Emergence of heterogeneity in acute leukemias

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1 Stiehl et al. Biology Direct (2016) 11:51 DOI /s RESEARCH Open Access Emergence of heterogeneity in acute leukemias Thomas Stiehl 1,2,3*, Christoph Lutz 4 an Anna Marciniak-Czochra 1,2,3 Abstract Backgroun: Leukemias are malignant proliferative isorers of the bloo forming system. Sequencing stuies emonstrate that the leukemic cell population consists of multiple clones. The genetic relationship between the ifferent clones, referre to as the clonal hierarchy, shows high interiniviual variability. So far, the source of this heterogeneity an its clinical relevance remain unknown. We propose a mathematical moel to stuy the emergence an evolution of clonal heterogeneity in acute leukemias. The moel allows linking properties of leukemic clones in terms of self-renewal an proliferation rates to the structure of the clonal hierarchy. Results: Computer simulations imply that the self-renewal potential of the first emerging leukemic clone has a major impact on the total number of leukemic clones an on the structure of their hierarchy. With increasing epth of the clonal hierarchy the self-renewal of leukemic clones increases, whereas the proliferation rates o not change significantly. The emergence of eep clonal hierarchies is a complex process that is facilitate by a cooperativity of ifferent mutations. Conclusion: Comparison of patient ata an simulation results suggests that the self-renewal of leukemic clones increases with the emergence of clonal heterogeneity. The structure of the clonal hierarchy may serve as a marker for patient prognosis. Reviewers: This article was reviewe by Marek Kimmel, Tommaso Lorenzi an Tomasz Lipniacki. Keywors: Acute leukemia, Heterogeneity, Clonal evolution, Mathematical moeling, Patient prognosis, Mutation, Self-renewal, Clonal hierarchy Backgroun Acute leukemias are clonal iseases of the bloo forming (hematopoietic) system. They lea to expansion of malignant cells an resulting impairment of bloo cell formation. Over the last years evience has accumulate that many leukemia subtypes are maintaine by a subpopulation of cells with stem cell-like properties [1 3]. These cells are referre to as leukemic stem cells (LSCs) or leukemia initiating cells (LICs) an potentially trigger relapse of the isease [4, 5]. Recent sequencing stuies have confirme that the leukemic cell population is compose of ifferent clones [6 8]. The size an number *Corresponence: Thomas.Stiehl@iwr.uni-heielberg.e 1 Institute of Applie Mathematics, Heielberg University, Im Neuenheimer Fel 205, Heielberg, Germany 2 Interisciplinary Center for Scientific Computing, Heielberg University, Im Neuenheimer Fel 205, Heielberg, Germany Full list of author information is available at the en of the article of clones follows a complex evolution over the course of the isease [9 12]. Genetic heterogeneity of ifferent clones seems to result in functional ifferences, such as a ifferent engraftment potential in mice or ifferent proliferation rates [13, 14]. Nevertheless, a irect link between genotype an cell function is still missing [13]. Genetic instability is a hallmark of soli cancers but a relatively rare event in acute leukemias. The number of somatic mutations etecte in acute leukemias is small compare to most other cancers [15, 16]. Nevertheless acute leukemias show a consierable interiniviual genetic heterogeneity an a complex genetic relationship among the ifferent clones. The clonal architecture of leukemias shows high interiniviual variability [12], see Fig. 1 for examples. The source of this variability is so far unknown. 2016TheAuthor(s). Open Access This article is istribute uner the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricte use, istribution, an reprouction in any meium, provie you give appropriate creit to the original author(s) an the source, provie a link to the Creative Commons license, an inicate if changes were mae. The Creative Commons Public Domain Deication waiver ( applies to the ata mae available in this article, unless otherwise state.

2 Stiehl et al. Biology Direct (2016) 11:51 Page 2 of 19 Fig. 1 Examples of the clonal architecture etecte in leukemic patients. Each tree correspons to one patient. The cell at the top correspons to the leukemic founer cell which acquires aitional mutations an gives rise to multiple leukemic clones. The examples are taken from [12]. Reconstruction of the clonal architecture from genetic measurements is not always unique. In case of ambiguity only one possibility is shown in the figure Clonal evolution in leukemias is a complex process. Hematopoiesis is known to be a tightly regulate process subject to several nonlinear feeback mechanisms [17]. Leukemic cells of many patients have the potential to interact with hematopoietic feeback signals [18, 19]. This may inclue leukemic cell stimulation by hematopoietic factors as well as alteration of the concentration of feeback signals by leukemic cells. Experiments further suggest the interaction of leukemic cells with the bone marrow microenvironment [20, 21]. Nonlinear interactions between hematopoiesis an the leukemic cell expansion on one han an the limite bone marrow space on the other may influence clonal selection [22]. The fact that important cell parameters such as proliferation rates or self-renewal probability cannot be measure in vivo further limits our unerstaning of leukemia evolution. Especially the following questions are so far unresolve: What is the source of interiniviual ifferences of the clonal hierarchy of leukemias? What is the functional ifference in terms of self-renewal an proliferation rates between cells at the top of the hierarchy an their escenants which have acquire aitional mutations? How o clones that appear early uring the isease iffer from clones that appear later? How o properties of leukemic cells present at one point in time influence the structure of the clonal hierarchy in the future? How o mutation rates influence the structure of the clonal hierarchy? These questions are of clinical relevance, since properties of the leukemic stem cells are important eterminants of isease ynamics, therapy resistance an relapse [14, 22, 23]. Deciphering of the clonal architecture using genomic methos has become more efficient an less expensive in recent years. Preiction of patient prognosis base on genetic markers alone is not straightforwar, since leukemogenetic hits vary consierably among patients an the interplay of the ifferent etecte mutations is complex an only partially unerstoo [24, 25]. Therefore, the question arises if the structure of the clonal architecture provies aitional insights into cell properties an patient prognosis. We propose a mathematical moeling approach to provie potential answers to these questions. Mathematical moels allow to systematically stuy the impact of cell parameters such as mutation rates, proliferation rates an self-renewal probability on the clonal hierarchy of cells. Simulation of the clonal selection process provies hints about which cell properties are linke to selective avantage an how