Abstract
An important step towards understanding the mechanistic basis of the central dogma is the quantitative characterization of the dynamics of nucleicacidbound molecular motors in the context of the living cell. To capture these dynamics, we develop lagtime analysis, a method for measuring in vivo dynamics. Using this approach, we provide quantitative locusspecific measurements of fork velocity, in units of kilobases per second, as well as replisome pause durations, some with the precision of seconds. The measured fork velocity is observed to be both locus and time dependent, even in wildtype cells. In this work, we quantitatively characterize known phenomena, detect brief, locusspecific pauses at ribosomal DNA loci in wildtype cells, and observe temporal fork velocity oscillations in three highlydivergent bacterial species.
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Introduction
At a singlemolecule scale, all cellular processes are both highly stochastic as well as subject to a crowded cellular environment where they typically compete with a large number of potentially antagonistic processes that share the same substrate^{1,2}. In spite of these challenges, essential processes must be robust at a cellular scale to facilitate efficient cellular proliferation. Understanding how these processes are regulated to achieve robustness remains an important and outstanding biological question^{3,4,5,6,7,8,9}. However, a central challenge in investigating these questions is the quantitative characterization of the activity of enzymes in the context of the living cell. For instance, although singlemolecule assays can resolve the pausing of molecular motors on nucleicacid substrates in the context of in vitro measurements^{10,11}, performing analogous measurements in the physiologicallyrelevant environment of the cell poses a severe challenge to the existing methodologies^{12}.
In this paper, we develop an approach, lagtime analysis, that facilitates the quantitative characterization of dynamics, with resolution of seconds, in the context of the living cell. The approach exploits exponential growth as the stopwatch to capture dynamics in exponentially proliferating cellular cultures^{13} and unlike competing approaches, it can circumvent the difficulties and potential artifacts introduced by cell synchronization^{14} or fluorescent labeling. Lagtime analysis exploits the same data as markerfrequency analysis, but it directly measures the locusspecific fork velocity, in units of kilobases per second, and the duration of replisome pauses in seconds. Lagtime analysis facilitates detailed comparisons to be made, not just between different loci in a single cell, but between wildtype and mutant cells as well as between bacterial species. Unlike a recent competing analysis, no detailed stochastic models or simulations are employed^{15}. We apply this approach to analyze three model bacterial systems: Bacillus subtilis, Vibrio cholerae, and Escherichia coli. In B. subtilis, we analyze transcriptioninduced replication antagonism which is the main determinant of replisome dynamics in a set of mutants with retrograde (reverseoriented) fork motion. An analysis of V. cholerae provides evidence that fork number is an important determinant of fork velocity, but also provides clear evidence that fork velocity is time dependent. To explore this timedependence, we analyze the fork velocity in E. coli which provides strong evidence for temporally oscillating fork velocity, consistent with a recent report^{15}. Finally, we demonstrate that these oscillations are observed in all three organisms. In summary, the observed phenomena demonstrate the central importance of characterizing central dogma processes in the context of the living cell, where their activity is regulated and modulated by the cellular environment.
Results
The bacterial cell cycle
The bacterial cell cycle is divided into three periods^{16,17}: The B period is analogous to the G_{1} phase of the eukaryotic cell cycle, corresponding to the period between cell birth and replication initiation. The C period is analogous to the S phase (and early M phase) in which the genome is replicated and simultaneously and sequentially segregated^{18}. The D period is analogous to a combination of phases G_{2} and lateM, corresponding to a period of time between replication termination and cell division, including the process of septation (i.e., cytokinesis).
The demographics of cellcycle periods of exponentially growing bacterial cells were first quantitatively modeled by Cooper and Helmstetter in an influential paper^{19} and then refined by multiple authors^{20,21,22}. In the Methods Section, we generalize these models to demonstrate that markerfrequency analysis quantitatively measures the cellcycle replication dynamics. The key results are summarized below.
Lagtime analysis
Our strategy will be to use exponential growth as the stopwatch with which we resolve cellcycle dynamics. In short, cells with greater cellcycle progression (i.e., age) are depleted in the population, equivalent to an independent, exponentially proliferating species that lags newborn cells by a time equal to its age^{13} (see Fig. 1 for a schematic illustration of the approach). Lagtime analysis is the measurement of this time lag. In principle, this approach can be applied to characterize the dynamics of any biological molecules or complexes; however, for concreteness, we will focus on replication dynamics. This process is of great biological interest and nextgeneration sequencing provides a powerful tool for digital, as well as genomewide, quantitation of the number of genomic loci.
In markerfrequency experiments, the number of each sequence N(ℓ) in a steadystate, asynchronously growing population is determined by mapping nextgenerationsequencing reads to the reference genome. This marker frequency can be reinterpreted as a measurement of the lag time τ(ℓ):
where N(ℓ) is the observed number of the locus at genomic position ℓ, N_{0} is the observed number of the origin in the culture, and k_{G} is the growth rate. This relation can be understood as a consequence of the exponential growth law^{13}.
