A mechanistic first--passage time framework for bacterial cell-division timing
AA mechanistic first–passage time framework for bacterialcell-division timing
Khem Raj Ghusinga, ∗ Cesar A. Vargas-Garc´ıa, † and Abhyudai Singh ‡ University of Delaware, Newark, Delaware, USA 19716. (Dated: October 10, 2018)
Abstract
How exponentially growing cells maintain size homeostasis is an important fundamental prob-lem. Recent single-cell studies in prokaryotes have uncovered the adder principle, where cells onaverage, add a fixed size (volume) from birth to division. Interestingly, this added volume differsconsiderably among genetically-identical newborn cells with similar sizes suggesting a stochasticcomponent in the timing of cell-division. To mechanistically explain the adder principle, we con-sider a time-keeper protein that begins to get stochastically expressed after cell birth at a rateproportional to the volume. Cell-division time is formulated as the first-passage time for proteincopy numbers to hit a fixed threshold. Consistent with data, the model predicts that while themean cell-division time decreases with increasing size of newborns, the noise in timing increaseswith size at birth. Intriguingly, our results show that the distribution of the volume added be-tween successive cell-division events is independent of the newborn cell size. This was dramaticallyseen in experimental studies, where histograms of the added volume corresponding to differentnewborn sizes collapsed on top of each other. The model provides further insights consistent withexperimental observations: the distributions of the added volume and the cell-division time whenscaled by their respective means become invariant of the growth rate. Finally, we discuss variousmodifications to the proposed model that lead to deviations from the adder principle. In summary,our simple yet elegant model explains key experimental findings and suggests a mechanism forregulating both the mean and fluctuations in cell-division timing for size control. ∗ [email protected] † [email protected] ‡ [email protected]; http://udel.edu/˜absingh/ a r X i v : . [ q - b i o . S C ] D ec ntroduction One common theme underlying life across all organisms is recurring cycles of growth ofa cell, and its subsequent division into two viable progenies. How an isogenic population ofproliferating cells maintains a narrow distribution of cell size, a property known as cell sizehomeostasis, has been a topic of research for long time [1–15]. Over the years, a number oftheories have been proposed to explain cell size control from a phenomenological viewpoint.Recent single-cell experiments have shown that several prokaryotes such as
Escherichia coli,Caulobacter crescentus, Bacillus subtilis , Pseudomonas aeruginosa , and
Desulfovibrio vul-garis Hildenborough [9, 12, 14, 16] follow what is called an adder principle. The adder modelstates that a cell, on average, adds a constant size between birth and division, irrespectiveof its size at birth [6, 7, 9, 10, 12–14, 17]. The adder mechanism has also been found to bepresent in the G1 phase of budding yeast
Saccharomyces cerevisiae [10], suggesting that itis possibly employed by a wide range of organisms.Despite a significant development in the understanding at phenomenological level, themolecular mechanisms underpinning the cell size control are not well understood [15, 18].A prevalent class of models posits that a protein acts as a time-keeper between subsequentoccurrences of an important event in the cell cycle [6, 11, 14, 19–24]. This protein is synthe-sized at a rate proportional to instantaneous volume (size) and the event of interest, whichcould either be the division itself [11, 14, 19–21] or commitment to division after a con-stant time [6, 22–24], takes place when the protein reaches a certain threshold. It has beenpreviously shown that this simple biophysical mechanism can lead to the adder principleof cell size control in the mean sense [6, 21, 24]. Interestingly, recent experimental data onbacteria has quite fascinating stochastic component as well. For instance, not only the meanbut the distribution of the volume added between two division events itself is independentof the volume at birth for a given growth condition. Also, in different growth conditions,the distributions of the added volume collapse if scaled by their respective means [12]. Itremains to be seen whether these stochastic traits can be produced by the aforementionedmolecular mechanism.A plausible source of the stochasticity is the noise in gene expression wherein randomnessin transcription and translation results in significant cell to cell variation in protein levels[25–29]. Here, we consider a time-keeper protein between two consecutive division events2nd show that its probabilistic expression can indeed manifest as the stochastic componentin cell size. An overview of the mechanism and it leading to added volume distributionindependent of the initial cell volume are depicted in Fig. 1. Considering a cell of givenvolume at birth, we assume that its volume grows exponentially until the time at which thetime-keeper protein triggers the division. The production of the protein is started right afterthe birth of a cell. The production rate of the protein scales with the volume, i.e., increasesexponentially (Fig. 1(a)). Cell division time is modeled as the first-passage time for proteincopy numbers to attain a certain threshold (Fig. 1(b)). The distribution of the first-passagetime is then used to compute the distribution of the volume added between birth to division(Fig. 1(c),(d)). Our results further show that distributions of the added volume, and celldivision time have scale-invariant forms: they collapse upon rescaling with their respectivemeans in different growth conditions. Lastly, we discuss implications of these findings inidentification of the time-keeper protein. We also deliberate upon various modifications tothe proposed model that result in deviations from the adder principle.