these properties evolve over time. The moeling approach allows linking the position of a clone within the hierarchy to functional properties, such as self-renewal an proliferation rates, an to compare it to functional properties of clones locate at ifferent positions in the hierarchy. The structure of the clonal architecture obtaine in moel simulations is compare to experimental ata from the literature an thus allows linking observations at the level of population ynamics to the cell function in terms of self-renewal an proliferation rates. Different mathematical concepts have been propose to moel mutations. Depening on the focus of interest, ifferent approaches can be applie. Moran process [26, 27] is use to escribe populations with size that is constant in time. Branching processes are use to escribe acquisition of mutations in growing populations. Examples inclue the accumulation of passenger an river mutations, interaction among river mutations or accumulation of mutations uring DNA copying [28 31]. In case of a large population an a continuous trait space, iffusion [32, 33] or integral kernels [34 36] have been use to escribe the effect of mutations. A rigorous relationship between processes at the level of single cells an limit escriptions in terms of eterministic or probabilistic equations is provie in ref. [37]. Examples for eterministic approaches to stuy mutations in iscrete or continuously structure population moels are given in ref. [38 40]. Computer simulations of iniviual-base moels an cellular automata provie a framework to stuy the impact of cellular processes on the whole population. Examples for iniviual-base cancer moels can be foun in ref. [41, 42]. This work is structure as follows. In the Methos section, we introuce the mathematical moel. In the Results section, we present simulation results an their biological interpretation. The Discussion section conclues with a iscussion of clinical implications of the results obtaine.

3 Stiehl et al. Biology Direct (2016) 11:51 Page 3 of 19 We ahere to the following terminology. Clonal architecture (clonal hierarchy) is unerstoo as the genetic relationship between ifferent clones. We represent the clonal architecture as a tree. Progeny of a noe has acquire one aitional mutation compare to its mother noe. As a clone we unerstan all genetically ientical stem an non-stem cells. A clone consisting of at least 1 % of the total cell mass is enote as a significant clone. The threshol of 1 % has been chosen base on the sensitivity of sequencing methos [43]. Methos The moel is efine as a system of nonlinear orinary ifferential equations escribing time evolution of hematopoietic cells an leukemic clones. Experimental ata imply that hematopoietic an leukemic cells interact, e.g., through feeback signals or the bone marrow microenvironment [18 21]. Therefore, the moel takes into account both healthy an leukemic cells. The presente moel is an extension of the moels of healthy hematopoiesis [44 46] an acute leukemias [22, 23, 47]. The main novelty lies in consiering a time epenent number of leukemic clones an in tracking the structure of the clonal hierarchies. During the course of the isease new clones arise ue to mutations which are acquire by leukemic cells. Properties of new clones are chosen from ranom istributions that epen on the properties of the cells that give rise to them. To moel stochastic extinction of clones with favorable properties, we take into account their extinction probabilities using the theory of branching processes. Compare to the work presente in [40], which focuses on neutral mutations in non-stem cells without feeback regulation or competition, we are intereste in the evolution of non-neutral stem cell mutations uner competitive pressure of a nonlinear feeback mechanism. An overview of the moel is provie in Fig. 2a. Moel structure Base on the classical unerstaning of the hematopoietic system [48] bloo cell formation is consiere as a stepwise process, with cells sequentially traversing an orere sequence of iscrete maturation states (compartments). We treat each compartment as a well-mixe tank an escribe its evolution using an orinary ifferential equation. The large count of cells in the hematopoietic system justifies this approach [48]. Since most leukemias are iseases of the white bloo cells, we only consier the white cell lineage of the healthy hematopoietic system. The moel escribes the interaction of the healthy cell lineage with an arbitrary number of leukemic clones. We assume that each lineage or clone consists of two ifferent cell types, namely cells that are able to ivie (stem an progenitor cells) an cells that have lost the ability to ivie (mature cells or post-mitotic leukemic blasts). Each cell type is characterize by the following cell properties: Proliferation rate, escribing the frequency of cell ivisions per unit of time. In case of post-mitotic cells the proliferation rate is consiere equal to zero. Fraction of self-renewal (self-renewal rate), escribing the fraction of progeny cells returning to the compartment occupie by the parent cells that gave rise to them. Death rate, escribing the fraction of cells ying per unit of time. For simplicity, we assume that iviing cells o not ie an that non-iviing cells ie at constant rates. We enote the compartment of iviing healthy cells as c 1 an that of mature cells as c 2.Wecounttheleukemic clones starting from 1. The respective compartments of the ith leukemic clone are enote as l1 i an li 2 resp. The proliferation rate of the healthy cells is enote as p c an that of the mitotic cells of the ith leukemic clone p i l.the respective fractions of self-renewal are enote a c an a i l. Death rates of the non-iviing compartments are c an i l. Feeback regulation of healthy hematopoiesis Formation of healthy bloo cells is subject to a tight regulation, meiate by a system of lineage- an stage-specific cytokines. If there is a nee for more bloo cells of a certain type, the concentration of cytokines increases an stimulates formation of mature cells [17, 49]. For simplicity, we consier only one feeback loop. We enote s(t) the value of the feeback signal at time t. Weset 1 1+kc 2 (t) s(t) =,wherek is a positive constant epening on prouction an elimination of cytokines [44]. This expression can be erive from cytokine kinetics [44]. It takes into account that the concentrations of important cytokines such as EPO an G-CSF epen on the concentration of mature cells [49]. The feeback signal assumes values between 0 an 1. On the basis of our earlier work an compatibility with clinical ata [44, 46], we assume feeback inhibition of the fraction of self-renewal by mature cells. The fraction of self-renewal of the healthy cells is assume to be equal to a c (t) = â c s(t) an that of leukemic cells of clone i to a i l (t) = âi l s(t). The parameters â c an â i l canbeinterpreteasthemaximalpossiblefraction of self-renewal. Numerical solutions of the moel of hematopoiesis subject to this feeback were valiate on the basis of clinical observations an show goo agreement with patient ata upon recovery from bone marrow transplantation [46].