In a deterministically timed model, the measured lag time would be equal to the replication time relative to initiation. In reality, the timing of all processes in the cell cycle is stochastic. We previously showed that the measured lag time is related to the distribution of durations in single cells by the exponential mean^{13}:
where \({{\mathbb{E}}}_{t}\) is the expectation over stochastic time t with distribution t ~ p_{i}( ⋅ ).
Determination of replisomepause durations
Replisomepause durations or the lag time difference between the replication of any two loci can be computed using the difference of lag times between the two loci:
We emphasize that the observed difference in lag time is the exponential mean of the stochastic time difference, which has important consequences for slow processes.
Determination of the fork velocity
For fast processes, like singlenucleotide incorporation, the exponential mean leads to a negligible correction (see Methods); therefore, the fork velocity has a simple interpretation: it is the slope of the genomic position versus lagtime curve:
or equivalently it is the ratio of the growth rate to the logslope:
which can be directly determined from the marker frequency.
Lagtime analysis reveals V. cholerae replication dynamics
To explore the application of lagtime analysis to characterize replication dynamics, we begin our analysis in the bacterial model system Vibrio cholerae, which harbors two chromosomes: Chromosome 1 (Chr1) is 2.9 Mb and Chromosome 2 (Chr2) is 1.1 Mb. The origin of Chr1, oriC1, fires first and roughly the first half of replication is completed before the replicationinitiatorRctBbindingsite crtS is replicated, triggering Chr2 initiation at oriC2^{23,24,25}. Chr1 and Chr2 then replicate concurrently for the rest of the C period (see Fig. 2a).
To demonstrate the power of lagtime analysis, we compute the marker frequency, lag time, and fork velocities. To measure pause times and replication velocities, we generate a piecewise linear model with a resolution set by the Akaike Information Criterion (AIC). The AICoptimal model for fast growth (in LB) had 39 knots, spaced by 100 kb, generating 38 measurements of locus velocity across the two chromosomes. The replication dynamics for growth in LB is shown in Fig. 2. For tabulated velocities, see Supplementary Data 1.
The measurement of the duration of fast processes
We focus first on the duration of time between crtS replication and the initiation of oriC2. Fluorescence microscopy imaging reveals that this wait time is very short^{26}, but it is very difficult to quantify since, the precise timing of the replication of the crtS sequence is difficult to determine by fluorescence imaging; however, this is a natural application for lagtime analysis. To measure the difference in lag time between crtS replication and oriC2 replication, we use Equation (3) to compute the replication time difference from the relative copy numbers. For this analysis, we generate a piecewise linear model with knots at the crtS and oriC2 loci. The measured lag time is
a pause duration which is clearly resolved in the lagtime plot shown in Fig. 2e.
The fork velocity is locus dependent
It is qualitatively clear from the forkdistanceversustime plot (Fig. 2e) that the fork velocity is locus dependent, since the trajectory is not straight. To test this question statistically, we compare the 39knot model to the null hypothesis (constant fork velocity), which is rejected with a pvalue of p ≪ 10^{−30} and therefore the data cannot be described by a single fork velocity (see Table 1). The resulting velocity profiles are shown in Fig. 2c, f.
Bilateral symmetry supports a timedependent mechanism
Our understanding of the replication process motivated two general classes of mechanisms: (i) timedependent and (ii) locusdependent mechanisms. Timedependent mechanisms, like a dNTPlimited replication rate, affect all forks uniformly and therefore loci equidistant from the origin should have identical fork velocities:
where ℓ is the genetic position relative to the origin. In contrast, in a locusdependent mechanism, like replicationconflictinduced slowdowns, the slow regions are expected to be randomly distributed over the chromosome. In this scenario we expect to see no bilateral symmetry between arms (see Methods).
A bilateral symmetry between the arms is clearly evident in the data (the mirror symmetry about the origin in Fig. 2b, c and is manifest in the lagtime analysis as the coincidence between the left and right arm trajectories and velocities in Fig. 2e, f. To quantitate this symmetry, we divide the variance of the fork velocity into symmetric and antisymmetric contributions (see Supplementary Method 4). A timedependent mechanism would generate a f_{S} = 100% symmetric variance, whereas a locusdependent mechanism would be expected to generate equal symmetric and antisymmetric variance contributions (f_{S} = 50%). V. cholerae Chr1 and Chr2 have f_{S} = 76% symmetry, consistent with a timedependent mechanism playing a dominant but not exclusive role in determining the fork velocity (see Table 1).
The replisome pauses briefly at rDNA in B. subtilis
To explore the possibility that locusdependent mechanisms could play a dominant role in determining the fork velocity profile, we next characterized the fork dynamics in the context of replication conflicts, where the antagonism between active transcription and replication, have been reported to stall the replisome by a locusspecific mechanism^{9,27}. In B. subtilis, there are seven highly transcribed rDNA loci on the right arm and only a single locus of the left arm. Consistent with the notion of rDNAinduced pausing, the ter locus is positioned asymmetrically on the genome, at 172° rather than 180°, leading the right arm of the chromosome to be shorter than the left arm (see Fig. 3a). In spite of the difference in length, both arms terminate roughly synchronously, implying that the average fork velocity is lower on the right arm, consistent with putative fork pausing at the rDNA loci. Are these conflictinduced pauses present in wildtype cells where the replication and transcription are codirectional? We have previously reported evidence based on singlemolecule imaging that they are^{12}, but there is as of yet no other unambiguous supporting evidence.
