BACKGROUND
mutation accumulation is thought to drive aging, carcinogenesis, and the increased incidence of birth defects with parental age.
mutations can be accrued as the result of exogenous dna damage caused by radiation or mutagens, or as the result of errors in dna replication.
an intricate cell machinery maintains the genome by detecting and repairing both dna lesions and replication errors  <cit> , strongly suggesting that minimization of mutation accumulation is an important performance objective for cells and organisms.
yet both eukaryotes and prokaryotes accumulate mutations at a rate higher than set by physical limits – as shown strikingly in the case of prokaryotes by the existence of anti-mutator mutants with lower mutation rates than wild-type  <cit> .
although in the case of some eukaryotes higher-than-optimal mutation rates are likely due in part to low population sizes causing genetic drift  <cit> , a more general possible explanation is that genome maintenance comes at a substantial cost in terms of metabolic resources or delays in dna replication .
strategies that do not incur a strong metabolic or speed penalty would thus likely be actively sought out by evolution.
stem cells are expected to play a major role in strategies to minimize the accumulation of mutations in tissues.
since stem cells stand at the top of cell lineages, they can help minimize this accumulation by maintaining a high-quality genome and periodically refreshing a pool of cells that accumulate mutations at a higher rate but that are only transiently present in the tissue.
stem cells can maintain a high-quality genome in essentially two ways.
one possibility is for stem cells to be intrinsically more resistant to mutation accrual , or to undergo more active or less error-prone dna damage repair – likely at the cost of increased metabolic expenditures or slow dna replication.
the other, independent possibility is simply for stem cells to cycle less frequently, and therefore incur fewer replication-dependent mutations over the organism’s lifespan.
asking whether and how organisms implement this strategy, which was proposed by cairns  <cit> , requires a theoretical approach that asks how it should be implemented in practice, and an experimental approach that asks whether theoretical predictions are met.
previous studies with a theoretical emphasis have explored particular principles governing the ratio between the speed at which stem cells cycle and the speed at which their differentiating descendants cycle.
for example, one study defined a performance objective as minimizing the chance of multiple mutational “hits” causing cancer, not considering the speed of development, and assumed an intrinsic difference in mutation rates between stem cells and their differentiating descendants  <cit> ; slower stem-cell cycling was reported to be favored when the stem-cell mutation rate was orders of magnitude lower than that for other cells.
another study focused on speed of development as a performance objective, not considering mutation accumulation, and found that the relative stem-cell cycle speed should be high during the first phase of development before abruptly switching to a lower value, following the “bang-bang” principle of control theory  <cit> .
because both mutation minimization and speed of development are performance objectives relevant to biological systems, here we ask how the slow stem-cell cycling principle outlined by cairns applies when considering these objectives jointly.
the model self-renewing organ we use for this purpose – the c. elegans hermaphroditic germ line – is such that both performance objectives are accessible, as detailed below.
a number of experimental studies have addressed cell cycle properties of stem cells in various contexts.
in vertebrates, although stem cells are thought to reside often in a quiescent state, many organs maintain stem-cell populations that cycle fast .
such fast-cycling populations appear to be supported by “reserve” populations that cycle less frequently and that are, for example, mobilized upon injury  <cit> .
multiple stem-cell subpopulations can thus exist in the same organ; since their discovery is often prompted by the use of new markers or combinations of markers, more are likely to be discovered in the future.
these multiple subpopulations, whose properties and relative contributions to tissue homeostasis can be difficult to measure over extended periods of time and often generate debate , make it challenging to test quantitatively whether cell cycle control follows the pedigree-depth minimization principle.
such a test is thus best performed in an experimental model system where the contribution of all proliferating cells can be readily assayed.
the c. elegans germ line provides a stem-cell model system that is highly amenable to stem-cell cycle studies .
this germ line is contained in tube-like gonadal arms, with stem cells located at the distal end within a mitotic zone .
the stem cells ensure self-renewal throughout life, compensating for cell loss to spermatogenesis, which occurs during larval development, and oogenesis and apoptosis that occur during adulthood.
the mz contains cycling cells and expresses factors driving the cell cycle – such as the worm homologue of cyclin e, cye- <dig>  <cit>  – throughout the  <dig> cell rows that it spans.
the mz is patterned along its distal–proximal axis, notably by counteracting gradients of the pumilio homologues fbf- <dig> and fbf- <dig>  which promote the stem-cell fate  <cit> , and of factors such as gld- <dig> that promote differentiation  <cit>  .
these factors define steps of differentiation within the mz, at rows ~6– <dig> and ~ <dig> from the distal end  <cit> , before the overt meiosis observed at row ~ <dig>  cells do not undergo active migration from one zone to the other, but rather are displaced along the distal–proximal axis; their differentiation state progresses accordingly.
the spatial layout of the mz is important because it obviates the need for fine markers to assay differentiation states – distance to the distal end is a reliable differentiation marker – and because it makes it straightforward to assay the proliferative contribution to the tissue of all cell subpopulations.
although no spatial differences in cell cycle length were found in previous studies  <cit> , variation in m-phase index hints at different cell cycle behavior along the distal–proximal axis  <cit> .fig.
<dig> organization of the c. elegans hermaphroditic gonadal arm.
a mitotic zone  contains stem cells at the distal end, which ensure organ self-renewal as cells are consumed proximally for spermatogenesis  or oogenesis and apoptosis .
differentiation of mitotic cells is controlled by opposing factors such as fbf-1/ <dig> and gld- <dig>  expressed in opposing gradients.
the cell cycle regulator cyclin e <dig>  is expressed throughout the mz.
subregions are shown that are considered in cell cycle analysis: distal-most mitotic zone , medial mitotic zone , and proximal mitotic zone .
cell position can be measured by the number of rows to the distal end
because of its predominantly selfing mode of reproduction, the c. elegans mutation rate is expected to be low: a high mutation rate would have led to rapid extinction of the species via mueller’s ratchet  <cit> .
indeed, the c. elegans mutation rate was found to be ~ <dig> × 10− <dig>  <cit>  or ~10− <dig>  <cit>  per site per generation, slightly lower than the human rate .
the c. elegans gonad thus provides a highly suitable model system to ask how organs minimize the accumulation of mutations, and what role stem cells play in that minimization.
to address the role of stem cells in minimizing mutation accumulation, we built models of cell cycling and mutation accrual, and optimized their parameters computationally.
we find that when taking into account constraints on speed of development and reproduction, c. elegans germ-line stem cells should cycle more slowly than their differentiating counterparts, but that the difference should only be approximately twofold.
using a new quantitative analysis technique, we show that this prediction is borne out experimentally.
we further show that slower stem-cell cycling could be due at least in part to fine-tuning along the distal–proximal axis of expression of the cell cycle regulator cye- <dig>  consistent with a previously identified motif in the germ-line gene regulatory network whose potential significance is highlighted by our approach.
RESULTS
slow-cycling progenitors can minimize replication-dependent mutations by balancing pedigree trees
many organs are generated and subsequently self-renew by amplification of a progenitor cell through multiple rounds of cell division.
the magnitude of the accumulation of dna replication-dependent mutations that results from this amplification is heavily dependent on the cell cycle control strategy that is followed.
accumulation of replication-dependent mutations is best understood by considering the pedigree of all cells that descend from the primordial progenitor .
this pedigree forms a structure known in computer science as a binary tree, where in this case each cell has either zero or two descendants.
we define the pedigree depth of a cell as the number of divisions separating a cell from the primordial germ cell.
the average number of replication-dependent mutations in an organ is then proportional to the average pedigree depth.
average pedigree depth is minimized when trees are balanced, i.e.
when no pairs of cells at the bottom of the tree have pedigree depths that differ by more than one  <cit> .
the performance of cell cycle control strategies in terms of replication-dependent mutation accumulation can thus be assayed by the balance in the cell pedigree trees that they produce.fig.
<dig> slow-cycling stem cells allow for an advantageous tradeoff between pedigree-depth minimization and early production of differentiated cells.
a average pedigree depth  of differentiated cells, defined as the average number of divisions between differentiated cells and the founding progenitor, is minimized by balanced trees , but differentiated cells  are not produced until all cells have finished dividing.
in this optimal configuration, p.d.
= log <dig> where the total number of cells to be produced n =  <dig>  β is the tree balance as defined by  <cit>  .
the gray outline indicates sister subtrees that are the least balanced .
the time axis units are given in rounds of cell division.
the individual pedigree depth of differentiated cells is shown as the inset number.
b early production of differentiated cells can be obtained by successive rounds of asymmetric divisions of a progenitor cell , at the cost of a substantial increase in average pedigree depth.
c pedigree trees can be shaped to allow for early differentiated cell production without incurring a large pedigree-depth penalty.
d–f pedigree tree shape can be controlled by modulating the cycling speed of a stem cell located at the distal end of a model tubular organ.
cells are pushed out toward the proximal end as a result of proliferation, and differentiate when reaching a threshold distance from the distal end .
α is the ratio of the cycling speed of non-stem cells to the cycling speed of the stem cell .
inset numbers show cell pedigree depth as in .
d if only the stem cell cycles, the pedigree tree is similar to that in  and the average pedigree depth is high.
e if the stem-cell cycles are ~ <dig> % slower than other cells in the mz, the pedigree-depth tree is more balanced.
f it is not beneficial for the stem cell to cycle more slowly than in : pedigree depth increases as a result of the increased cycling that other cells in the mz must undergo to produce the desired cell number.
g there exists a single optimal value of α that minimizes the average pedigree depth within the context of models shown in ; the optimal α increases as the total number n of cells to be produced increases .
in other words, the more cells in total are to be produced, the slower stem cells should cycle to preserve the low pedigree depth
fig.
<dig> experimental analysis of c. elegans germ-cell cycling.
a time course of larval germ-cell proliferation at its onset.
a fit assuming exponential growth gave a cell cycle length of  <dig>  h for early germ-line development.
b–e spatial cytometry reveals qualitative differences in cell cycle behavior along the distal–proximal axis of the c. elegans germ-line mz.
b cell cycle phase indices change as a function of distance to the distal end , both at the l <dig> stage and at l <dig> +  <dig> day; in particular, the g <dig> index is higher distally at the expense of the s-phase index.
cell cycle phase indices were determined by pulse-fixing worms with the s-phase label edu and quantification of dna contents.
thin lines show  <dig> % bootstrap confidence band.
arrows show the position at which the g <dig> index starts to rise, which was used to define the proximal end of the mmz.
c–f different progression of edu-positive and edu-negative cell populations at l <dig>  or l <dig> +  <dig> day .
c, e cell cycle progression after edu pulse-chase differs between dmmz  and mmz .
dna content histograms are shown for edu-positive cells  and edu-negative cells , for a range of chase times .
overall, dna content histograms cycle as expected as cells progress through the cycle; the original dna content histogram is approximately reconstituted by 5– <dig> h. but crucially, dmmz and mmz histograms show statistically significant differences  that suggest that mmz cells cycle faster; for example, at l <dig>  the higher incidence of low dna content, edu-positive cells at the  <dig> h chase time in the mmz suggests that these cells underwent division earlier than in the dmmz.
d, f independent analysis of edu pulse-chase data confirms that mmz cycles faster than dmmz.
the fraction of edu-labeled mitoses  in the dmmz and mmz is shown for the same chase times as in .
significant differences, as expected for faster mmz cycling, are apparent at l <dig> for the  <dig> h,  <dig> h, and  <dig> h time points  and at l <dig> +  <dig> day for the  <dig> h,  <dig> h, and  <dig> h time points
fig.