4 Stiehl et al. Biology Direct (2016) 11:51 Page 4 of 19 Fig. 2 Overview of the mathematical moel. a Moel structure: The moel inclues one hematopoietic cell line an an arbitrary number of leukemic clones. Leukemic an healthy cells interact by feeback signals. Due to mutations new clones with ifferent properties arise. Mutation rates of leukemic an healthy cells are enote as ν an γ resp. b Example simulation: The panel shows the time course of mitotic leukemic cells. The horizontal axis shows the time since appearance of the first leukemic cell. The simulation ens when the mature healthy cell count is below 5 % of its steay state value. This correspons to the eath of the patient. Each color represents one clone Moel equations for the hematopoietic system Thefluxtomitosisofhealthycellsattimet equals p c c 1 (t). During mitosis, a parent cell is replace by two progeny cells. The outflux from mitosis at time t equals 2p c c 1 (t), of which the fraction 2â c s(t)p c c 1 (t) stays in compartment 1 (process referre to as self-renewal). The fraction 2 ( 1 â c s(t) ) p c c 1 (t) moves to compartment 2 (process referre to as ifferentiation). We obtain the following system of orinary ifferential equations t c 1(t) = ( 2â c s(t) 1 ) p c c 1 (t) t c 2(t) = 2 ( 1 â c s(t) ) p c c 1 (t) c c 2 (t) 1 s(t) = 1 + kc 2 (t) with the initial conitions c 1 (0), c 2 (0) given. Moel of leukemia We assume that healthy an leukemic cells respon to the same feeback signals. This assumption is supporte by the fining that leukemic cells express receptors for hematopoietic cytokines [18] an that they interact with the bone marrow microenvironment [20, 21]. Due to cytokine clearance by receptor meiate enocytosis [17, 49] leukemic cells contribute to the cytokine clearance. In the presence of leukemic cells, the feeback signal is given by 1 s(t) = 1 + kc 2 + k n(t) i=1 li 2 (t). Here, n(t) enotes the number of leukemic clones presentattimet. This expression has been erive in ref. [47] for the special case of one leukemic clone. The propose feeback mechanism has been valiate base on clinical ata [23]. Moel simulations suggest that the choice of iniviual k values for each leukemic clones, i.e., moeling the signal as s(t) = 1 1+kc 2 + n(t) has no sig- i=1 ki l2 i (t) nificant impact on the quantities consiere in this stuy. For n leukemic clones we obtain the following system of equations: t c 1(t) = ( 2â c s(t) 1 ) p c c 1 (t) t c 2(t) = 2 ( 1 â c s(t) ) p c c 1 (t) c c 2 (t) t l1 1 (t) = ( 2â 1 l s(t) 1) p 1 l l1 1 (t) t l1 2 (t) = 2 ( 1 â 1 l s(t)) p 1 l l1 1 (t) 1 l l1 2 (t)... t ln 1 (t) = ( 2â n l s(t) 1) p n l ln 1 (t) t ln 2 (t) = 2 ( 1 â n l s(t)) p n l ln 1 (t) n l ln 2 (t) s(t) = kc 2 (t) + k n i=1 l i 2 with the initial conitions c 1 (0),..., l2 n (0) given. Mutations We assume that mutations occur uring genome replication which takes place before mitosis. We consier the rate to be ientical for all clones an constant in time. This is supporte by the fact that genomic instability is a relatively rare event in leukemias [15, 16]. The flux to mitosis of leukemic clone i at time t is given as p i l li 1 (t). We assume that a fraction ν of the prouce progeny has a mutation.