To detect putative short pauses at the rDNA loci in wildtype B. subtilis, a lownoise dataset was essential. We therefore examined a number of different datasets, including our own, to search for a dataset with the lowest statistical and systematic noise. A markerfrequency dataset for a nearly wildtype strain growing on minimal media was identified for which the noise level was extremely low (see Supplementary Method 3B). The lagtime analysis is shown in Fig. 3b. Replication pauses should result in discrete steps in the lag time; however, no clearly defined steps are visible in the lagtime plot. The pauses are either absent or too small to be clearly visible without statistical analysis.
To achieve optimal statistical resolution, we used the AIC modelselection framework^{28,29} on four competing hypotheses: In Model 1, fork velocities are constant and equal on both arms with no pauses. In Model 2, fork velocities are constant but unequal on the left and right arms with no pauses. In Model 3, fork velocities are constant and equal on the left and right arms with equalduration pauses at each rDNA locus. In Model 4, fork velocities are constant and unequal on the left and right arms with equalduration pauses at each rDNA locus. AIC selected Model 3 (equal arm velocities with rDNA pauses) and a pause duration of:
is observed. The pause models were strongly supported over the nonpause models (ΔAIC_{23} = 4.3 and ΔAIC_{43} = 9.4). Therefore, statistical analysis supports the existence of short slowdowns (i.e., pauses) at the rDNA, even if these features are not directly observable without statistical analysis. In highernoise datasets, the statistical inference was ambiguous.
Strong, headon conflicts lead to long pauses
Although we have just demonstrated that endogenous codirectional conflicts are detected statistically, they do not lead to a clear unambiguous signature. In contrast, strong, exogenous headon conflicts in which the replisome and transcriptional machinery move in opposite directions can lead to particularly potent conflicts and even cell death^{3,4,5,6,7,8,9,30}. The ability to engineer conflicts at specific loci facilitates the use of lagtime analysis for measuring the duration of the replication pauses.
To measure the pause durations due to headon conflicts, we analyze the marker frequency from a strain, rrnIHG(inv), generated by Srivatsan and coworkers with three rDNA genes (rrnIHG) inverted so that they are transcribed in the headon orientation. Markerfrequency datasets were reported for this strain in two growth conditions: minimal supplemented with casamino acids, in which the strain grows at an intermediate growth rate, and unsupplemented minimal media, in which the strain grows at a slow growth rate^{31}. (Mutant cells cannot proliferate in rich media, presumably because the transcription conflicts are so severe^{31}.) In both slow and intermediate growth conditions, a clearly resolved step at the headon locus is observed in the markerfrequency and lagtime analysis (Fig. 3b), exactly analogous to the simulated pause (see Methods).
To determine the pause durations in the two growth conditions, we again consider a model with an unknown pause duration (at the inverted rDNA locus) and constant but unequal fork velocities on the left and right arms. The observed lagtime pauses are
for the slow and intermediate growth rates, respectively.
Although lagtime analysis reports a precise pause duration, it is important to remember that the observed lag time corresponds to the exponential mean of the stochastic state lifetime, Equation (2), including cells that arrest and therefore never complete the replication process. Equation (18) accounts for the pause generated by this arrested cell fraction. Srivatsan and coworkers report that 10% of the cells are arrested in intermediate growth, which accounts for 8.3 min of the lag time, leaving an estimated pause time of Δτ_{pause} = 1.4 ± 0.9 min for nonarrested cells, which is roughly consistent with the pause time observed in slow growth conditions.
Slow retrograde replication in B. subtilis
Are all conflictinduced slowdowns consistent with long pauses at a small number of rDNA loci? Wang et al. have previously engineered a headon strain, 257°::oriC, with less severe conflicts by moving oriC down the left arm of the chromosome to 257°^{32} (see Fig. 3a). The resulting strain has a very short left arm and a very long right arm, the first third of which is replicated in the retrograde (i.e., reverse to wildtype) orientation. This retrograde region contains only a single rDNA locus. All other regions are replicated in the antegrade (i.e., endogenous) orientation.
Consistent with the analysis of Wang et al., we position knots to divide the chromosome into three regions with three distinct slopes: an early antegrade region A1 (the short left arm) with logslope α_{A1} = 0.34 ± 0.01 Mb^{−1}, a retrograde region R with logslope α_{R} = 0.63 ± 0.01 Mb^{−1} and a late antegrade region A2 with logslope α_{A2} = 0.26 ± 0.01 Mb^{−1}, that replicates after the left arm terminates (see Fig. 3c). Due to the higher percentage of headon genes in the R region compared with the A1 region, the conflict model predicts more rapid replication in region A1 versus R. Consistent with this prediction, the ratio of replication velocities is:
revealing a strong replicationdirection dependence. The slope appears relatively constant, consistent with a model of uniformlydistributed slow regions rather than a small number of long pauses as observed in the reversal of the rDNA locus rrnIHG. Our quantitative analysis is consistent with the interpretation of Wang et al.^{32}.