<dig> quantitative cell cycle models that allow for a cell cycle gradient across the mz provide a good fit to experimental data, and show ~ <dig> -fold slower cycling of stem cells.
a dna content histograms of edu-positive cells derived from best-fit simulations of cell cycling to l <dig> +  <dig> day experimental data  overlaid with the same experimental data , at  <dig> h,  <dig> h, and  <dig> h .
experimental data were derived from a total of n =  <dig> gonadal arms.
b fractions of edu-labeled mitoses derived from l <dig> +  <dig> day experimental data  or from best-fit simulations .
c, d best-fit cell cycle parameters show faster cell cycling at the proximal end of the mmz  than at the distal dmmz  both at l <dig>  and l <dig> +  <dig> day , and both when fitting dna content histograms  or fractions of labeled mitoses .
each dot on the graph corresponds to a bootstrap sample; ellipses contain  <dig> % of bootstrap samples and are located off the diagonal, which corresponds to equal cell cycle speeds across the distal–proximal axis.
jitter was added to bootstrap samples to aid visualization .
e, f distal cells have longer g <dig> than proximal cells.
stacked bars show the length of each cell cycle phase along the distal–proximal axis, as computed using best-fit parameters.
note that absolute cell cycle lengths cannot be directly derived from fig.
3b
fig.
<dig> cyclin e levels are graded across the dmmz and mmz, and are differentially dependent on cell cycle phase in the dmmz and mmz.
a example of cye- <dig> staining pattern in a gonadal arm at l <dig> +  <dig> day .
cye- <dig> levels appear to start low in the distal region, rise, and then fall in the proximal region.
b quantification of nuclear cye- <dig> levels using  <dig> cells segmented from  <dig> gonadal arms.
each dot represents a cell; the red line is the average at each cell row, with a  <dig> % bootstrapped confidence interval.
c, d cells with typical g <dig> morphology  have higher cye- <dig> content than their neighbors .
e scatterplot of nuclear cye- <dig> content vs. dna content, showing that cells with lower dna content – i.e.
early in the cell cycle – have moderately higher levels of cye- <dig> than cells with higher dna content.
density colored via “jet” lookup table , and piecewise-linear trend line computed as described in “methods”.
f, g variation of cye- <dig> content with cell cycle phase is lesser for cells in the dmmz  than in the mmz .
the difference between dmmz and mmz is statistically significant .
arrows show two clusters at low and high dna content.
h, i quantification of nuclear cye- <dig> profile as in , but considering only cells with low  or high  dna content
the strategy that minimizes pedigree depth and thus replication-dependent mutation accumulation has significant drawbacks.
this strategy produces a balanced pedigree tree by maintaining an expanding pool of progenitors in which all cells keep cycling at the same rate until the time the organ has reached its final set number of cells .
it precludes the differentiation of cells before that time, requires a large pool of progenitors, and is impractical for organs that must undergo self-renewal throughout life.
early cell differentiation and small progenitor pools are made possible by the naive alternative strategy that consists of maintaining a lineage of asymmetrically dividing progenitors – but this comes at the cost of an unbalanced pedigree tree and thus increased pedigree depth .
a third strategy is possible that compromises between the two previous strategies: a population of long-lived slow-cycling progenitors divides asymmetrically to self-renew and to give rise to faster-cycling progenitors that only persist transiently before differentiating.
this strategy, which we refer to as the pedigree-depth quasi-minimization strategy hereafter, can lead to a highly balanced pedigree tree while allowing early production of differentiated cells and small progenitor pool size .
organ spatial structure and cell cycle length distribution can be exploited for beneficial shaping of pedigree trees
how can organs control differentiation of fast- and slow-cycling progenitors to implement the pedigree-depth quasi-minimization strategy?
this compromise strategy requires control of the transition from the fast-cycling to the slow-cycling state, and control of the number of cycles the faster-cycling population undergoes before differentiation.
many organs have a spatial structure with stem cells located in a niche and cells outside of the niche undergoing differentiation .
this structure can provide for simple control of both the transition between the stem cell and differentiated states and the transition between slow- and fast-cycling states, if cells are displaced from the niche as a result of proliferation, and if the transitions are controlled by distance to the niche.
we considered a simple model organ organized along a single axis, in which cell proliferation pushes cells away from the niche and in which cells speed up in their cell cycle as their distance to the niche increases – but leave the cell cycle and differentiate after reaching a threshold distance .
a cell cycle speed ratio of ~ <dig>  between stem cells and differentiating cells yields lower pedigree depth than both lower and higher ratios .
the optimal cell cycle speed ratio increases as the total number of cells to be produced increases .
an approximately twofold slowdown in stem-cell cycle length optimizes c. elegans germ-line mutation accumulation
what is the optimal compromise between minimization of mutation accumulation and early production of differentiated cells, and what is the resulting optimal stem-cell cycling speed?
the answers to these questions depend on the relative costs of mutation accumulation and of delaying the production of differentiated cells.
we tackle this problem within the context of the c. elegans hermaphroditic gonadal arm, which over the reproductive lifetime of an individual produces ~ <dig> cells that differentiate by entering meiosis.
cells leaving the mz ensure compensation of germ-cell loss to apoptosis and gametogenesis, maintaining gonadal arm cell numbers at a rough steady state of ~ <dig> during adulthood .
only  <dig> meiotic cells give rise to gametes on average; others contribute to oocyte growth by streaming cytoplasmic content  <cit>  and can undergo apoptosis.
the germ-line mutation rate is low  and timing of reproduction is critical to worm fitness  <cit> .
therefore, both minimization of mutation accumulation and early production of differentiated cells are important performance objectives for the worm germ line.
we first sought to establish whether the mz’s tubular organization can efficaciously minimize pedigree depth when combined with a cell cycle gradient.
the minimal average pedigree depth of the ~ <dig> germ cells produced over the lifetime of a gonadal arm is log <dig> =  <dig> .
this minimal value can only be reached by keeping all cells in a cycling state until the time the population number reaches its final value; the body of a young adult c. elegans hermaphrodite could most likely not fit such a high number of germ cells.
we thus asked whether average pedigree depth of differentiated cells can be minimized to a value close to its theoretical minimum even with an mz of limited size.
we used the simulations outlined in box  <dig> and detailed in “methods.” the length of the mitotic cell cycle was modeled as a linear gradient, varying from  <dig>  h at the proximal edge of the mitotic zone to a value at the distal end that was free to vary above a minimum of  <dig>  h .
the value at the distal end was allowed to vary between each of four ranges of developmental stages ; however, the mz length and width did not vary between developmental stages.
thus, this simulation had six free parameters: mz width and length , and distal cell cycle length for each developmental stage.
these six parameters were optimized as described in “methods” to minimize pedigree depth of the first  <dig> differentiated cells.
the minimal pedigree depth, achieved with an mz comprising  <dig> cells, was  <dig>  ; this is close to the theoretical minimum of  <dig> .box  <dig> simulation setup.
agent-based simulations used to characterize the dependence of pedigree depth on the spatiotemporal profile of cell cycle lengths comprised control of cell cycle length by position along the distal–proximal axis, cell movement through the mitotic and meiotic zones, and eventual differentiation or apoptosis.
the spatial domain of the mz was defined by a lattice of positions that could be occupied by at most one cell at a time.
the lattice was rectangular , or had a shape defined from experimental measurements.
the lattice was seeded with a single primordial cell located at the distal end.
as this cell divided, its descendants filled the mz first width-wise and then length-wise, with daughter cells being pushed laterally or proximally as cells behind them  divided.
to mimic the cone-like structure of the gonad, cells at either end of a given row could be displaced in a way that they wrapped around to the other end of the same row .
once daughter cells were pushed beyond the last mz row, they exited the mitotic cell cycle and differentiated by entering the meiotic zone.
the meiotic zone was modeled as a first-in-first-out queue, with cells entering at the distal end as they left the mz, and exiting at the proximal end as they underwent apoptosis or matured as an oocyte.
the length of the mitotic cell cycle was modeled as a linear gradient, controlled at the distal end of the mz and at its proximal end by two parameters with value greater than  <dig>  h .
depending on the kind of simulation, cell length values at the control points were allowed to change at various developmental stages ; in this case, the cell cycle length was linearly interpolated along the time axis in addition to the space axis.
notes on optimizations:
1: optimization  <dig> was run with deleterious mutation rate u =  <dig> , and optimization  <dig> with u =  <dig>
2: optimization  <dig> assumes a mutation rate dependent on cell cycle speed
3: optimization  <dig> assumes that distal cells preserve an immortal dna strand
each row shows results for one kind of problem, defined by the objective function to optimize , the parameters that are free to evolve within bounds during the optimization procedure , and other characteristics grouped under “optimization setup.” results shown are derived using a grid search  α is the ratio of cell cycle lengths between the distal end of the dmmz and the proximal end of the mmz .
see additional file 1: table s <dig> for an extended version of this table, including credible intervals derived from markov chain monte carlo
we next asked whether the cell cycle gradient present along the distal–proximal axis played a substantial role in minimizing pedigree depth in optimization  <dig>  we performed a second optimization that was identical to the first except that cell cycle length was fixed both in space and in time, and constrained total mz cell number to be no more than the optimum for optimization  <dig> ; the minimal pedigree depth was  <dig>  .
note that temporal freedom in cell cycle length does not make any difference to pedigree depth in the absence of spatial freedom, since a fixed number of cells are produced.
the difference between optimizations  <dig> and  <dig> can thus be fully ascribed to the lack of spatial freedom in optimization  <dig>  we next performed a third optimization that lifted the total mz cell number constraint in optimization  <dig>  and found that even without that constraint, the minimal pedigree depth was  <dig> , still higher than when a cell cycle gradient is allowed .
lastly, when changing optimization  <dig> to fix mz cell number to  <dig> – the approximate number of actively cycling cells determined from experimental data  – the pedigree depth was  <dig>  , a high value that further underscores the importance of cell cycle gradients.