5 Stiehl et al. Biology Direct (2016) 11:51 Page 5 of 19 Therefore, 2p i l li 1 (t)ν mutate cells are prouce at time t, of which 2â i l s(t)pi l li 1 (t)ν are in the mitotic compartment an 2(1 â i l s(t))pi l li 1 (t)ν belong to the post-mitotic compartment. The influx of mutate mitotic cells from the clone i is efine as α i (t) = 2â i l s(t)pi l li 1 (t)ν. Thenumber of non-mutate cells is given by 2p i l li 1 (t)(1 ν), of which 2â i l s(t)pi l li 1 (t)(1 ν) are mitotic cells an the remainer, 2(1 â i l s(t))pi l li 1 (t)(1 ν), belongs to the non-iviing compartment. We obtain the following set of equations escribing ynamics of clone i: t li 1 (t) = 2âi l s(t)pi l li 1 (t)(1 ν) pi l li 1 (t) t li 2 (t) = 2(1 âi l s(t))pi l li 1 (t) i l li 2 (t) α i (t) = 2â i l s(t)pi l li 1 (t)ν A similar system of equations has been obtaine in [40]. Since l2 i is consiere to be post-mitotic, we o not istinguish between cells that acquire a mutation uring the ivisions an those that i not. The influx α(t) of mutate mitotic cells of all leukemic clones at time t is given by α(t) = n(t) i=1 α i(t), wheren(t) is the number of leukemic clones present at time t. We consier the rate α(t) astherateofaninhomogeneous Poisson process. Poisson processes escribe rare events [50], therefore, they are a suitable framework to escribe mutations. We use the Poisson process to etermine the time points of the mutations. At the respective time points, one cell acquires a new mutation an gives rise to a new clone. This founer cell is chosen from the present clones accoring to their contribution α i to the total rate α. Self-renewal an proliferation rates of the new clone are chosen base on the parameters of the founer cell. We assume that the traits (self-renewal an proliferation rates) of the new clone are normally istribute with a preefine variance an the mean value corresponing to the parameters of the founer cell. Since biological parameters are restricte to a preefine interval, we use truncate normal istributions. A suitable interval for proliferation rates is between one ivision per year an one ivision per ay [46] an the fraction of self-renewal is by efinition between zero an one. At the time of its birth a new clone consists of one mitotic an zero post-mitotic cells. Due to the stochasticity of cell fate ecisions or ue to cell eath it is possible that the new clone becomes extinct. For example, if the newly generate mitotic cell ivies an gives rise to two ifferentiate progeny, the new clone will eventually extinct, since there exist no more mitotic cells. We use the theory of Galton-Watson processes to calculate the probability of extinction of new clones. We aopt the methoology from ref. [51], which is similar to the approach use in ref. [31]. We notice that a clone eventually becomes extinct if it has no mitotic cells. If a mitotic cell ivies, with the probability a 2 both progeny are mitotic cells, with the probability 2(1 a)a one progeny is a mitotic cell an with probability (1 a) 2 both progeny are fully ifferentiate. By a we enote the fraction of self-renewal of the mitotic cells. The probability generating function for the number of mitotic progeny is f (x) = a 2 x 2 + 2a(1 a)x + (1 a) 2. If we assume in aition that the parent cell ies with a probability uring ivision the probability generating function is f (x) = (1 )(a 2 x 2 + 2a(1 a)x + (1 a) 2 ) +. If we assume that cells of the new clone ivie at iscrete times it, i N, wheret is the average generation time, we can use the theory of Galton-Watson processes to calculate the extinction probability. We know that the extinction probability is the smaller solution of f (x) = x [28]. We neglect cell eath an obtain for the extinction probability p e (a) = 2a2 2a + 1 2a 2 (2a 2 ) 2a a 2 (1 a)2 a 2. We notice that p e < 1ifa > 0.5. For each new clone we calculate the extinction probability base on its self-renewal rate at the time of emergence. With probability 1 p e thenewcloneisintroucetothesystem by aing two equations to the system escribing ynamics of mitotic an post-mitotic cells of the new clone. With probability p e the new clone is not introuce to the system since it becomes extinct. Accoring to biological ata suggesting that all leukemic cells are erive from one leukemic or preleukemic clone [12], we neglect mutations in the healthy cells. Instea we introuce one leukemic founer clone at t = 0tothesystem an stuy the ynamics of the founer clone an the clones arising from it. Parameters of the founer clone are assume to be normally istribute with the means equal to parameters of healthy hematopoietic cells. Parametrization The parameters of the hematopoietic system are aopte from ref. [22]. In the following we shortly escribe the calibration. For etails see ref. [22] an the references therein. The numbers of myeloi mitotic ( c 1 ) an post-mitotic ( c 2 ) cells an the neutrophil clearance ( 2 ) are taken from literature. Analytical expressions of the steay state cell counts allow to calculate p c base on c 1, c 2 an 2.Theparameter k can be expresse as a function of known quantities an â c. To obtain an appropriate value for â c,wesimulatestem cell transplantation. We know that after transplantation of aoseof3to cells per kg boy weight, patients recoverto neutrophils per liter of bloo within 15 ays. To set the initial conition we assume that the ratio

6 Stiehl et al. Biology Direct (2016) 11:51 Page 6 of 19 of myeloi to erythroi cells in the transplant is as in the healthy marrow between 2:1 an 4:1. We choose a 1 such that we observe recovery after 2-3 weeks. This results in the following parameters: â c = 0.87, p c = 0.45/ay, c = 2.3/ay, k = , l = 0.5/ay. These parameters refer to healthy cells. Parameters of the leukemic cells are chosen ranomly accoring to normal istributions. For the simulations the rate ν is varie between an Stanar eviations for the normal istributions are varie between an 0.1. The stanar eviations an mutation rates use to obtain the figures are specifie in the figure captions. Stopping criteria for simulations are either ecline of healthy bloo cells to less than 5 % of the healthy steay state value or a simulate time span that excees 100 years of age for a given patient. An example simulation is epicte in Fig. 2b. Results Simulations over wie parameter ranges, incluing variation of mutation rates over several orers of magnitue, show that the phenomena presente below are robust with respect to the parameter choice. Self-renewal rate of significant clones increases uring the course of the isease We first ask how cell properties evolve uring the course of the isease. For this purpose, we compare self-renewal an proliferation