Rapid late replication due to genomic asymmetry
This dataset has a striking feature that is not emphasized in previous reports. Late antegrade fork velocity is faster than early antegrade velocity:
Although this effect is weaker than the replicationdirection dependence discussed above, Equation (10), its size is still comparable. An analogous latetime speedup is seen in two other ectopic origin strains (see the Supplementary Figs. 8 and 10).
One potential hypothesis is that a locusdependent mechanism slows the fork in the A1 region relative to the A2 region; however, no velocity difference is evident in these regions in the wildtype cells (Fig. 3b). Alternatively, one could hypothesize that there is some form of communication between forks that leads to a slowdown in region A1 due to the slowdown in region R; however, no coincident slowdown is observed in rrnIHG(inv) at a position opposite the rrnIHG locus, inconsistent with this hypothesis. Another possible hypothesis is that latetime replication is always rapid; however, no significant speedup is observed in either wildtype B. subtilis (Fig. 3a) or V. cholerae cells at the end of the replication process (Fig. 3b and Fig. 2e). However, there is one extremely important difference between 257°::oriC and the wildtype strains: Due to the asymmetric positioning of the origin and replication traps at the terminus (Fig. 3a), there is only a single active replication fork as the A2 region is replicated. We therefore hypothesize that the fork velocity is inversely related to active fork number.
Fork number determines velocities in V. cholerae
To explore the effects of changes in the fork number on fork velocity, it is convenient to return to V. cholerae. In slow growth conditions, the cells start the C period with a pair of replication forks, for which the forknumber model predicts faster fork velocity, and finish the replication cycle with two pairs of forks, predicting slower fork velocity.
Although the structure of the velocity profile is more complex than predicted by the forknumber model alone, the observed fork velocity is broadly consistent with its predictions. If a mean fork velocity is computed before and after oriC2 initiates, the ratio is:
which is quantitatively consistent with the hypothesis that more forks lead to a slowdown in replication and the size of the effect is comparable to what is observed in B. subtilis, Equation (11).
A mutant V. cholerae strain has been constructed that facilitates a nontrivial test of the forknumber model: In the monochromosomal strain MCH1, Chr2 is recombined into Chr1 at the terminus of Chr1, resulting in a single monochromosome (Chr 1–2) (see Fig. 4a). Both the wildtype and MCH1 strains have essentially identical sequence content, implying the locusdependent model would predict identical replication velocities; however, all replication in MCH1 occurs with only a single set of forks whereas the wildtype strain replicates the latter half of the C period with two pairs of forks, one pair on each chromosome.
The measured fork velocities are shown in Fig. 4b and support the forknumber model: MCH1 replicates the sequences after crtS at roughly 1.6 times the fork velocity of the wildtype cells, consistent with the forknumber model. Alternatively, we can consider the same quantitation of fork velocity we considered above: The ratio of fork velocities of loci replicated before crtS to those replicated afterwards:
therefore only a very small slowdown is observed after crtS is replicated in MCH1, even though exactly the same sequences are replicated, again consistent with the forknumber model.
The fork velocity oscillates in E. coli
Although experiments in V. cholerae clearly support the forknumber model, there is significant variability that cannot be explained by this model alone. Are timedependent variations in fork velocity also observed in organisms that replicate a single chromosome? To answer this question, we worked in the gramnegative model bacterium Escherichia coli, which harbors a single 4.6 Mb chromosome. A large collection of markerfrequency datasets have already been generated for both rapid and slow growth conditions by the Rudolph lab^{33}. As with the B. subtilis markerfrequency datasets, we selected those that had the lowest statistical and systematic noise (see the Supplementary Methods 3B).
The fork velocities are shown in Fig. 5. As before, statistically significant variation is observed in the fork velocity as a function of position (see Table 1 and Supplementary Method 6). As discussed above in the context of V. cholerae, we had initially hypothesized that this variation might be a consequence of rDNA position or some other locusdependent mechanism; however, there are three arguments against this hypothesis: (i) The slowvelocity regions are not coincident with rDNA locations (Fig. 5a) or relative GC content (Supplementary Fig. 1). (ii) Consistent with the timedependent model, 84% (and 59%) of the observed variation in the fork velocity is symmetric for fast (and slow) growth. (iii) We would expect that a locusdependent model would predict slow regions that are consistent between fast and slow growth, which is not observed (see the purple arrows in Fig. 5a). We therefore conclude that the dominant mechanism for determining the fork velocity is a timedependent mechanism, consistent with our observations for V. cholerae.
Lagtime analysis is particularly informative with respect to the mechanism of variation in the fork velocity: Although there is no alignment in the velocity with respect to locus position (Fig. 5a), there is clear alignment of the fork velocity variation with respect to lag time (Fig. 5b), not only between the left and right arms of the chromosome, but between slow and fast growth conditions. The oscillations do not align with respect to locus position (Fig. 5a) since the difference in average fork velocity leads the slow and fast temporal periods to correspond to different locus positions under slow and fast growth conditions.