overall, germ-cell pedigree depth can be efficaciously minimized by slow cycling of stem cells and differentiation of cells pushed away from the niche.
we next focused on germ cells that become gametes, because only they can transmit mutations to future generations.
the majority of germ cells do not undergo gametogenesis but instead undergo apoptosis  <cit>  .
the production of the ~ <dig> germ cells with a gamete fate is intertwined through time with the production of ~ <dig> meiotic cells whose eventual fate is not gametogenesis.
the minimal average pedigree depth of gametes is thus larger than the minimal average pedigree depth for  <dig> cells that are the only descendants of a primordial progenitor – i.e.
log <dig> =  <dig>  – and likely smaller than the minimal average pedigree depth for  <dig> cells – i.e.
log <dig> =  <dig> .
we extended our model to take into account the extensive apoptosis that occurs in the germ line, and asked how well the pedigree depth of the germ cells that escape apoptosis and undergo gametogenesis can be minimized.
using an apoptosis probability derived from experimental data and that increases with germ-line replicative age  and loose constraints on mz size, we identified a minimal gamete pedigree depth of  <dig> , achieved with an mz that was  <dig> cell diameters long and one cell diameter wide .
this gamete pedigree depth, achieved with a steep cell cycle gradient that is further discussed below, is substantially lower than the theoretical minimum of  <dig>  for the production of  <dig> cells.
it is also lower than the minimal pedigree depth of the first  <dig> cells, because differentiated cells produced early in development  are more likely to become gametes than differentiated cells produced later.
we next ran an optimization identical to optimization  <dig> except that it constrained mz dimensions to  <dig> ×  <dig> rows, closer to experimental mz dimensions, which led to a minimal increase of optimal pedigree depth to  <dig>  .
the cell cycle gradient present along the distal–proximal axis plays a substantial role in minimizing pedigree depth: with a cell cycle length fixed in space and time, the minimal pedigree depth was  <dig>  without constraints on mz size .
note that again temporal freedom in cell cycle length does not make any difference to pedigree depth in the absence of spatial freedom, because inputs to the simulations that are not optimized over are defined in terms of cell divisions rather than elapsed time .
overall, gamete pedigree depth can be efficaciously minimized by slow stem-cell cycling even when the size of the stem-cell pool is constrained.table  <dig> apoptosis probabilities used in mutation accumulation simulations
probabilities are given as a function of the sequence index of cells leaving the mz , and computed from cell cycle and germ-cell count data as described in “methods”; primary spermatocytes do not undergo apoptosis
having established that the simple rules we used for control of germ-cell cycling and differentiation make it possible to minimize gamete pedigree depth, we turned to the tradeoff between pedigree-depth minimization and speed of reproduction.
the cell cycle speed profiles reported above that minimize gamete pedigree depth  would cause slower germ-line development than is experimentally observed.
assuming that a minimal gonad size is required before oogenesis begins, for example because of the role of meiotic cells in streaming content to oocytes  <cit> , a slower development rate delays the onset of reproduction.
using a matrix population model , we computed that the slower reproductive schedule imparted by the optimal cell cycle profile derived from optimization  <dig> leads to a sixfold increase in population doubling time compared to a flat cell cycle length profile fixed at  <dig>  h .
this delay would lead to a fitness loss sufficiently large for natural selection to act upon, since mutants with a developmental delay as small as  <dig>  h can be outcompeted by wild-type organisms  <cit> .
to ask where the optimum between pedigree-depth minimization and fast reproduction lies, we defined a fitness value that captures the effects of delays in the reproductive schedule due to slow stem-cell cycling and of long-term mutational load stemming from replication-dependent mutations .
the impact of replication-dependent mutations depends on the deleterious mutation rate u, for which only indirect estimates are available .
using u =  <dig>  following  <cit>  or the substantially higher value u =  <dig>  following  <cit> , we found that the distribution of cell cycle speeds that maximizes this fitness is one where all cells cycle essentially as fast as permissible – which comes at the cost of sub-optimal germ-cell pedigree depths .
therefore, within the context of the c. elegans germ line, the pressure for slow stem-cell cycling to minimize pedigree depth is strong only as long as this slow cycling does not delay the reproductive schedule.
we thus asked which mz dimensions and cell cycle profile minimized pedigree depth while allowing for a speed of germ-line development and reproduction that were at least as high as determined experimentally by germ-cell counts and rates of oocyte production .
we ran an optimization identical to optimization  <dig>  except that we introduced development and reproduction rate constraints, reduced the maximum mz cell number to enhance computational tractability, and fixed cell cycle length at larval stage l <dig> to its experimentally determined value of  <dig>  h. the latter change was important because a short l <dig> germ-cell cycle is favored by optimizations, which get close to the  <dig>  h limit ; yet the germ-cell cycle at that stage is longer , possibly because of physical constraints beyond the scope of our simulations .
this optimization minimized pedigree depth to a value of  <dig>  .
for comparison, a flat profile produced progeny with average pedigree depth of  <dig> ; the advantage afforded by the cell cycle gradient is thus ~ <dig>  fewer divisions in the germ-cell lineage per generation, i.e.
~ <dig>  divisions per day given the c. elegans generation time of 2– <dig> days.
optimal mz dimensions were  <dig> cells long by  <dig> cells wide , and the optimal cell cycle gradient amplitude was approximately twofold .
experimentally determined dimensions are  <dig>  cells along the long axis  and  <dig>  cells on average along the short axis .
optimal dimensions derived from simulations are thus in remarkably close agreement with experimental measurements.
predicted cell cycle lengths are very similar whether they are derived using the rectangular geometry predicted to be optimal, or the experimentally characterized cone-like geometry that also contains pre-meiotic cells that have left the cell cycle and whose prevalence is forced to follow the experimentally characterized profile along the distal–proximal axis .
these predicted cell cycle lengths are also in close agreement with experimental measurements, as shown in the next section.table  <dig> meiotic cell production constraints enforced on optimizations 10–13
the numbers are derived from the average reproductive schedule measured from n =  <dig> worms by counting progeny laid at days  <dig> to  <dig> of adulthood, and dividing the total of number of progeny by  <dig> for primary spermatocytes , and the daily counts by  <dig> for oocytes
finally, we asked how the optimal cell cycle profile is affected by two putative mechanisms that alter the accrual of replication-dependent mutations and their distribution to daughter cells.
we first considered a model according to which the accrual of mutations is inversely proportional to the speed of the cell cycle, e.g.
because a slower cell cycle could leave more time for proofreading and correction of replication errors.
this leads to a cell cycle length gradient that is marginally steeper than when mutation accrual is independent of cell cycle speed .
we next considered an “immortal strand” model, in which stem cells located in row  <dig> do not retain mutations caused by dna replication, which are instead passed on to daughter cells.
according to this model, it is optimal for stem cells to cycle quickly, because mutations are flushed out of the tissue by stem-cell cycling .
this prediction is at odds with experimental measurements shown in the next section, and we thus do not consider it further.
overall, considering the performance objectives of mutation minimization and timely reproduction allows us to make qualitative and quantitative predictions about the germ-line cell cycle profile.
first, starting from l <dig>  cells in the distal mz should cycle more slowly than cells in the proximal mz ; this difference should persist in adulthood.
second, the cell cycle speed difference between distal and proximal mzs should be of the order of twofold.
to test these predictions, we set out to quantify cell cycle speed along the mz distal–proximal axis in worms at the l <dig> and young adult stages.
cell cycle speed varies ~ <dig> -fold along the distal–proximal axis
to measure germ-line cell cycle speed, we performed pulse-chase labeling using the thymidine analog edu, which is incorporated by cells in s phase.
to distinguish between different cell types within the mz, we quantified fluorescence intensities of cells segmented from confocal images of intact gonadal arms; this allowed us to record the distance of each cell to the distal end as well as dna and edu contents .
we first examined the distribution of cell cycle phase indices along the distal–proximal axis of the gonad .
a sharp increase in the proportion of g <dig> cells occurs at rows  <dig>  or  <dig> .
this is consistent with an increasing proportion of cells having entered the early stages of meiosis as they move proximally from that position, and with the ~ <dig> cells distal to that position providing most of the proliferative activity in adults  <cit>  .
to focus on mitotic cells, we first analyzed the distal-most  <dig> rows  or  <dig> rows .
in the following, we further subdivide this region into a distal region referred to as distal-most mz  comprising rows 1– <dig>  or 1– <dig>  and a more proximal region referred to as medial mz  comprising rows 9– <dig>  or 7– <dig> .
we first looked for qualitative differences in cell cycling along the distal–proximal axis of the gonad.
comparisons of dna content for edu-negative and edu-positive populations in the dmmz and mmz regions revealed differences consistent with the mmz cycling more quickly than the dmmz .
to confirm this result we analyzed the data in an independent way, scoring the fraction of labeled mitoses   <cit> .
we also found significant differences compatible with faster cycling of the mmz .
we next quantified the differences in cell cycle speed between the dmmz and mmz regions.
this quantification must account for the fact that cells from the dmmz feed into the mmz, minimizing the apparent differences between these regions; the mmz thus cannot be analyzed independently.
we therefore fitted experimentally derived dna content histograms and flms to simulations of germ-cell cycling that assumed a linear gradient of cell cycle lengths spanning those regions .
these simulations were identical to those used for pedigree-depth optimization in terms of rules for cell division and ensuing cell displacements, but different in that they were initiated with a pre-filled mz instead of a single progenitor, only covered the period of time corresponding to the cell cycle experiment analyzed , did not allow for changes in cell cycle length parameters over that period, and kept track of progression through the g <dig>  s, g <dig>  and m phases of the cycle.
the best-fit simulation data provided a close fit to the experimental data , supporting the validity of our model.
the average ratio of cell cycle speeds between the distal dmmz and proximal mmz was  <dig>   and  <dig>   at the l <dig> and l <dig> +  <dig> day stages, respectively .
importantly, this result is supported by two independent analysis techniques: one based on the flm, which has been used before without distinguishing between subpopulations along the distal-proximal axis  <cit> , and the new technique we report based on dna content histograms  that makes use of all cells instead of only rare m-phase cells .
flm-based analysis of the proximal mz  suggests a flat cell cycle profile for cells that have not left the mitotic cycle .
therefore, our experimental analysis verified the theoretical prediction that an approximately twofold cell cycle speed gradient should exist along the distal–proximal axis.
to begin asking how cell cycle length is regulated across the distal–proximal axis of the gonad, we computed the estimated distribution of cycle lengths based on our best-fit simulations .
the length of g <dig> showed a clear reduction along the distal–proximal axis , while the other phases did not .