rates of the significant clones of 600 simulate patients. The significant clones emerge at ifferent time points ue to mutations. We count the significant clones in the orer of the time of their emergence. Simulation results imply that in more than 95 % of patients the self-renewal rate of the secon emerging significant clone is larger than the self-renewal rate of the first emerging significant clone. The self-renewal rate of the thir significant clone is larger than the self-renewal rate of the secon significant clone an so forth. Interestingly, proliferation rates o not iffer significantly among significant clones. This fining confirms that high self-renewal rate is beneficial for expansion of clones. Previous simulation stuies have shown that high self-renewal rate might be relate to poor prognosis an high resistance to therapy [22, 23]. Simulation results imply that clones become more aggressive uring the course of the isease. In the remaining 5 % of simulate patients, the following phenomena have been observe: (i) in very rare cases (less than 1 %) a slight reuction in the self-renewal rate is compensate by an increase in proliferation rate, (ii) two new clones emerge within a short time span, the selfrenewal rate of both clones is larger than the self-renewal rate of the parent clones, but one of the emerging clones has a slightly reuce self-renewal rate compare to the other. In this case both new clones have growth avantage at the time of their origin an, therefore, grow to a significant size. The increasing self-renewal of clones over time is the result of a selection process. The selection was stuie numerically in [22] an prove in [52] for a multi-clonal system in absence of mutations. When anewclonearisesuetoamutation,itsself-renewalan proliferation rates can be larger or smaller than the corresponing rates of the parent clone. They are chosen accoring to normal istributions with constant stanar eviation an means equal to the parameter values of the parent clone. It is not straightforwar to preict whether progeny clones with higher self-renewal, higher proliferation or both have more competitive avantages. Simulation results show that among the newly arising clones with ranom proliferation an self-renewal values those with high self-renewal are more competitive than those with high proliferation. The results are epicte in Fig. 3. Properties of the first clone etermine if there is no outbreak of the isease, a monoclonal isease, or clonal iversity The number of significant clones varies among the patients [12]. We compare self-renewal an proliferation rates of the significant clones in simulate patients to investigate how these properties impact the total number of significant clones per patient. As mentione above, we suppose that all leukemic clones originate from a single founer clone which itself is erive from healthy hematopoietic cells. Simulation results imply that the selfrenewal rate of the founer clone has a major impact on the total number of significant clones emerging uring the course of the isease. If the founer clone has a high self-renewal rate it has the ability to expan fast. This fast expansion leas to a clinical isease an potential eath of the patient before aitional significant clones can emerge. In case of very small self-renewal rate, the founer clone expans slowly an the isease oes not become significant throughout the life span. In case of intermeiate self-renewal rate of the founer clone, multiple significant clones arise. If the founer clone gives rise to a clone that grows to a significant size over time, the self-renewal rate of this secon significant clone etermines whether a thir significant clone will arise. If the self-renewal rate of the secon significant clone is high, fast expansion an progression of the isease follow. The remaining life time of the patient is too short for emergence of aitional significant clones. If the self-renewal rate of the secon significant clone has intermeiate values, isease progression is slow an more significant clones can emerge. This principle is extene to a higher number of leukemic clones: If the self-renewal rate of the first n significant clones is intermeiate, the probability is high that aitional clones will emerge. If one clone among the first n significant clones has a high

7 Stiehl et al. Biology Direct (2016) 11:51 Page 7 of 19 Fig. 3 Self-renewal rate of significant clones increases uring the course of the isease. The figure is base on 600 simulate patients. a Time evolution of one simulate patient who evelope 4 clones uring the course of the isease. The first, secon, thir an fourth significant clone are epicte using ifferent colors. b Self-renewal rates of the first, secon, thir an fourth significant leukemic clone of the subgroup of patients harboring 4 significant clones at the en of the simulation (reuction of healthy cell count by 95 %). The self-renewal rates iffer significantly (p < 0.05 in t-test) between first an secon, secon an thir, thir an fourth clone. c Self-renewal rates of the first, secon an thir significant leukemic clone of the subgroup of patients harboring 3 clones at the en of simulations. The self-renewal rates iffer significantly between the clones (p < 0.05 in t-test). Proliferation rates of the first, secon an thir significant leukemic clone of the subgroup of patients harboring 3 clones at the en of simulation. The proliferation rates o not iffer significantly between the clones (p > 0.05 in t-test).parameters: mutation rate ν = , self-renewal an proliferation rates of the new clones are normally istribute with the means of the istributions equal to proliferation an self-renewal rates of the parent clone an stanar eviation equal to The central mark is the meian, the eges of the box are the 25th an 75th percentiles, points are rawn as outliers if they are larger than q (q 3 q 1 ) or smaller than q 1 1.5(q 3 q 1 ),whereq 1 an q 3 are the 25th an 75th percentiles, respectively self-renewal rate, progression is fast an no more clones emerge. Clones with small self-renewal rate never grow to a significant size. Proliferation rates of the clones o not have a significant impact on the total number of clones. Simulations show that if we restrict self-renewal rate of the leukemic founer clone to intermeiate values, e.g., between an 0.875, the number of clones per patient increases. The results are summarize in Fig. 4. Self-renewal rate increases with an increasing epth of clones In leukemia patients, clonal hierarchies show consierable interiniviual variations [12]. We ask how the properties of clones influence the epth of the clonal hierarchy. We assume that the founer clone has epth 1. Clones that have acquire k aitional mutations in comparison to the founer clone have epth 1 + k. Depthofa clonal hierarchy is unerstoo as the maximal epth of its

