproduce the material. cancer. Nature 422, 494 (2003). To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ 55. Rozhok, A. I., Salstrom, J. L. & DeGregori, J. Stochastic modeling reveals an evolutionary mechanism underlying elevated rates of childhood leukemia. Proc. Natl Acad. Sci. USA 113, 1050–1055 (2016). r The Author(s) 2017

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