we thus conclude that distal-most cells cycle more slowly for the most part because they spend more time in g <dig>
a cyclin e gradient exists in the distal mz that does not depend on cell cycle phase
to begin identifying mechanisms potentially responsible for slower stem-cell cycling in the c. elegans germ line, we quantified the spatial expression profile of the cell cycle regulator cye- <dig>  we focused on this regulator because it is expressed in the mz and is required for germ-cell cycling  <cit>  and because of its intriguing regulation: it is repressed by the proximal, differentiation-promoting factor gld- <dig>  <cit> , but its transcript is also bound by the repressor fbf- <dig>  <cit> , which acts to promote stem-cell fate distally.
nuclear cye- <dig> expression follows a biphasic gradient within the mz, with a peak at row  <dig> .
a gradient of cye- <dig> thus spans the region comprising rows 1– <dig>  in which we showed that a cell cycle gradient exists.
the difference between the dmmz and mmz is modest  but statistically significant .
average nuclear cye- <dig> levels thus correlate positively with cell cycle speed.
since in most cell types cye- <dig> levels oscillate with cell cycle phase, we asked whether lower cye- <dig> levels in distal-most cells could be explained by their longer g <dig> phase.
we first ascertained whether in the c. elegans germ-line, cye- <dig> expression levels oscillate with cell cycle phase.
we quantified cye- <dig> contents in rows 1– <dig>  and found that cells at the beginning of the cycle indeed express moderately higher cye- <dig> , but that this phase dependence on cell cycle of cye- <dig> levels is for the most part contributed by the mmz and not the dmmz .
we next asked whether the cye- <dig> gradient we observed along the distal–proximal axis was predominantly contributed by cells at a specific phase of the cycle, but found no difference in overall cye- <dig> profiles when considering only cells at the beginning or at the end of the cycle as defined by dna content .
we conclude that cye- <dig> expression levels are regulated in a way that is partly independent of cell cycle phase.
although evidence that is more direct awaits further study, this is consistent with the idea that cye- <dig> may play a causative role in changes in cell cycle length along the distal–proximal axis.
discussion
potential alternative explanations for slow stem-cell cycling
our simulations of replication-dependent mutation accumulation predicted that, to minimize this accumulation while meeting constraints on speed of reproduction, the c. elegans mz should have dimensions of  <dig> ×  <dig> cell rows and should have an approximately twofold cell cycle length gradient across the distal–proximal axis.
while these theoretical predictions were borne out experimentally, there are alternative, non-mutually exclusive potential explanations for the presence of a cell cycle length gradient.
for example, changes in cell cycle speed could be a side effect of cells progressing through differentiation, or could even be part of the mechanism that promotes differentiation  <cit> .
but the change commonly observed in the course of differentiation is a lengthening of the cell cycle , in contrast to the shortening of the cell cycle that we observed in c. elegans germ cells initiating differentiation.
it is also possible that a slower cell cycle allows for more efficient dna repair, a lower dna replication error rate, or lower metabolic demands on the cell that minimize production of dna-damaging free radical species.
indeed, such slower cycling could be a requirement for the lower stem-cell mutation rate posited in some models  <cit> .
data are lacking to use these ideas to extract quantitative predictions on the relationship between the extent of cell cycle lengthening and a reduction in mutation rate.
we showed that our quantitative predictions of cell cycle length ratios were largely unchanged by the additional assumption that mutation rate is inversely proportional to cell cycle length, and that the pedigree-depth quasi-minimization strategy is still effective at further reducing mutation accumulation.
since our quantitative predictions match experimental data closely, the pedigree-depth quasi-minimization strategy is a strong candidate for explaining how the speed of stem-cell cycling was tuned by evolution.
other strategies to minimize mutation accumulation
we note that there are a number of strategies other than cell cycle control to minimize mutation accumulation.
another potential strategy is asymmetric segregation of immortal strands of dna by stem cells  <cit> .
by retaining the unreplicated dna strands at each division, stem cells could segregate replication errors to their differentiating descendants and thus suppress the accumulation of mutations in the stem-cell compartment.
this strategy has been proposed to apply in different contexts to all chromosomes  <cit> , some chromosomes  <cit> , or not at all  <cit> .
how does the pedigree-depth quasi-minimization strategy interact with the immortal strand strategy, which does not rely on control of cell cycle length?
our results show that if this strategy were followed by the c. elegans germ line, the cell cycle length profiles should be very different from those we observed experimentally: stem cells, which would not accumulate mutations, should cycle quickly .
for organs that rely on a large pool of stem cells, if an immortal strand strategy applies, slow cycling of cells at the top of the lineage hierarchy would be beneficial as the stem-cell pool expands during development  <cit> , but once the stem-cell compartment is fully developed stem cells would cycle quickly.
an independent strategy to minimize the accumulation of mutations, whether they were incurred from errors in dna replication or not, is for cells that accrued mutations to senesce  <cit>  or undergo apoptosis  <cit> .
in the c. elegans germ line, extensive apoptosis occurs in older adults.
while this apoptosis could be explained by the elimination of nurse cells  <cit>  or the need to reduce competition between developing germ cells  <cit> , it appears that apoptosis could preferentially eliminate damaged cells in certain contexts  <cit> .
this idea could be further explored in the future with tools to estimate the mutational load in populations of cells before and after they have been purged of apoptotic cells.
extension to other organs
the pedigree-depth quasi-minimization strategy extends to other tissues.
in the following, we consider three differences between the c. elegans gonad and other self-renewing organs that are relevant to pedigree-depth quasi-minimization.
first, a difference with many vertebrate organs is speed of development.
while small developmental delays are expected to have a strong, deleterious effect on fitness in an organism with a short life cycle and a boom–bust lifestyle such as c. elegans, they are likely to have a smaller impact on organisms with a longer life cycle.
such organisms are thus expected to favor low mutation accumulation over high speed of development at least to some extent, since pedigree-depth quasi-minimization will come at a lessened cost.
notably, however, it has been proposed that the development of mouse intestinal crypts is designed to minimize the time to formation of a mature crypt  <cit> .
this strongly suggests that the tradeoff we have investigated between mutation minimization and speed of development is of broad relevance to animals other than c. elegans.
second, a large difference lies in the number of cells to be produced over an individual’s lifetime – with a c. elegans gonadal arm producing ~ <dig> cells and a human testis or hematopoietic system over  <dig>  <cit>  and  <dig>  <cit> , respectively.
because of these differences, the pedigree-depth quasi-minimization strategy predicts that stem cells in vertebrates should have a slower cycling speed relative to their differentiating descendants than in c. elegans.
while in many contexts the contribution of various stem-cell populations remains to be established, the presence of sporadically cycling “reserve” populations  <cit>  is consistent with this idea.
the pedigree-depth quasi-minimization strategy similarly predicts a negative correlation between stem-cell cycling speed and number of cells to be produced over a lifetime; this correlation holds true when comparing hematopoiesis in a number of mammalian species  <cit> .
third, different organs may have different optimal distributions of mutations in the cells that they produce.
in the context of somatic tissues, an important expected benefit of mutation minimization is reduction of cancer frequency.
since multiple “hits” are thought to be required for malignant transformation  <cit> , it might be advantageous for a tissue to minimize the number of cells that carry two or more mutations  <cit> , even if that came at the cost of an overall increase in mutation frequency.
but mutator mutations likely play a significant role in tumorigenesis  <cit> , and control of stem-cell lineage might be better used to minimize the frequency at which the first mutator mutation occurs, since the carcinogenic effects of such a mutation might be difficult to counteract.
in the context of the germ line, the performance objective assumed in the present study was minimization of the average number of mutations in progeny.
the mutation frequency in c. elegans is low , suggesting that the problem of multiple mutations per progeny might not be of practical relevance – quantification of mutation distributions in progeny from old hermaphrodites could confirm this or provide data to guide modifications to the performance objective.
overall, the pedigree-depth quasi-minimization strategy is of broad relevance but would gain from being fine-tuned once the combined effects of multiple mutations carried by the same cell are better understood.
control of cell cycle length to minimize pedigree depth
our study identified two cell cycle phases that show substantial variation in their duration.
s phase is shorter during larval development than in adulthood, and g <dig> is longer in distal cells than in proximal cells both during larval development and in adulthood.
lengthening of g <dig> in preference to other cell cycle phases is consistent with mutation minimization, as replicated chromosomes offer the possibility of error-free damage repair with homologous recombination using the sister chromatid  <cit> .
regulation of g <dig> length has been reported in other contexts  <cit> .
why the s phase lengthens as well as g <dig> when germ lines transition to the adult stage is less clear.
we speculate that a longer s phase could be less error prone because it allows more time for error-free repair before trans-lesion synthesis occurs  <cit> ; the s phase could be shorter during larval development because the benefits of faster development outweigh the costs of decreased dna replication fidelity, which is consistent with our findings and those of  <cit> .
what role does cyclin e <dig> play in control of cell cycle length?
our data contribute two new observations that expand understanding of that role.
first, we extend previous reports that cyclin e <dig> is expressed throughout the cell cycle  by showing with finer quantification that cyclin e <dig> expression levels do change with cell cycle phase .
interestingly, a similar finding has been made in mammalian embryonic stem cells using apc activity as a readout  <cit> , extending earlier reports highlighting the lack of robust oscillations of cell cycle regulators in these cells  <cit> .
second, and more importantly, we show that cyclin e <dig> levels are graded along the distal–proximal axis of the c. elegans germ line in a way that is not solely dependent on changes in the lengths of cell cycle phases.
this suggests that cye- <dig> could play an upstream role in controlling overall cell cycle length, which is also compatible with the complex regulation of cyclin e by the mitosis-promoting factors fbf-1/ <dig> and the meiosis-promoting factor gld- <dig>
a role of cye- <dig> in regulating cell cycle length along the distal–proximal axis could appear at first sight surprising: cyclin e is better known for its role in driving g <dig> progression  <cit> , but a minimal fraction of cells are in g <dig> along the distal–proximal axis – even in the very proximal mz, where cyclin e <dig> levels drop significantly – and it is g <dig> whose length is modulated along that axis.
a role of cye- <dig> in regulating the length of g <dig> is possible given that cdk <dig> is known to play a role in progression through s phase and to m phase  <cit> .
this cdk <dig> role is thought to rely normally on complex formation with cyclin a <dig>  <cit> , but continued expression of cyclin e <dig> past g <dig> in cycling mz cells could allow activity of a cyclin e/cdk <dig> complex past g <dig>  although in the c. elegans germ line cye- <dig> is the cell cycle regulator whose interplay with differentiation regulators is best documented  <cit> , b-type cyclins could also play an important role in control of cell cycle length as they are also potential targets of both fbf-1/ <dig> and gld- <dig>  <cit> .
overall, it appears that there is a complex interplay between the cell cycle machinery and regulators of differentiation.
the design principle highlighted in this study provides one potential reason for the need for fine cell cycle control as cells proceed through differentiation.