Forkvelocity oscillations are observed in three organisms
Temporal oscillations in the fork velocity are an unexpected phenomenon. Are these features a systematic error with a single dataset? First we note that these oscillations are present in two E. coli growth conditions (LB and minimal). This phenomenon would be on sounder footing if similar oscillations are observed in two evolutionarily distant species: the gramnegative V. cholerae and grampositive B. subtilis. If this phenomenon is observed, to what extent are the oscillations of similar character (e.g., phase, amplitude, and period)?
We compared the lagtimedependent fork velocity for all three species. In B. subtilis, we have already discussed a rDNAinduced pausing on the right arm, which could complicate the interpretation of the data. We therefore consider the fork velocity on just the left arm. For E. coli and V. cholerae, we compute the average velocity as a function of lag time between the two arms. Since the different organisms and growth conditions have different mean fork velocities, we compare the fork velocity relative to the overall mean. The results are shown in Fig. 6 and Table 2.
All three organisms show oscillations with the same qualitative features: Each fork velocity has roughly the same phase: The velocity begins high, before decaying. The relative amplitudes, roughly 0.5 peaktopeak, are all comparable with the largestamplitude oscillations observed in V. cholerae and the smallest in E. coli. When the relative velocities are compared, it is striking how much consistency there is between growth conditions in E. coli and B. subtilis. Finally, the period of oscillation is comparable but distinct in all three organisms, ranging from 10 to 15 min. The oscillation characteristics are summarized in a table in Fig. 6 and Table 2.
Discussion
The focus of this paper is on the development of lagtime analysis, which uses exponential growth as the timer to characterize replication dynamics. Previous markerfrequency analyses have often reported a logslope (e.g., refs. ^{32,34}), which is closely related to the fork velocity. What new insights does the measurement of the fork velocity offer over this closely related approach? The fork velocity approach has two important advantages: (i) The first advantage is a conceptual one. The underlying quantity of interest is velocity (or rate per base pair). This is the quantity that is measured in vitro and is relevant in a mechanistic model. In contrast, the logslope is an emergent quantity that is only relevant in the context of exponential growth. (ii) The second advantage is concrete: Although logslope measurements allow ratiometric comparisons between fork velocity at different loci in the same dataset, they cannot be used to make comparisons across datasets. Any comparison of the logslope between cells with different growth rate (e.g., due to changes in growth conditions, mutations, species, etc.) are meaningless. For instance, the logslopes of the wildtype and MCH1 V. cholerae strains are very different even though the changes in the fork velocity are small. Our wideranging comparisons between growth conditions, mutants, and organisms demonstrate the power of reporting fork velocity over the logslope.
Although our focus has been on replication in bacterial cells, an important question is to what extent our approach could be adapted to eukaryotic cells. First, we emphasize that the lagtime analysis is directly applicable without modification to the eukaryotic context. As such, the timing of the replication of loci can be analyzed; however, since the S phase is typically a smaller fraction of the cell cycle and the genomes of eukaryotic cells are larger, deeper sequencing will be required to achieve the same resolution we demonstrate in the context of bacterial cells. One significant potential refinement to this approach is the use of cell sorting (sortseq) to enrich for replicating cells which can greatly increase the signaltonoise ratio^{35,36}; however, this approach appears to lead to significant flattening near earlyfiring origins, as we have observed in other contexts (Supplementary Method 3B), and therefore increasing sequencing depth is probably the most promising approach for eukaryotic systems when quantitative characterization is a priority (see Methods Equations (22) and (23) for an estimate of resolution).
Although lagtime analysis can easily be extended to the eukaryotic context, the measurement of the fork velocity will require some care. A critical assumption in our analysis is that replication forks move unidirectionally at any particular locus, i.e., it can be either rightward or leftward moving but not both (see Supplementary Method 3I). Fork traps prevent this bidirectionality in many bacterial cells. For loci in the terminus region, although the replication timing can be determined with high precision, the bidirectionality of the fork movement prevents the measurement of fork velocities in these regions. This is a more important limitation in eukaryotic cells where the number of origins is much greater; however, if regions of the chromosome can be found where fork movement is unidirectional, e.g., sufficiently close to earlyfiring origins, fork velocity measurements could be made in eukaryotic cells. For instance, these conditions appear to be met for a significant fraction of the Saccharomyces cerevisiae genome^{36}. With significant increases in sequencing depth, we expect analogous replication phenomenology, including pausing and locus and timedependent fork velocities, will be observed in eukaryotic systems using lagtime analysis.
As we prepared this manuscript, we became aware of a competing group which also uses markerfrequency analysis to test a specific hypothesis: the fork velocity is oscillatory in E. coli^{15}, consistent with our own observations. Although our reports share some conclusions, this competing approach requires detailed models for the cell cycle and the fork velocity, along with explicit stochastic simulations. We demonstrate an approach to measure fork velocities independent of model assumptions or detailed hypotheses for the fork velocity, without the need for numerical simulation and complete with the ability to perform an explicit and tractable error analysis.