8 Stiehl et al. Biology Direct (2016) 11:51 Page 8 of 19 Fig. 4 Impact of self-renewal rate on the total number of significant clones. Data from 600 simulate patients. a Overview over panels (b) (). b We compare self-renewal rate of the first significant clone in two patient groups. Group 1: patients harboring only one significant clone throughout the isease. Group 2: patients harboring more than one significant clone. The self-renewal rate of the first significant clone is significantly higher in group 1. Leftmost plot in(b): If the self-renewal rate of leukemic clones is close to the self-renewal rate of healthy cells, no significant clones emerge. c We compare the self-renewal rate of the secon significant clone in two patient groups. Group 1: patients harboring two significant clones throughout the isease. Group 2: patients harboring more than two significant clones. The self-renewal rate of the secon significant clone is significantly higher in group 1. We compare the self-renewal rates of the thir emerging significant clone in two patient groups. Group 1: patients harboring three significant clones throughout the isease. Group 2: patients harboring more than three significant clones. The self-renewal rate of the thir significant clone is significantly higher in group 1. e Explanation of atain (b) ():Clones with highself-renewal rateslea to fast progression of the isease an eath before new significant clones can emerge. Clones with small self-renewal rates never grow to a significant size. Clones with intermeiate self-renewal rates grow with moerate spee an their offspring have enough time to grow to a significant size. Proliferation rates o not iffer significantly between all consiere groups. Parameters: mutation rate ν = , self-renewal an proliferation rates of the new clones are normally istribute with the mean of the istributions equal to proliferation an self-renewal rates of the parent clone an stanar eviation equal to Significance: p < 0.05 in t-test

9 Stiehl et al. Biology Direct (2016) 11:51 Page 9 of 19 clones. Simulations inicate that the self-renewal rate of the clones increases with their epths. This is plausible: To be able to give rise to new clones, a given clone has to achieve a critical mass of cells. Only then it is probable that cells of this clone mutate an give rise to offspring clones. To achieve the critical mass, a clone nees properties favorable for efficient expansion. This occurs if an only if its self-renewal rate is increase compare to its mother clone, since otherwise the mother clone outgrows its offspring. Simulations show that a eep clonal hierarchy requires a step-wise increase of self-renewal rate with each mutation. The step size etermines how eep the hierarchy will be. If the step size is too small, offspring clones grow slowly an it takes long time until they an their offspring grow to a significant size. In this case the parent clone remains ominant in size an is responsible for isease progression. If the step size is large, the offspring clones grow fast an the patient ies before potential new offspring achieve a significant size. In both cases the hierarchy is flat. Only if self-renewal rate increases by steps of intermeiate size, eep hierarchies are observe. In this case, the offspring clones have a sufficient growth avantage compare to their parents but they grow slow enough for their offspring to achieve a significant size an to give rise to new clones. Self-renewal rate of the significant clone which appears first has a major impact on the epth of the hierarchy. If it has a high self-renewal rate, the isease progresses fast an the patient ies before offspring achieve a critical mass. In case of small self-renewal rate of the first clone, eep hierarchies can emerge, suppose that it gives rise to offspring with higher self-renewal rates. Analogously the properties of a clone of epth 2 etermine whether a clone of epth 3 can emerge. The proliferation rate has no impact on the epth of the hierarchy. High self-renewal rate is potentially linke to poor prognosis an fast progression of the isease [22, 23]. If we consier the maximum of self-renewal capacity over all significant clones, the simulations imply that it increases significantly with the epth of the clonal hierarchy. Therefore, our stuy suggests that the epth of the clonal hierarchy coul be consiere a prognostic parameter. Since in our simulations eep hierarchies are linke to high self-renewal rates, our results suggest that eep hierarchies coul be linke to poor prognosis. Interestingly, there is no correlation between the total number of significant clones an the maximal self-renewal rates of the significant clones. The results are summarize in Fig. 5. Cooperativity of mutations might explain emergence of eep hierarchies In the patients investigate in ref. [12], hierarchies of epths between 3 an 5 have been etecte. Due to the finings escribe in the previous paragraph, emergence of such hierarchies requires a coorinate increase of the self-renewal rate with each acquire mutation. Appearance of clones with too high self-renewal rates leas to fast progression an eath before eep hierarchies can be establishe. Therefore, the existence of eep hierarchies is not compatible with mutations that lea to large changes of cell properties. Inee, if we assume that traits of mutate clones are uniformly istribute in the trait space, eep hierarchies are never observe in simulations. This observation raises the question which probability istributions are suitable choices to moel the effect of mutation in the trait space. We have investigate the assumption that the traits of the new clone follow normal istributions with means equal to the traits of the cell that gave rise to the new clone. Depening on the assume stanar eviations of the normal istributions we can observe hierarchies of varying epths. If the stanar eviations are too large, the hierarchies remain flat, since aggressive clones that lea to fast progression an eath emerge early in the isease. If the stanar eviations are too small, the traits of the offspring clones are very close to the traits of the parent clones. Therefore, the offspring clones have little growth avantage compare to their parent clones an consequently nee long time to grow to a significant size an to prouce offspring growing to a significant size. In these cases the hierarchy of significant clones remains flat. Only if the stanar eviation is within a limite range, a consierable number