CONCLUSIONS
to address the role of stem cells in minimizing mutation accumulation, we built models of cell cycling and mutation accrual and optimized their parameters computationally.
we found that when taking into account constraints on speed of development and reproduction, c. elegans germ-line stem cells should cycle more slowly than their differentiating counterparts, but the difference should only be approximately twofold.
we additionally predicted optimal mz size dimensions of  <dig> ×  <dig> cell rows.
using a new, quantitative analysis technique, we showed that our predictions were borne out experimentally.
our results provide the first quantitative test of the slow stem-cell cycling strategy originally proposed by  <cit> .
these results strongly support the idea that mutation minimization is a relevant performance objective , and highlight an important limitation in the slow-cycling strategy.
we further showed that slower stem-cell cycling could be due at least in part to fine-tuning along the distal–proximal axis of expression of the cell cycle regulator cye- <dig>  consistent with the presence of a previously identified motif in the germ-line gene regulatory network whose potential significance is highlighted by our approach.

 mutation accumulation is thought to drive aging, carcinogenesis, and the increased incidence of birth defects with parental age. mutations can be accrued as the result of exogenous dna damage caused by radiation or mutagens, or as the result of errors in dna replication. an intricate cell machinery maintains the genome by detecting and repairing both dna lesions and replication errors, strongly suggesting that minimization of mutation accumulation is an important performance objective for cells and organisms. yet both eukaryotes and prokaryotes accumulate mutations at a rate higher than set by physical limits – as shown strikingly in the case of prokaryotes by the existence of anti-mutator mutants with lower mutation rates than wild-type. although in the case of some eukaryotes higher-than-optimal mutation rates are likely due in part to low population sizes causing genetic drift, a more general possible explanation is that genome maintenance comes at a substantial cost in terms of metabolic resources or delays in dna replication. strategies that do not incur a strong metabolic or speed penalty would thus likely be actively sought out by evolution. stem cells are expected to play a major role in strategies to minimize the accumulation of mutations in tissues. since stem cells stand at the top of cell lineages, they can help minimize this accumulation by maintaining a high-quality genome and periodically refreshing a pool of cells that accumulate mutations at a higher rate but that are only transiently present in the tissue. stem cells can maintain a high-quality genome in essentially two ways. one possibility is for stem cells to be intrinsically more resistant to mutation accrual, or to undergo more active or less error-prone dna damage repair – likely at the cost of increased metabolic expenditures or slow dna replication. the other, independent possibility is simply for stem cells to cycle less frequently, and therefore incur fewer replication-dependent mutations over the organism’s lifespan. asking whether and how organisms implement this strategy, which was proposed by cairns, requires a theoretical approach that asks how it should be implemented in practice, and an experimental approach that asks whether theoretical predictions are met. previous studies with a theoretical emphasis have explored particular principles governing the ratio between the speed at which stem cells cycle and the speed at which their differentiating descendants cycle. for example, one study defined a performance objective as minimizing the chance of multiple mutational “hits” causing cancer, not considering the speed of development, and assumed an intrinsic difference in mutation rates between stem cells and their differentiating descendants; slower stem-cell cycling was reported to be favored when the stem-cell mutation rate was orders of magnitude lower than that for other cells. another study focused on speed of development as a performance objective, not considering mutation accumulation, and found that the relative stem-cell cycle speed should be high during the first phase of development before abruptly switching to a lower value, following the “bang-bang” principle of control theory. because both mutation minimization and speed of development are performance objectives relevant to biological systems, here we ask how the slow stem-cell cycling principle outlined by cairns applies when considering these objectives jointly. the model self-renewing organ we use for this purpose – the c. elegans hermaphroditic germ line – is such that both performance objectives are accessible, as detailed below. a number of experimental studies have addressed cell cycle properties of stem cells in various contexts. in vertebrates, although stem cells are thought to reside often in a quiescent state, many organs maintain stem-cell populations that cycle fast. such fast-cycling populations appear to be supported by “reserve” populations that cycle less frequently and that are, for example, mobilized upon injury. multiple stem-cell subpopulations can thus exist in the same organ; since their discovery is often prompted by the use of new markers or combinations of markers, more are likely to be discovered in the future. these multiple subpopulations, whose properties and relative contributions to tissue homeostasis can be difficult to measure over extended periods of time and often generate debate, make it challenging to test quantitatively whether cell cycle control follows the pedigree-depth minimization principle. such a test is thus best performed in an experimental model system where the contribution of all proliferating cells can be readily assayed. the c. elegans germ line provides a stem-cell model system that is highly amenable to stem-cell cycle studies. this germ line is contained in tube-like gonadal arms, with stem cells located at the distal end within a mitotic zone. the stem cells ensure self-renewal throughout life, compensating for cell loss to spermatogenesis, which occurs during larval development, and oogenesis and apoptosis that occur during adulthood. the mz contains cycling cells and expresses factors driving the cell cycle – such as the worm homologue of cyclin e, cye- – throughout the cell rows that it spans. the mz is patterned along its distal–proximal axis, notably by counteracting gradients of the pumilio homologues fbf- and fbf- which promote the stem-cell fate, and of factors such as gld- that promote differentiation. these factors define steps of differentiation within the mz, at rows ~6– and ~ from the distal end, before the overt meiosis observed at row ~ cells do not undergo active migration from one zone to the other, but rather are displaced along the distal–proximal axis; their differentiation state progresses accordingly. the spatial layout of the mz is important because it obviates the need for fine markers to assay differentiation states – distance to the distal end is a reliable differentiation marker – and because it makes it straightforward to assay the proliferative contribution to the tissue of all cell subpopulations. although no spatial differences in cell cycle length were found in previous studies, variation in m-phase index hints at different cell cycle behavior along the distal–proximal axis.fig. organization of the c. elegans hermaphroditic gonadal arm. a mitotic zone contains stem cells at the distal end, which ensure organ self-renewal as cells are consumed proximally for spermatogenesis or oogenesis and apoptosis. differentiation of mitotic cells is controlled by opposing factors such as fbf-1/ and gld- expressed in opposing gradients. the cell cycle regulator cyclin e is expressed throughout the mz. subregions are shown that are considered in cell cycle analysis: distal-most mitotic zone, medial mitotic zone, and proximal mitotic zone. cell position can be measured by the number of rows to the distal end because of its predominantly selfing mode of reproduction, the c. elegans mutation rate is expected to be low: a high mutation rate would have led to rapid extinction of the species via mueller’s ratchet. indeed, the c. elegans mutation rate was found to be ~ × 10− or ~10− per site per generation, slightly lower than the human rate. the c. elegans gonad thus provides a highly suitable model system to ask how organs minimize the accumulation of mutations, and what role stem cells play in that minimization. to address the role of stem cells in minimizing mutation accumulation, we built models of cell cycling and mutation accrual, and optimized their parameters computationally. we find that when taking into account constraints on speed of development and reproduction, c. elegans germ-line stem cells should cycle more slowly than their differentiating counterparts, but that the difference should only be approximately twofold. using a new quantitative analysis technique, we show that this prediction is borne out experimentally. we further show that slower stem-cell cycling could be due at least in part to fine-tuning along the distal–proximal axis of expression of the cell cycle regulator cye- consistent with a previously identified motif in the germ-line gene regulatory network whose potential significance is highlighted by our approach.  slow-cycling progenitors can minimize replication-dependent mutations by balancing pedigree trees many organs are generated and subsequently self-renew by amplification of a progenitor cell through multiple rounds of cell division. the magnitude of the accumulation of dna replication-dependent mutations that results from this amplification is heavily dependent on the cell cycle control strategy that is followed. accumulation of replication-dependent mutations is best understood by considering the pedigree of all cells that descend from the primordial progenitor. this pedigree forms a structure known in computer science as a binary tree, where in this case each cell has either zero or two descendants. we define the pedigree depth of a cell as the number of divisions separating a cell from the primordial germ cell. the average number of replication-dependent mutations in an organ is then proportional to the average pedigree depth. average pedigree depth is minimized when trees are balanced, i.e. when no pairs of cells at the bottom of the tree have pedigree depths that differ by more than one. the performance of cell cycle control strategies in terms of replication-dependent mutation accumulation can thus be assayed by the balance in the cell pedigree trees that they produce.fig. slow-cycling stem cells allow for an advantageous tradeoff between pedigree-depth minimization and early production of differentiated cells. a average pedigree depth of differentiated cells, defined as the average number of divisions between differentiated cells and the founding progenitor, is minimized by balanced trees, but differentiated cells are not produced until all cells have finished dividing. in this optimal configuration, p.d. = log where the total number of cells to be produced n = β is the tree balance as defined by. the gray outline indicates sister subtrees that are the least balanced. the time axis units are given in rounds of cell division. the individual pedigree depth of differentiated cells is shown as the inset number. b early production of differentiated cells can be obtained by successive rounds of asymmetric divisions of a progenitor cell, at the cost of a substantial increase in average pedigree depth. c pedigree trees can be shaped to allow for early differentiated cell production without incurring a large pedigree-depth penalty. d–f pedigree tree shape can be controlled by modulating the cycling speed of a stem cell located at the distal end of a model tubular organ. cells are pushed out toward the proximal end as a result of proliferation, and differentiate when reaching a threshold distance from the distal end. α is the ratio of the cycling speed of non-stem cells to the cycling speed of the stem cell. inset numbers show cell pedigree depth as in. d if only the stem cell cycles, the pedigree tree is similar to that in and the average pedigree depth is high. e if the stem-cell cycles are ~ % slower than other cells in the mz, the pedigree-depth tree is more balanced. f it is not beneficial for the stem cell to cycle more slowly than in: pedigree depth increases as a result of the increased cycling that other cells in the mz must undergo to produce the desired cell number. g there exists a single optimal value of α that minimizes the average pedigree depth within the context of models shown in; the optimal α increases as the total number n of cells to be produced increases. in other words, the more cells in total are to be produced, the slower stem cells should cycle to preserve the low pedigree depth fig. experimental analysis of c. elegans germ-cell cycling. a time course of larval germ-cell proliferation at its onset. a fit assuming exponential growth gave a cell cycle length of h for early germ-line development. b–e spatial cytometry reveals qualitative differences in cell cycle behavior along the distal–proximal axis of the c. elegans germ-line mz. b cell cycle phase indices change as a function of distance to the distal end, both at the l stage and at l + day; in particular, the g index is higher distally at the expense of the s-phase index. cell cycle phase indices were determined by pulse-fixing worms with the s-phase label edu and quantification of dna contents. thin lines show % bootstrap confidence band. arrows show the position at which the g index starts to rise, which was used to define the proximal end of the mmz. c–f different progression of