Although our initial investigations were dependent on explicit numerical simulations of stochastic models, the use of lagtime analysis not only circumvents the need to perform these numerical simulations, but demonstrates that stochastic models are equivalent to deterministic models as well as providing a framework to understand the effects of stochasticity on the growth of populations through the use of the exponential mean, Equation (2)^{13}. This significant simplification will make lagtime analysis both widely applicable as well as accessible to other investigators who lack specialized analytical skills and modeling expertise.
Our measurements of the replication velocity reveal that there are multiple important determinants that result in complex velocity profiles. Previous work had already demonstrated that increases (or decreases) in dNTP pool levels lead to concomitant decreases (or increases) in the C period duration, consistent with a dNTPlimited model of the replication velocity^{37,38,39,40}. Our data are broadly consistent with these previous results, but in a subcellular context: (i) The forknumber model, in which fork velocities decrease as the number of active forks increase, is clearly consistent with a mechanism in which the nucleotide pool levels, although highly regulated^{41}, cannot completely compensate for the increased incorporation rate associated with multiple forks. (ii) The observation of the fork velocity oscillations is also consistent with an analogous failure of the regulatory response to compensate, this time temporally. The initial fall in the fork velocity is consistent with a model in which dNTP levels initially fall as replication initiates and nucleotides begin to be incorporated into the genome. Reduction in the dNTP levels causes a regulatory response to increase dNTP synthesis by ribonucleotide reductase^{41}, but the finite response time of the network could lead to dynamic overshoot in the regulatory feedback, leading to oscillations^{42}. Ref. ^{15} has also argued that this oscillatingdNTPlevel model would lead to timedependent oscillations in the mutation rate which are consistent with the originmirrorsymmetric distribution of the mutation observed in E. coli. However, this interesting phenomenon and this hypothesized mechanism will require further investigation.
A key clue to the potential significance of the forkvelocity oscillations comes from their observation, not only in E. coli, but also in B. subtilis and V. cholerae, three highly divergent species, as well as their observation under multiple growth conditions. Although it has long been assumed that homeostatic regulation keeps key cellular metabolites in a relatively narrow range, our observations, as well as the recent reports of oscillations in other key nucleotides in bacteria (e.g., ATP in E. coli^{43}), suggest that key metabolites are in fact subject to significant temporal oscillations even in the context of steadystate logphase growth. These observations, if their ubiquity is supported by future work, may require a significant revision of our understanding of the metabolic environment of the cell.
Retrograde fork motion, where the fork moves in the opposite direction from wildtype cells, lead to the largest changes in fork velocity observed. To what extent is the observed slowdown a consequence of a few longduration pauses versus a regionwide slowdown? In regions which exclude the rDNA, the effect appears well distributed. However, it is important to note that the genomic resolution of lagtime analysis is still much too low to resolve individual transcriptional units. We anticipate that with increased sequencing depth as well as improvements in sample preparation, this approach could detect genomic structure in the fork velocity at the resolution of individual transcriptional units. Although we did analyze a number of mutants with retrograde fork movement in V. cholerae and E. coli (analysis not shown), the competing effect of increased fork number as well as the genomic instability of these strains made these experiments difficult to interpret quantitatively, since fork number and direction were both affected in these strains^{32,44}. We concluded qualitatively that retrograde replication direction appears not to play as large a role in these gramnegative bacteria as it does in grampositive B. subtilits, consistent with previous evidence^{31,32,45,46,47}. However, we expect lagtime analysis could be used to characterize even small effects of the retrograde fork orientation in morecarefully engineered strains, analogous to those that we analyzed in the context of B. subtilis^{31,32}.
Previous reports^{31,32}, including our own^{12,48,49,50,51}, had reported longduration replicationconflict induced pauses, especially in mutant strains where the orientation of rDNA^{31} or other highly transcribed genes^{48} are inverted to give rise to a headon conflict between replication and transcription. The contribution of lagtime analysis in this context is multifold: First, we provide a quantitative number in the context of the veryshortduration pauses for codirectional transcription in wildtype cells. This analysis supports a longstanding hypothesis that the right arm of the B. subtilis chromosome is shorter than the left arm to compensate for pausing at the rDNA loci that arm predominately located on this arm.
We also report quantitative measurements for the longer pauses that results from headon conflicts in mutants where highly transcribed genes are inverted. Our analysis gives us the ability to quantitatively differentiate the contributions of fork pausing and arrest in the analysis of the marker frequency, which was previously impossible. Our measurement of a timescale of minutes is consistent with our previous in vivo singlemolecule measurements in which we report transcriptiondependent disassembly of the core replisome^{12}. Could the observed forkvelocity oscillations be misinterpreted as pauses? The observed lagtime offset between the two arms (e.g., Fig. 3b) is not predicted by forkvelocity oscillations.