of patients with eep hierarchies is observe in the simulations. In acute leukemias, where genetic instability is rare, generation of large numbers of mutate cells an selection of those which exactly fit the properties require to establish eep hierarchies is not a realistic scenario, since mutation rates are relatively low, compare to other cancers. Leukemias show high interiniviual genetic variability. The assumption that all escribe mutations will lea to exactly those changes in the self-renewal rates that are require to establish eep hierarchies seems also improbable. If we assume that the stanar eviation of the normal istribution accoring to which the traits of the offspring are chosen increases with each mutation, eep hierarchies becomeamorefrequentevent.thisisplausiblesince small stanar eviations for the first mutation avoi emergence of clones that show fast expansion an subsequent eath of the patient. Step-wise increase of stanar eviation with each mutation allows the offspring to gain sufficient growth avantage compare to the parent clones that they can grow to a significant size. At the same time large jumps leaing to aggressive clones remain rare. The assumption that jump sizes in the trait space increase with the number of mutate genes in a cell seems plausible from a biological point of view. Cells are known

10 Stiehl et al. Biology Direct (2016) 11:51 Page 10 of 19 Fig. 5 Impact of the self-renewal rate on the epth of the hierarchy. The figure is base on 600 simulate patients. a Examples for hierarchies of ifferent epths. Colors are use to visualize clones of ifferent epths. b Self-renewal rate of significant clones increases significantly (p < 0.05in t-test) with the epth of the clones in hierarchies. Consiere are only patients with clonal hierarchies of epth 3. c Properties of the first clone in the hierarchy ecies about the epth of the hierarchy. Only if self-renewal rate of the first clone is small enough eep hierarchies emerge. If self-renewal rate of the first clone is high, isease progression an eath occur before eep hierarchies can establish. Comparisonof self-renewal rates of significant an insignificant clones of epth 2. Self-renewal rates of insignificant clones is significantly smaller than self-renewal ratesof significant clones. This emonstrates that clones o not become significant if their self-renewal rate is too small. Some of the insignificant clones show high self-renewal rates. These clones have originate late uring the isease an coul not grow to a significant size before eath of the patients. If proliferation rate is very slow, clones with high self-renewal cannot grow to a significant size. Proliferation rate has no impact on the epth of the hierarchy. Parameters: mutation rate ν = , self-renewal an proliferation rates of the new clones are normally istribute with the mean of the istributions equal to proliferation an self-renewal rates of the parent clone an stanar eviation equal to to have reunant pathways for regulation of important functions. Perturbation of one pathway by a mutation might therefore lea to only small jumps in the trait space, whereas subsequent perturbation of multiple pathways may lea to larger jumps in the trait space. This means that the presence of a mutation facilitates the occurrence of large effects ue to an aitional mutation. In this sense the ifferent mutations are cooperative. The importance of cooperativity is unerline by the following simulation experiment: We assume that the probability of large jumps in the trait space increases with the number of accumulate mutations. We moel these effect using normal istributions with increasing stanar eviations σ 1 < σ 2 < σ 3..., i.e., the size of the jump in the trait space ue to the first mutation is given by a normal istribution with stanar eviation σ 1,thejump ue to the secon mutation by a normal istribution with stanar eviation σ 2 etc. We simulate the emergence of clonal hierarchies uner these assumptions. We repeat simulations uner moifie assumptions, for example, we

11 Stiehl et al. Biology Direct (2016) 11:51 Page 11 of 19 assume that for all mutations the size of the jump in the trait space is given by a normal istribution with stanar eviation equal to σ 1 or equal to σ 2 etc. We run simulations for all possible permutations of σ 1, σ 2, σ 3... Comparison of simulation results shows that the number of patients harboring hierarchies of epth 4 or more is maximize if stanar eviations increase from one mutation to another. The results are epicte in Fig. 6. Impact of the mutation rates an probability istributions on the clonal hierarchies We stuie the architecture of clonal hierarchies for several mutation rates. For increase mutation rates the total number of clones increases. Interestingly, the number of significant clones increases only moerately if the mutation rates are varie over several orers of magnitue; for example, if the rate increases from to , the mean number of all clones increases by a factor of 8, whereas the mean number of significant clones increases only by 1. In all cases the number of significant clones was smaller than 15 an for 80 % of the patients smaller than 10. This is in line with the observation of clone numbers in experimental stuies [11, 12]. This fining unerlines the role of competition between the ifferent clones. The competition selects among an increasing total number of clones always a small number of significant clones. Simulation results imply that patients with less aggressive clones an without isease outbreak are overrepresente in case of small mutation rates. Patients with highly aggressive clones an fast isease progression are over-represente in case of high mutation rates. This is plausible: The higher the mutation rate, the more clones are generate per unit of time. The probability that at least one clone per patient has favorable growth properties increases with the number of generate clones. Similarly the probability that highly aggressive clones an fast isease progression occur increases with increasing mutation rate. For all mutation rates we observe that clonal hierarchies are flat in case of fast isease progression an in case of very slow isease progression compare to cases with intermeiate isease progression. Increase mutation rates act in favor of eep hierarchies. Nevertheless this effect is mil an the mean epth increases by 1 if the mutation rate increases by a factor of Fig. 6 Impact of cooperativity between mutations on epth of the hierarchy. The figure is base on 100 simulate patients. The number of patients harboring a clonal hierarchy of epth 4 or more is maximize, if the jumps in the trait spaces increase with the number of mutations. Parameters: Self-renewal an proliferation rates of the leukemic founer clone are rawn from normal istributions with mean values equal to proliferation an self-renewal rates of healthy cells an stanar eviation σ 1 = First mutation: self-renewal an proliferation rates of the new clone are normally istribute with the means of the istributions equal to proliferation an self-renewal rates of the founer clone an stanar eviation σ 2 = 5 σ 1. Secon mutation: self-renewal an proliferation rates of the new clone are normally istribute with the means of the istributions equal to proliferation an self-renewal rates of the parent clone an stanar eviation σ 3 = 20 σ 1. Thir an higher mutations: self-renewal an proliferation rates of the new clone are normally istribute with the means of the istributions equal to proliferation an self-renewal rates of the parent clone an stanar eviation σ 3 = 100 σ 1. Mutation rate ν =