edu-positive and edu-negative cell populations at l or l + day. c, e cell cycle progression after edu pulse-chase differs between dmmz and mmz. dna content histograms are shown for edu-positive cells and edu-negative cells, for a range of chase times. overall, dna content histograms cycle as expected as cells progress through the cycle; the original dna content histogram is approximately reconstituted by 5– h. but crucially, dmmz and mmz histograms show statistically significant differences that suggest that mmz cells cycle faster; for example, at l the higher incidence of low dna content, edu-positive cells at the h chase time in the mmz suggests that these cells underwent division earlier than in the dmmz. d, f independent analysis of edu pulse-chase data confirms that mmz cycles faster than dmmz. the fraction of edu-labeled mitoses in the dmmz and mmz is shown for the same chase times as in. significant differences, as expected for faster mmz cycling, are apparent at l for the h, h, and h time points and at l + day for the h, h, and h time points fig. quantitative cell cycle models that allow for a cell cycle gradient across the mz provide a good fit to experimental data, and show ~ -fold slower cycling of stem cells. a dna content histograms of edu-positive cells derived from best-fit simulations of cell cycling to l + day experimental data overlaid with the same experimental data, at h, h, and h. experimental data were derived from a total of n = gonadal arms. b fractions of edu-labeled mitoses derived from l + day experimental data or from best-fit simulations. c, d best-fit cell cycle parameters show faster cell cycling at the proximal end of the mmz than at the distal dmmz both at l and l + day, and both when fitting dna content histograms or fractions of labeled mitoses. each dot on the graph corresponds to a bootstrap sample; ellipses contain % of bootstrap samples and are located off the diagonal, which corresponds to equal cell cycle speeds across the distal–proximal axis. jitter was added to bootstrap samples to aid visualization. e, f distal cells have longer g than proximal cells. stacked bars show the length of each cell cycle phase along the distal–proximal axis, as computed using best-fit parameters. note that absolute cell cycle lengths cannot be directly derived from fig. 3b fig. cyclin e levels are graded across the dmmz and mmz, and are differentially dependent on cell cycle phase in the dmmz and mmz. a example of cye- staining pattern in a gonadal arm at l + day. cye- levels appear to start low in the distal region, rise, and then fall in the proximal region. b quantification of nuclear cye- levels using cells segmented from gonadal arms. each dot represents a cell; the red line is the average at each cell row, with a % bootstrapped confidence interval. c, d cells with typical g morphology have higher cye- content than their neighbors. e scatterplot of nuclear cye- content vs. dna content, showing that cells with lower dna content – i.e. early in the cell cycle – have moderately higher levels of cye- than cells with higher dna content. density colored via “jet” lookup table, and piecewise-linear trend line computed as described in “methods”. f, g variation of cye- content with cell cycle phase is lesser for cells in the dmmz than in the mmz. the difference between dmmz and mmz is statistically significant. arrows show two clusters at low and high dna content. h, i quantification of nuclear cye- profile as in, but considering only cells with low or high dna content the strategy that minimizes pedigree depth and thus replication-dependent mutation accumulation has significant drawbacks. this strategy produces a balanced pedigree tree by maintaining an expanding pool of progenitors in which all cells keep cycling at the same rate until the time the organ has reached its final set number of cells. it precludes the differentiation of cells before that time, requires a large pool of progenitors, and is impractical for organs that must undergo self-renewal throughout life. early cell differentiation and small progenitor pools are made possible by the naive alternative strategy that consists of maintaining a lineage of asymmetrically dividing progenitors – but this comes at the cost of an unbalanced pedigree tree and thus increased pedigree depth. a third strategy is possible that compromises between the two previous strategies: a population of long-lived slow-cycling progenitors divides asymmetrically to self-renew and to give rise to faster-cycling progenitors that only persist transiently before differentiating. this strategy, which we refer to as the pedigree-depth quasi-minimization strategy hereafter, can lead to a highly balanced pedigree tree while allowing early production of differentiated cells and small progenitor pool size. organ spatial structure and cell cycle length distribution can be exploited for beneficial shaping of pedigree trees how can organs control differentiation of fast- and slow-cycling progenitors to implement the pedigree-depth quasi-minimization strategy? this compromise strategy requires control of the transition from the fast-cycling to the slow-cycling state, and control of the number of cycles the faster-cycling population undergoes before differentiation. many organs have a spatial structure with stem cells located in a niche and cells outside of the niche undergoing differentiation. this structure can provide for simple control of both the transition between the stem cell and differentiated states and the transition between slow- and fast-cycling states, if cells are displaced from the niche as a result of proliferation, and if the transitions are controlled by distance to the niche. we considered a simple model organ organized along a single axis, in which cell proliferation pushes cells away from the niche and in which cells speed up in their cell cycle as their distance to the niche increases – but leave the cell cycle and differentiate after reaching a threshold distance. a cell cycle speed ratio of ~ between stem cells and differentiating cells yields lower pedigree depth than both lower and higher ratios. the optimal cell cycle speed ratio increases as the total number of cells to be produced increases. an approximately twofold slowdown in stem-cell cycle length optimizes c. elegans germ-line mutation accumulation what is the optimal compromise between minimization of mutation accumulation and early production of differentiated cells, and what is the resulting optimal stem-cell cycling speed? the answers to these questions depend on the relative costs of mutation accumulation and of delaying the production of differentiated cells. we tackle this problem within the context of the c. elegans hermaphroditic gonadal arm, which over the reproductive lifetime of an individual produces ~ cells that differentiate by entering meiosis. cells leaving the mz ensure compensation of germ-cell loss to apoptosis and gametogenesis, maintaining gonadal arm cell numbers at a rough steady state of ~ during adulthood. only meiotic cells give rise to gametes on average; others contribute to oocyte growth by streaming cytoplasmic content and can undergo apoptosis. the germ-line mutation rate is low and timing of reproduction is critical to worm fitness. therefore, both minimization of mutation accumulation and early production of differentiated cells are important performance objectives for the worm germ line. we first sought to establish whether the mz’s tubular organization can efficaciously minimize pedigree depth when combined with a cell cycle gradient. the minimal average pedigree depth of the ~ germ cells produced over the lifetime of a gonadal arm is log =. this minimal value can only be reached by keeping all cells in a cycling state until the time the population number reaches its final value; the body of a young adult c. elegans hermaphrodite could most likely not fit such a high number of germ cells. we thus asked whether average pedigree depth of differentiated cells can be minimized to a value close to its theoretical minimum even with an mz of limited size. we used the simulations outlined in box  and detailed in “methods.” the length of the mitotic cell cycle was modeled as a linear gradient, varying from h at the proximal edge of the mitotic zone to a value at the distal end that was free to vary above a minimum of h. the value at the distal end was allowed to vary between each of four ranges of developmental stages; however, the mz length and width did not vary between developmental stages. thus, this simulation had six free parameters: mz width and length, and distal cell cycle length for each developmental stage. these six parameters were optimized as described in “methods” to minimize pedigree depth of the first differentiated cells. the minimal pedigree depth, achieved with an mz comprising cells, was; this is close to the theoretical minimum of.box simulation setup. agent-based simulations used to characterize the dependence of pedigree depth on the spatiotemporal profile of cell cycle lengths comprised control of cell cycle length by position along the distal–proximal axis, cell movement through the mitotic and meiotic zones, and eventual differentiation or apoptosis. the spatial domain of the mz was defined by a lattice of positions that could be occupied by at most one cell at a time. the lattice was rectangular, or had a shape defined from experimental measurements. the lattice was seeded with a single primordial cell located at the distal end. as this cell divided, its descendants filled the mz first width-wise and then length-wise, with daughter cells being pushed laterally or proximally as cells behind them divided. to mimic the cone-like structure of the gonad, cells at either end of a given row could be displaced in a way that they wrapped around to the other end of the same row. once daughter cells were pushed beyond the last mz row, they exited the mitotic cell cycle and differentiated by entering the meiotic zone. the meiotic zone was modeled as a first-in-first-out queue, with cells entering at the distal end as they left the mz, and exiting at the proximal end as they underwent apoptosis or matured as an oocyte. the length of the mitotic cell cycle was modeled as a linear gradient, controlled at the distal end of the mz and at its proximal end by two parameters with value greater than h. depending on the kind of simulation, cell length values at the control points were allowed to change at various developmental stages; in this case, the cell cycle length was linearly interpolated along the time axis in addition to the space axis. notes on optimizations: 1: optimization was run with deleterious mutation rate u =, and optimization with u = 2: optimization assumes a mutation rate dependent on cell cycle speed 3: optimization assumes that distal cells preserve an immortal dna strand each row shows results for one kind of problem, defined by the objective function to optimize, the parameters that are free to evolve within bounds during the optimization procedure, and other characteristics grouped under “optimization setup.” results shown are derived using a grid search α is the ratio of cell cycle lengths between the distal end of the dmmz and the proximal end of the mmz. see additional file 1: table s for an extended version of this table, including credible intervals derived from markov chain monte carlo we next asked whether the cell cycle gradient present along the distal–proximal axis played a substantial role in minimizing pedigree depth in optimization we performed a second optimization that was identical to the first except that cell cycle length was fixed both in space and in time, and constrained total mz cell number to be no more than the optimum for optimization; the minimal pedigree depth was. note that temporal freedom in cell cycle length does not make any difference to pedigree depth in the absence of spatial freedom, since a fixed number of cells are produced. the difference between optimizations and can thus be fully ascribed to the lack of spatial freedom in optimization we next performed a third optimization that lifted the total mz cell number constraint in optimization and found that even without that constraint, the minimal pedigree depth was, still higher than when a cell cycle gradient is allowed. lastly, when changing optimization to fix mz cell number to – the approximate number of actively cycling cells determined from experimental data – the pedigree depth was, a high value that further underscores the importance of cell cycle gradients. overall, germ-cell pedigree depth can be efficaciously minimized by slow cycling of stem cells and differentiation of cells pushed away from the niche. we next focused on germ cells that become gametes, because only they can transmit mutations to future generations. the majority of germ cells do not undergo gametogenesis but instead undergo apoptosis. the production of the ~ germ cells with a gamete fate is intertwined through time with the production of ~ meiotic cells whose eventual fate is not gametogenesis. the minimal average pedigree depth of gametes is thus larger than the minimal average pedigree depth for cells that are the only descendants of a primordial progenitor – i.e. log = – and likely smaller than the minimal average pedigree depth for cells – i.e. log =. we extended our model to take into account the extensive apoptosis that occurs in the germ line, and asked how well the pedigree depth of the germ cells that escape apoptosis and undergo gametogenesis can be minimized. using an apoptosis probability derived from experimental data and that increases with germ-line replicative age and loose constraints on mz size, we identified a minimal gamete pedigree depth of, achieved with an mz that was cell diameters long and one cell diameter wide. this gamete pedigree depth, achieved with a steep cell cycle gradient