In this paper, we introduce a method for quantitively characterizing cellular dynamics by lagtime analysis. Although more broadly applicable, we focus our analysis on the characterization of replication dynamics using nextgeneration sequencing to quantitate DNA locus copy number genomewide. The approach has the ability to make precise, even at the resolution of seconds, measurements of time differences and pause durations, as well as the ability to quantitatively measure fork velocities in vivo in physiological units of kb s^{−1}, at genomic resolutions of roughly 100 kb. Importantly, unlike markerfrequency analysis, our approach allows direct quantitative comparisons to be made between growth conditions, mutant strains, and even different organisms. The resulting measurements of replication dynamics reveal complex phenomenology, including temporal oscillations in the fork velocity as well as evidence for multiple mechanisms that determine the fork velocity. The lagtime analysis has great potential for application beyond bacterial systems as well as the potential to significantly increase in resolution and sensitivity as sequencing depth and sample preparation improve.
Methods
Strains used in this study
Detailed information about the strains used in this study are included in Supplementary Table 1.
Introduction to markerfrequency analysis
Our focus will be on markerfrequency analysis, which measures the total number of a genetic locus in an asynchronous population. The model was generalized to predict the marker frequency N(ℓ) of a locus a genomic distance ℓ away from the origin^{20,21,22}:
where N_{0} is the number of origins, which grows exponentially in time with the rate of mass doubling of the culture, k_{G}. Since the origin is replicated first, the number of origins is always largest compared to the numbers of other loci. Quantitatively, the copy number is predicted to decay exponentially with logslope:
where k_{G} is the population growth rate and v is the fork velocity, typically expressed in units of kilobases per second. To derive this result, two critical assumptions were made: (i) the timing of the cell cycle is deterministic and (ii) the fork velocity is constant^{19,20}.
Initially, our naïve expectation was that the interplay between the significant stochasticity of the cellcycle timing with the asynchronicity of the culture would prevent markerfrequency analysis from being used as a quantitative tool for characterizing cellcycle dynamics. For instance, significant stochasticity is observed in the duration of the B period^{52} (i.e., the duration of time between cell birth and the initiation of replication). Does this stochasticity lead to a failure of the logslope relation, Equation (15)?
Stochastic simulations support the logslope relation
To explore the role of stochasticity and a locusdependent fork velocity in shaping the marker frequency, we simulated the cell cycle using a stochastic simulation. Our aim was not to perform a simulation whose mechanistic details were correct, but rather to study how strong violations of the CooperHelmstetter assumptions, in particular how stochasticity, as strong or stronger than that observed, influenced the observed marker frequency and the logslope relation, Equation (15). In short, we used a Gillespie simulation^{53} where the B period duration and the lifetime of replisome nucleotide incorporation steps are exponentially distributed, and we added regions of the genome where the incorporation rate was fast as well as a single slow step on one arm. See Fig. 7a and Supplementary Notes 1–6 for a detailed description of the model, as well as movies of the marker frequency approaching steadystate growth, starting from a singlecell progenitor (Supplementary Movies 1 and 2).
To our initial surprise, the stochasticity of the model had no effect on the predicted logslope of the locus copy number (see Fig. 7b). In spite of the stochastic duration of the B period and the locusdependence, the marker frequency still decays exponentially with the same decay length locally, i.e.,:
where k_{G} was the empirically determined growth rate in the simulation and v(ℓ) was the local fork velocity at the locus with position ℓ.
We hypothesized that this result might be a special case of choosing an exponential lifetime distribution, since this is consistent with a stochastic realization of chemical kinetics. To test this hypothesis, we simulated several different distributions, including a uniform distribution, for the duration of the B period and the stepping lifetime for the replisome (as well as simulating multifork replication). In each case, the local logslope relation held, Equation (16), even as the growth rate and fork velocities changed with the changes in the underlying simulated growth dynamics. We therefore hypothesized that Equation (16) was a universal law of cellcycle dynamics and independent of Cooper and Helmstetter’s original assumptions.
The exponentialmean duration
Motivated by this empirical evidence, we exactly computed the population demography in a class of stochastically timed cell models^{13}. In short, we showed that there is an exact correspondence between these stochastically timed models and deterministically timed models in exponential growth. The relationship between the corresponding deterministic lifetime τ_{i} of a state i and the underlying distribution p_{i} in the stochastic model is the exponential mean, Equation (2)^{13}. The exponential mean biases the mean towards short times, the growth rate k_{G} determines the strength of this bias, and the biological mechanism for this bias is due to the enrichment of young cells relative to old cells in an exponentially growing culture^{13}.
To understand the consequences of this result, we consider two special cases of this exponential mean. For processes with lifetimes short compared to the doubling time, Equation (2), can be Taylor expanded to show that the exponential mean is:
the regular arithmetic mean μ_{t} with a leadingorder correction proportional to the product of the growth rate and variance \({\sigma }_{t}^{2}\). In the context of singlenucleotide incorporation, this correction is on order onepartinamillion and therefore can be ignored. As a consequence, Equation (16), corresponding to the transitions between states with shortlifetimes, is unaffected by the stochasticity, exactly as we observed in our simulations.