12 Stiehl et al. Biology Direct (2016) 11:51 Page 12 of This observation can be explaine by the fact that high mutation rates lea to increase numbers of leukemic clones. Therefore, the probability that a clone gives rise to at least one offspring with favorable growth properties increases. As iscusse above, probability istributions accoring to which the traits of new clones are etermine have an important effect on the epth of the hierarchy. If uniform istributions over the possible parameter range are chosen, eep hierarchies are very rarely observe. Also the total number of significant clones is ecrease. Similarly, if stanar eviations of normal istributions increase over a certain threshol, the mean number of significant clones slightly ecreases, e.g., the number of significant clones ecreases by 1 if the stanar eviations are increase from 0.01 to Comparison to ata We compare the structure of the clonal hierarchy obtaine by simulations of our moel with the clonal hierarchies in 30 patients from [12]. The patients ata are base on genetic stuies. To take into account the limitations of the experimental methos, we compare the ata only to significant clones observe in the numerical simulations. For more than 60 % of the patients the clonal hierarchies are reprouce by our moel. Besies, we observe both hierarchies obtaine in numerical simulations that are not foun in the patients ata an hierarchies in the ata which coul not be reprouce numerically. The latter coul be explaine by ynamic variation over the hierarchies in time. The hierarchy at iagnosis only reflects the situation at one time point. In simulation results we only consiere the hierarchies at three time points per patient, namely at the time points when mature cell counts have ecrease by 5, 50 an 95 %. In approximately 30 % of the patients with hierarchies not reprouce by the simulations, patient ata coul be reprouce if one clone existing in the simulations with an insignificant size woul grow to a significant size. Examples are provie in Fig. 7. Discussion We propose a mathematical moel to stuy the emergence of clonal heterogeneity in acute leukemias. The moel consiers the interactions of multiple leukemic clones with healthy hematopoiesis an the emergence of new clones ue to mutations. We use computer simulations to stuy the impact of leukemic cell self-renewal an proliferation rates on the structure of the clonal hierarchy. At the same time, the moel provies insights into how the properties of clones at ifferent positions in the clonal hierarchy iffer. These questions are clinically relevant, since the patients prognosis an the treatment response may epen on the properties of the leukemic cells [23]. Moel simulations suggest that the self-renewal rate of leukemic clones has a major impact on the structure of the clonal hierarchy, whereas proliferation rates show no significant influence. The self-renewal rate of the emerging clones increases uring the course of the isease. There is evience that a high self-renewal rate of clones may be linke to poor prognosis [23]. In this sense, clones appearing later uring the isease are more aggressive than those present at the beginning of the isease. Similarly, simulations suggest that the self-renewal rates of clones increase with increasing epth of the hierarchy, whereas proliferation rates o not epen significantly on the epth of clones in the hierarchy. Simulations of large patient groups suggest that there might exist a significant relationship between the epth of the clonal hierarchy an the maximal self-renewal rate. This fining suggests to evaluate the epth of the clonal hierarchy as a potential marker for patient prognosis. Mutations etecte in acute leukemias act at ifferent regulatory levels. There is evience that many of them lea to increase self-renewal. Important examples for genes where mutations lea to increase selfrenewal are the chromatin moifiers TET2 [53], DNMT3A [54] an MLL [55] or the transcription factors C/EBPα [56], RUNX1/CBFβ [57, 58] an factors encoe by the HOX genes, e.g., as NUP98-HOXA9 [59]. Other examples inclue the isocitrate ehyrogenase IDH1 [60], the NRAS gene [61] or the multi-functional protein NPM1 [62]. Importantly, more than one of these mutations can occur in the same cell [63, 64]. This is in line with the step-wise increase in self-renewal observe in the moel simulations. Emergence of the clonal hierarchy is a ynamic process. Moel simulations show that the properties of the existing clones have an impact on the structure of the clonal hierarchy in the future. Presence of aggressive clones with high self-renewal rates leas to fast progression of the isease. The short remaining lifespan of the patient limits the number of new clones that can emerge an grow to a significant size. Therefore in presence of aggressive clones, the clonal hierarchies consist of a relatively small number of clones. On the other han, if the self-renewal rates of new clones is very close to the self-renewal rate of the parent clone, the new clone expans slowly an takes a long time to reach a significant size. Therefore, mutations causing only small changes in self-renewal rates also lea to small numbers of significant clones an flat hierarchies. The moel simulations suggest that the emergence of eep clonal hierarchies is a complex process. To give rise to offspring, a clone requires a critical mass, otherwise it is unlikely that a clone acquires new mutations. A eep hierarchy is create if new clones have high enough selfrenewal rates to grow to a critical mass before the patient ies, but not too high self-renewal rates to avoi fast progression an eath before the new clones can prouce their own offspring. Simulations imply that these