that is further discussed below, is substantially lower than the theoretical minimum of for the production of cells. it is also lower than the minimal pedigree depth of the first cells, because differentiated cells produced early in development are more likely to become gametes than differentiated cells produced later. we next ran an optimization identical to optimization except that it constrained mz dimensions to × rows, closer to experimental mz dimensions, which led to a minimal increase of optimal pedigree depth to. the cell cycle gradient present along the distal–proximal axis plays a substantial role in minimizing pedigree depth: with a cell cycle length fixed in space and time, the minimal pedigree depth was without constraints on mz size. note that again temporal freedom in cell cycle length does not make any difference to pedigree depth in the absence of spatial freedom, because inputs to the simulations that are not optimized over are defined in terms of cell divisions rather than elapsed time. overall, gamete pedigree depth can be efficaciously minimized by slow stem-cell cycling even when the size of the stem-cell pool is constrained.table apoptosis probabilities used in mutation accumulation simulations probabilities are given as a function of the sequence index of cells leaving the mz, and computed from cell cycle and germ-cell count data as described in “methods”; primary spermatocytes do not undergo apoptosis having established that the simple rules we used for control of germ-cell cycling and differentiation make it possible to minimize gamete pedigree depth, we turned to the tradeoff between pedigree-depth minimization and speed of reproduction. the cell cycle speed profiles reported above that minimize gamete pedigree depth would cause slower germ-line development than is experimentally observed. assuming that a minimal gonad size is required before oogenesis begins, for example because of the role of meiotic cells in streaming content to oocytes, a slower development rate delays the onset of reproduction. using a matrix population model, we computed that the slower reproductive schedule imparted by the optimal cell cycle profile derived from optimization leads to a sixfold increase in population doubling time compared to a flat cell cycle length profile fixed at h. this delay would lead to a fitness loss sufficiently large for natural selection to act upon, since mutants with a developmental delay as small as h can be outcompeted by wild-type organisms. to ask where the optimum between pedigree-depth minimization and fast reproduction lies, we defined a fitness value that captures the effects of delays in the reproductive schedule due to slow stem-cell cycling and of long-term mutational load stemming from replication-dependent mutations. the impact of replication-dependent mutations depends on the deleterious mutation rate u, for which only indirect estimates are available. using u = following or the substantially higher value u = following, we found that the distribution of cell cycle speeds that maximizes this fitness is one where all cells cycle essentially as fast as permissible – which comes at the cost of sub-optimal germ-cell pedigree depths. therefore, within the context of the c. elegans germ line, the pressure for slow stem-cell cycling to minimize pedigree depth is strong only as long as this slow cycling does not delay the reproductive schedule. we thus asked which mz dimensions and cell cycle profile minimized pedigree depth while allowing for a speed of germ-line development and reproduction that were at least as high as determined experimentally by germ-cell counts and rates of oocyte production. we ran an optimization identical to optimization except that we introduced development and reproduction rate constraints, reduced the maximum mz cell number to enhance computational tractability, and fixed cell cycle length at larval stage l to its experimentally determined value of h. the latter change was important because a short l germ-cell cycle is favored by optimizations, which get close to the h limit; yet the germ-cell cycle at that stage is longer, possibly because of physical constraints beyond the scope of our simulations. this optimization minimized pedigree depth to a value of. for comparison, a flat profile produced progeny with average pedigree depth of; the advantage afforded by the cell cycle gradient is thus ~ fewer divisions in the germ-cell lineage per generation, i.e. ~ divisions per day given the c. elegans generation time of 2– days. optimal mz dimensions were cells long by cells wide, and the optimal cell cycle gradient amplitude was approximately twofold. experimentally determined dimensions are cells along the long axis and cells on average along the short axis. optimal dimensions derived from simulations are thus in remarkably close agreement with experimental measurements. predicted cell cycle lengths are very similar whether they are derived using the rectangular geometry predicted to be optimal, or the experimentally characterized cone-like geometry that also contains pre-meiotic cells that have left the cell cycle and whose prevalence is forced to follow the experimentally characterized profile along the distal–proximal axis. these predicted cell cycle lengths are also in close agreement with experimental measurements, as shown in the next section.table meiotic cell production constraints enforced on optimizations 10–13 the numbers are derived from the average reproductive schedule measured from n = worms by counting progeny laid at days to of adulthood, and dividing the total of number of progeny by for primary spermatocytes, and the daily counts by for oocytes finally, we asked how the optimal cell cycle profile is affected by two putative mechanisms that alter the accrual of replication-dependent mutations and their distribution to daughter cells. we first considered a model according to which the accrual of mutations is inversely proportional to the speed of the cell cycle, e.g. because a slower cell cycle could leave more time for proofreading and correction of replication errors. this leads to a cell cycle length gradient that is marginally steeper than when mutation accrual is independent of cell cycle speed. we next considered an “immortal strand” model, in which stem cells located in row do not retain mutations caused by dna replication, which are instead passed on to daughter cells. according to this model, it is optimal for stem cells to cycle quickly, because mutations are flushed out of the tissue by stem-cell cycling. this prediction is at odds with experimental measurements shown in the next section, and we thus do not consider it further. overall, considering the performance objectives of mutation minimization and timely reproduction allows us to make qualitative and quantitative predictions about the germ-line cell cycle profile. first, starting from l cells in the distal mz should cycle more slowly than cells in the proximal mz; this difference should persist in adulthood. second, the cell cycle speed difference between distal and proximal mzs should be of the order of twofold. to test these predictions, we set out to quantify cell cycle speed along the mz distal–proximal axis in worms at the l and young adult stages. cell cycle speed varies ~ -fold along the distal–proximal axis to measure germ-line cell cycle speed, we performed pulse-chase labeling using the thymidine analog edu, which is incorporated by cells in s phase. to distinguish between different cell types within the mz, we quantified fluorescence intensities of cells segmented from confocal images of intact gonadal arms; this allowed us to record the distance of each cell to the distal end as well as dna and edu contents. we first examined the distribution of cell cycle phase indices along the distal–proximal axis of the gonad. a sharp increase in the proportion of g cells occurs at rows or. this is consistent with an increasing proportion of cells having entered the early stages of meiosis as they move proximally from that position, and with the ~ cells distal to that position providing most of the proliferative activity in adults. to focus on mitotic cells, we first analyzed the distal-most rows or rows. in the following, we further subdivide this region into a distal region referred to as distal-most mz comprising rows 1– or 1– and a more proximal region referred to as medial mz comprising rows 9– or 7–. we first looked for qualitative differences in cell cycling along the distal–proximal axis of the gonad. comparisons of dna content for edu-negative and edu-positive populations in the dmmz and mmz regions revealed differences consistent with the mmz cycling more quickly than the dmmz. to confirm this result we analyzed the data in an independent way, scoring the fraction of labeled mitoses. we also found significant differences compatible with faster cycling of the mmz. we next quantified the differences in cell cycle speed between the dmmz and mmz regions. this quantification must account for the fact that cells from the dmmz feed into the mmz, minimizing the apparent differences between these regions; the mmz thus cannot be analyzed independently. we therefore fitted experimentally derived dna content histograms and flms to simulations of germ-cell cycling that assumed a linear gradient of cell cycle lengths spanning those regions. these simulations were identical to those used for pedigree-depth optimization in terms of rules for cell division and ensuing cell displacements, but different in that they were initiated with a pre-filled mz instead of a single progenitor, only covered the period of time corresponding to the cell cycle experiment analyzed, did not allow for changes in cell cycle length parameters over that period, and kept track of progression through the g s, g and m phases of the cycle. the best-fit simulation data provided a close fit to the experimental data, supporting the validity of our model. the average ratio of cell cycle speeds between the distal dmmz and proximal mmz was and at the l and l + day stages, respectively. importantly, this result is supported by two independent analysis techniques: one based on the flm, which has been used before without distinguishing between subpopulations along the distal-proximal axis, and the new technique we report based on dna content histograms that makes use of all cells instead of only rare m-phase cells. flm-based analysis of the proximal mz suggests a flat cell cycle profile for cells that have not left the mitotic cycle. therefore, our experimental analysis verified the theoretical prediction that an approximately twofold cell cycle speed gradient should exist along the distal–proximal axis. to begin asking how cell cycle length is regulated across the distal–proximal axis of the gonad, we computed the estimated distribution of cycle lengths based on our best-fit simulations. the length of g showed a clear reduction along the distal–proximal axis, while the other phases did not. we thus conclude that distal-most cells cycle more slowly for the most part because they spend more time in g a cyclin e gradient exists in the distal mz that does not depend on cell cycle phase to begin identifying mechanisms potentially responsible for slower stem-cell cycling in the c. elegans germ line, we quantified the spatial expression profile of the cell cycle regulator cye- we focused on this regulator because it is expressed in the mz and is required for germ-cell cycling and because of its intriguing regulation: it is repressed by the proximal, differentiation-promoting factor gld-, but its transcript is also bound by the repressor fbf-, which acts to promote stem-cell fate distally. nuclear cye- expression follows a biphasic gradient within the mz, with a peak at row. a gradient of cye- thus spans the region comprising rows 1– in which we showed that a cell cycle gradient exists. the difference between the dmmz and mmz is modest but statistically significant. average nuclear cye- levels thus correlate positively with cell cycle speed. since in most cell types cye- levels oscillate with cell cycle phase, we asked whether lower cye- levels in distal-most cells could be explained by their longer g phase. we first ascertained whether in the c. elegans germ-line, cye- expression levels oscillate with cell cycle phase. we quantified cye- contents in rows 1– and found that cells at the beginning of the cycle indeed express moderately higher cye-, but that this phase dependence on cell cycle of cye- levels is for the most part contributed by the mmz and not the dmmz. we next asked whether the cye- gradient we observed along the distal–proximal axis was predominantly contributed by cells at a specific phase of the cycle, but found no difference in overall cye- profiles when considering only cells at the beginning or at the end of the cycle as defined by dna content. we conclude that cye- expression levels are regulated in a way that is partly independent of cell cycle phase. although evidence that is more direct awaits further study, this is consistent with the idea that cye- may play a causative role in changes in cell cycle length along the distal–proximal axis. discussion potential alternative explanations for slow stem-cell cycling our simulations of replication-dependent mutation accumulation predicted that, to minimize this accumulation while meeting constraints on speed of reproduction, the c. elegans mz should have dimensions of × cell rows and should have an approximately twofold cell cycle length gradient across the distal–proximal axis. while these theoretical predictions were borne out experimentally, there are alternative, non-mutually exclusive potential explanations for the presence of a cell cycle length gradient. for example, changes in cell cycle speed could be a side effect of cells progressing through differentiation, or could even be