Another important case to consider is the strong disorder limit, in which a small fraction of the population ϵ stochastically arrests, i.e., with lifetime ∞, while the other individuals have exponentialmean lifetime τ_{0}. Using the definition in Equation (2), it is straightforward to show that the deterministic lifetime is:
where T is the population doubling time and the second equality is an approximation for small ϵ. The exponentialmean duration is extended by the arrest, but remains finite. Therefore, an arrest of a subpopulation is indistinguishable from a longer duration pause in an exponentially proliferating population (see ref. ^{13}).
Markerfrequency demography
For a stochastic model with locusdependent fork velocity, we showed that Equations (14) and (15) generalize to
where we will call τ(ℓ) the lag time of a locus at position ℓ, which is equal to the sum of the differential lag times for each sequential step:
where δτ_{i} is the differential lag time for state i or the exponential mean of the state lifetime^{13}. In the continuum limit, it is more convenient to represent this sum as an integral:
where the fork velocity is defined: v(ℓ_{i}) ≡ 1 bp/δτ_{i}. To demonstrate that the generalized stochastic model predicts the logslope relation, Equation (16), the logslope can be derived by substituting Equation (21) into Equation (19), as was observed in the stochastic simulations, demonstrating the universality of Equation (4). We note that Wang and coworkers had previously derived an equivalent expression using the deterministic framework of the CooperHelmstetter model in the Material and Methods Section of ref. ^{31}.
Stochasticity has a minimal effect on the marker frequency
We initially had hypothesized that stochasticity should affect the marker frequency. As explained above, it is the rapidity of base incorporation that explains why stochasticity is dispensable in this context. The same argument does not apply to the B period which is comparable to the duration of the cell cycle. However, for the marker frequency, it is lagtime differences between the replication times of loci that is determinative, and therefore the lag time of the B period cancels from these lagtime differences. Although it is mostly irrelevant for understanding wildtype cell dynamics, stochasticity and an arrested subpopulation will play an important role in one phenomenon we analyze: replicationconflict induced pauses.
Time resolution
Due to the large number of reads achievable in nextgeneration sequencing, the time resolution will be high in carefully designed analyses. The number of reads is subject to counting or Poisson noise. It is therefore straightforward to estimate the experimental uncertainty in the lag time due to finite read number:
where we have used a read number inspired by the replicationconflict pausing example. This estimate suggests that under standard conditions, time measurements with an uncertainty of seconds are possible using this approach.
Forkvelocity resolution
To compute the slope in Equation (4), the logmarkerfrequency is fit to a piecewise linear function with equal spacing between knots (see Fig. 7b). There is an important tradeoff between genomic resolution (i.e., the genomic distance between knots) and fork velocity precision (i.e., the uncertainty in velocity measurement): Increasing the genomic distance between knots reduces the genomic resolution but also reduces the uncertainty in the velocity measurement. We therefore consider a series of models with increasing genomic resolution and use the Akaike Information Criterion (AIC) to select the optimal model^{28,29} (see Supplementary Methods 3J). This approach balances the desire to resolve features by increasing the genomic resolution with the loss of velocity precision.
Given a knot spacing, it is straightforward to estimate the relative error:
where n is the read depth in reads per base and Δℓ is the spacing between knots in basepairs. Therefore, for a canonical nextgenerationsequencing experiment, we can expect to achieve roughly 10% error in the fork velocity for 100 kb genomic resolution. Note that in our error analysis, we have included only the error from cell number N, not the error from the uncertainty in the cellcycle duration, which covaries between loci in a particular experiment.
Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
Data availability
The sequencing datasets generated during the current study are available from the NCBI Sequence Read Archive with the BioProject accession code PRJNA919081. The data from Galli et al.^{34} and MidgleySmith et al.^{54} are both available from the European Nucleotide Archive (ENA), with the accession codes PRJEB28538 and PRJEB25595, respectively. The digitized data from Wang et al.^{32} and Srivatsan et al.^{31} are available in the Source Data file. More detailed information about data availability is provided in Supplementary Table 2. Source data are provided with this paper.
Code availability
MATLAB scripts written for this study are available on the GitHub repository and on reasonable request.
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Acknowledgements
We would like to thank B. Traxler, A. Nourmohammad, J. Mougous, and J. Mittler for many useful conversations. We would like to thank P. Levin, J. Wang, L. Simmons, and S. Pigolotti for advice on our manuscript. We thank S. B. Peterson and A. Schaefer for help with V. cholerae. We would like to thank J. Wang, C. Possoz, F.X. Barre, and C. Rudolph for detailed conversations about their data. This work was supported by NIH grant R01GM128191, which was awarded to P.A.W. and H.M.
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D.H., A.E.J., T.W.L., H.M., and P.A.W. designed the experiments. D.H., A.E.J., and B.S.S. assembled input data and ran experiments. D.H. and P.A.W. developed the theory, wrote code, ran the model, and analyzed output data. D.H., A.E.J., H.M., and P.A.W. wrote the manuscript.
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Huang, D., Johnson, A.E., Sim, B.S. et al. The in vivo measurement of replication fork velocity and pausing by lagtime analysis. Nat Commun 14, 1762 (2023). https://doi.org/10.1038/s41467023374562
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DOI: https://doi.org/10.1038/s41467023374562
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