part of the mechanism that promotes differentiation. but the change commonly observed in the course of differentiation is a lengthening of the cell cycle, in contrast to the shortening of the cell cycle that we observed in c. elegans germ cells initiating differentiation. it is also possible that a slower cell cycle allows for more efficient dna repair, a lower dna replication error rate, or lower metabolic demands on the cell that minimize production of dna-damaging free radical species. indeed, such slower cycling could be a requirement for the lower stem-cell mutation rate posited in some models. data are lacking to use these ideas to extract quantitative predictions on the relationship between the extent of cell cycle lengthening and a reduction in mutation rate. we showed that our quantitative predictions of cell cycle length ratios were largely unchanged by the additional assumption that mutation rate is inversely proportional to cell cycle length, and that the pedigree-depth quasi-minimization strategy is still effective at further reducing mutation accumulation. since our quantitative predictions match experimental data closely, the pedigree-depth quasi-minimization strategy is a strong candidate for explaining how the speed of stem-cell cycling was tuned by evolution. other strategies to minimize mutation accumulation we note that there are a number of strategies other than cell cycle control to minimize mutation accumulation. another potential strategy is asymmetric segregation of immortal strands of dna by stem cells. by retaining the unreplicated dna strands at each division, stem cells could segregate replication errors to their differentiating descendants and thus suppress the accumulation of mutations in the stem-cell compartment. this strategy has been proposed to apply in different contexts to all chromosomes, some chromosomes, or not at all. how does the pedigree-depth quasi-minimization strategy interact with the immortal strand strategy, which does not rely on control of cell cycle length? our results show that if this strategy were followed by the c. elegans germ line, the cell cycle length profiles should be very different from those we observed experimentally: stem cells, which would not accumulate mutations, should cycle quickly. for organs that rely on a large pool of stem cells, if an immortal strand strategy applies, slow cycling of cells at the top of the lineage hierarchy would be beneficial as the stem-cell pool expands during development, but once the stem-cell compartment is fully developed stem cells would cycle quickly. an independent strategy to minimize the accumulation of mutations, whether they were incurred from errors in dna replication or not, is for cells that accrued mutations to senesce or undergo apoptosis. in the c. elegans germ line, extensive apoptosis occurs in older adults. while this apoptosis could be explained by the elimination of nurse cells or the need to reduce competition between developing germ cells, it appears that apoptosis could preferentially eliminate damaged cells in certain contexts. this idea could be further explored in the future with tools to estimate the mutational load in populations of cells before and after they have been purged of apoptotic cells. extension to other organs the pedigree-depth quasi-minimization strategy extends to other tissues. in the following, we consider three differences between the c. elegans gonad and other self-renewing organs that are relevant to pedigree-depth quasi-minimization. first, a difference with many vertebrate organs is speed of development. while small developmental delays are expected to have a strong, deleterious effect on fitness in an organism with a short life cycle and a boom–bust lifestyle such as c. elegans, they are likely to have a smaller impact on organisms with a longer life cycle. such organisms are thus expected to favor low mutation accumulation over high speed of development at least to some extent, since pedigree-depth quasi-minimization will come at a lessened cost. notably, however, it has been proposed that the development of mouse intestinal crypts is designed to minimize the time to formation of a mature crypt. this strongly suggests that the tradeoff we have investigated between mutation minimization and speed of development is of broad relevance to animals other than c. elegans. second, a large difference lies in the number of cells to be produced over an individual’s lifetime – with a c. elegans gonadal arm producing ~ cells and a human testis or hematopoietic system over and, respectively. because of these differences, the pedigree-depth quasi-minimization strategy predicts that stem cells in vertebrates should have a slower cycling speed relative to their differentiating descendants than in c. elegans. while in many contexts the contribution of various stem-cell populations remains to be established, the presence of sporadically cycling “reserve” populations is consistent with this idea. the pedigree-depth quasi-minimization strategy similarly predicts a negative correlation between stem-cell cycling speed and number of cells to be produced over a lifetime; this correlation holds true when comparing hematopoiesis in a number of mammalian species. third, different organs may have different optimal distributions of mutations in the cells that they produce. in the context of somatic tissues, an important expected benefit of mutation minimization is reduction of cancer frequency. since multiple “hits” are thought to be required for malignant transformation, it might be advantageous for a tissue to minimize the number of cells that carry two or more mutations, even if that came at the cost of an overall increase in mutation frequency. but mutator mutations likely play a significant role in tumorigenesis, and control of stem-cell lineage might be better used to minimize the frequency at which the first mutator mutation occurs, since the carcinogenic effects of such a mutation might be difficult to counteract. in the context of the germ line, the performance objective assumed in the present study was minimization of the average number of mutations in progeny. the mutation frequency in c. elegans is low, suggesting that the problem of multiple mutations per progeny might not be of practical relevance – quantification of mutation distributions in progeny from old hermaphrodites could confirm this or provide data to guide modifications to the performance objective. overall, the pedigree-depth quasi-minimization strategy is of broad relevance but would gain from being fine-tuned once the combined effects of multiple mutations carried by the same cell are better understood. control of cell cycle length to minimize pedigree depth our study identified two cell cycle phases that show substantial variation in their duration. s phase is shorter during larval development than in adulthood, and g is longer in distal cells than in proximal cells both during larval development and in adulthood. lengthening of g in preference to other cell cycle phases is consistent with mutation minimization, as replicated chromosomes offer the possibility of error-free damage repair with homologous recombination using the sister chromatid. regulation of g length has been reported in other contexts. why the s phase lengthens as well as g when germ lines transition to the adult stage is less clear. we speculate that a longer s phase could be less error prone because it allows more time for error-free repair before trans-lesion synthesis occurs; the s phase could be shorter during larval development because the benefits of faster development outweigh the costs of decreased dna replication fidelity, which is consistent with our findings and those of. what role does cyclin e play in control of cell cycle length? our data contribute two new observations that expand understanding of that role. first, we extend previous reports that cyclin e is expressed throughout the cell cycle by showing with finer quantification that cyclin e expression levels do change with cell cycle phase. interestingly, a similar finding has been made in mammalian embryonic stem cells using apc activity as a readout, extending earlier reports highlighting the lack of robust oscillations of cell cycle regulators in these cells. second, and more importantly, we show that cyclin e levels are graded along the distal–proximal axis of the c. elegans germ line in a way that is not solely dependent on changes in the lengths of cell cycle phases. this suggests that cye- could play an upstream role in controlling overall cell cycle length, which is also compatible with the complex regulation of cyclin e by the mitosis-promoting factors fbf-1/ and the meiosis-promoting factor gld- a role of cye- in regulating cell cycle length along the distal–proximal axis could appear at first sight surprising: cyclin e is better known for its role in driving g progression, but a minimal fraction of cells are in g along the distal–proximal axis – even in the very proximal mz, where cyclin e levels drop significantly – and it is g whose length is modulated along that axis. a role of cye- in regulating the length of g is possible given that cdk is known to play a role in progression through s phase and to m phase. this cdk role is thought to rely normally on complex formation with cyclin a, but continued expression of cyclin e past g in cycling mz cells could allow activity of a cyclin e/cdk complex past g although in the c. elegans germ line cye- is the cell cycle regulator whose interplay with differentiation regulators is best documented, b-type cyclins could also play an important role in control of cell cycle length as they are also potential targets of both fbf-1/ and gld-. overall, it appears that there is a complex interplay between the cell cycle machinery and regulators of differentiation. the design principle highlighted in this study provides one potential reason for the need for fine cell cycle control as cells proceed through differentiation.  to address the role of stem cells in minimizing mutation accumulation, we built models of cell cycling and mutation accrual and optimized their parameters computationally. we found that when taking into account constraints on speed of development and reproduction, c. elegans germ-line stem cells should cycle more slowly than their differentiating counterparts, but the difference should only be approximately twofold. we additionally predicted optimal mz size dimensions of × cell rows. using a new, quantitative analysis technique, we showed that our predictions were borne out experimentally. our results provide the first quantitative test of the slow stem-cell cycling strategy originally proposed by. these results strongly support the idea that mutation minimization is a relevant performance objective, and highlight an important limitation in the slow-cycling strategy. we further showed that slower stem-cell cycling could be due at least in part to fine-tuning along the distal–proximal axis of expression of the cell cycle regulator cye- consistent with the presence of a previously identified motif in the germ-line gene regulatory network whose potential significance is highlighted by our approach. 
computational simulations of mutation accumulation characterize a tradeoff between fast development and low mutation accumulation, and show that slow-cycling stem cells allow for an advantageous compromise to be reached. experimental measurements of cell cycle lengths derived using a new, quantitative technique are consistent with these predictions. stem cells are thought to play a critical role in minimizing the accumulation of mutations, but it is not clear which strategies they follow to fulfill that performance objective. slow cycling of stem cells provides a simple strategy that can minimize cell pedigree depth and thereby minimize the accumulation of replication-dependent mutations. this compromise is such that worm germ-line stem cells should cycle more slowly than their differentiating counterparts, but only by a modest amount. our findings shed light both on design principles that underlie the role of stem cells in delaying aging and on evolutionary forces that shape stem-cell gene regulatory networks. although the power of this strategy was recognized early on, a quantitative assessment of whether and how it is employed by biological systems is missing.
 stem cells are thought to play a critical role in minimizing the accumulation of mutations, but it is not clear which strategies they follow to fulfill that performance objective. slow cycling of stem cells provides a simple strategy that can minimize cell pedigree depth and thereby minimize the accumulation of replication-dependent mutations. although the power of this strategy was recognized early on, a quantitative assessment of whether and how it is employed by biological systems is missing.  here we address this problem using a simple self-renewing organ – the c. elegans gonad – whose overall organization is shared with many self-renewing organs. computational simulations of mutation accumulation characterize a tradeoff between fast development and low mutation accumulation, and show that slow-cycling stem cells allow for an advantageous compromise to be reached. this compromise is such that worm germ-line stem cells should cycle more slowly than their differentiating counterparts, but only by a modest amount. experimental measurements of cell cycle lengths derived using a new, quantitative technique are consistent with these predictions.  our findings shed light both on design principles that underlie the role of stem cells in delaying aging and on evolutionary forces that shape stem-cell gene regulatory networks. electronic supplementary material the online version of this article contains supplementary material, which is available to authorized users. issue-copyright-statement© the author 2015 