Model description
Considering a cell with volume at birth V , its volume at a time t after birth is given by V ( t ) = V exp( αt ) , (1)where α represents cell growth rate. As shown in Fig. 1(a), production of a protein isstarted right after the birth of a cell. A stochastic gene expression model describing thetime evolution of this protein is described below.Let x ( t ) denote the protein count at time t . Assuming constitutive transcription andmRNA half life is smaller than the cell cycle time, we make the translation burst approx-imation where each mRNA molecule degrades instantaneously after producing a burst ofprotein molecules [30–35]. Protein synthesis is given by the following model: Gene r ( t ) −−→ Gene + B i × P rotein. (2)The variable B i denotes the size of i th protein burst which, for each i ∈ { , , , · · · } , is in-dependently drawn from a positive-valued distribution. It essentially represents the numberof protein molecules synthesized in a single mRNA lifetime and typically follows a geometric3istribution [31, 33, 35–38]. Furthermore, the production rate r ( t ) of the protein or alterna-tively the burst arrival rate (transcription rate) is assumed to be volume dependent. Morespecifically, we consider r ( t ) as r ( t ) = k m V ( t ) = k m V e αt , (3)where k m is a proportionality constant. In this formulation, the production rate scaling withthe cell volume is an essential component of maintaining concentration homeostasis. Indeedsuch a dependency of transcription rate on cell volume has been observed in mammaliancells [39].Note that the protein count x ( t ) = n (cid:88) i = i B i , (cid:104) B i (cid:105) := b, (4)is a sum of independent and identically distributed random variables B i ’s, where n is thenumber of bursts arrived (transcription events) in time interval [0 , t ]. The key assumptionis that the cell divides when the protein level x ( t ) crosses a threshold X . We would like tomention that though the model described here is for gene expression, it can be used to modelany process involving accumulation of molecules wherein the production rate scales with thevolume. A general distribution for B i allows a wider range of processes to be covered. Thescope can be further widened by considering the parameters k m , b , and X to be functionsof the growth rate α . In the next section, we quantify the cell division time by modeling itas the first time taken by x ( t ) to cross X . Cell division time as a first-passage time problem
The first-passage time (
F P T ) for the stochastic process x ( t ) to cross a threshold X ismathematically defined as: F P T := inf { t : x ( t ) ≥ X | x (0) = 0 } . (5)For the stochastic gene expression model described in the previous section, time evolution of x ( t ) and corresponding first-passage times for cells with different initial volumes is depictedin Fig. 1(b). To characterize the distribution of F P T , we need to quantify two quantities:arrival time T n of a n th burst, and minimum number of burst (transcription) events N x ( t ) to cross X . The conditional probability density function of F P T for agiven cell volume at birth V is then given as f F P T | V ( t ) = ∞ (cid:88) n =1 f T n ( t ) f N ( n ) , (6)where f T n ( t ) represents the probability density function of T n , and f N ( n ) represents theprobability mass function of N .Since x ( t ) can only increase in time, the distribution of N can be determined by thedistribution of the burst size B using the relation: P rob ( N ≤ n ) = P rob (cid:32) n (cid:88) i =1 B i ≥ X (cid:33) . (7)As an specific yet physiologically relevant example, when the burst size distribution is con-sidered to be geometric, f N ( n ) is given by f N ( n ) = (cid:18) n + X − n − (cid:19) (cid:18) b + 1 (cid:19) n − (cid:18) bb + 1 (cid:19) X , (8)where b represents the mean (or expected value) of burst size B i [40, 41]. Furthermore,the probability density function of n th arrival event f T n is governed by the underlying burstarrival process. In our case, the transcription (burst arrival) rate is time dependent. There-fore, the burst arrival process is an inhomogeneous Poisson process for which the probabilitydensity function of the arrival times are available in literature (see [42, 43]). In particular,we have f T n ( t ) = ( R ( t )) n − ( m − r ( t ) exp( − R ( t )) . (9)The transcription or burst arrival rate r ( t ) is referred to as the intensity function of thecorresponding inhomogeneous Poisson process. Also, R ( t ) is the mean value function of theinhomogeneous poisson process R ( t ) := (cid:90) t r ( s ) ds = k m V α (cid:0) e αt − (cid:1) . (10)Using (9) in (6), the expression for the probability density function of F P T for a cell ofgiven volume at birth becomes f F P T | V ( t ) = ∞ (cid:88) n =1 ( R ( t )) n − ( n − r ( t ) exp( − R ( t )) f N ( n ) . (11)5he conditional F P T distribution in (11) qualitatively emulates the experimental observa-tions in [12] that the mean cell division time decreases as the cell size at birth is increased(refer to SI, section S1). This is intuitively expected as a cell with large size a birth will havea higher transcription rate as compared to a cell with small size. Hence, on average, thetime taken by the protein to reach the prescribed threshold is smaller in the larger cell. Themodel also predicts that the noise (quantified using coefficient of variation squared, CV )in the cell division time increases as new born cell volume is increased (see Fig. 2 (left)).This prediction is consistent with data from [12], as shown on the right part of Fig. 2 . Thenoise behavior can be explained by observing that on average a cell with smaller volume atbirth takes more time for division. Therefore the fluctuations are time averaged, leading toa smaller noise in division time as compared to that of a cell with larger volume at birth. Distribution of the volume added between divisions
In the previous section, we determined the distribution of the division time for a cell ofgiven initial volume V . Coupling this with the fact that volume of the cell grows exponen-tially in time, we can find the distribution of the volume added to V . More precisely, thevolume added from birth to division (denoted by ∆ V ) is given as∆ V = V (cid:0) e αF P T − (cid:1) . (12)The distribution F P T given in (11) can be used to find the distribution of ∆ V (see SI,section S2). Let f ∆ V ( v ) denote the probability density function of ∆ V , then we have f ∆ V ( v ) = ∞ (cid:88) n =1 (cid:0) k m vα (cid:1) n − ( n − k m α exp (cid:18) − k m vα (cid:19) f N ( n ). (13)One striking observation is that f ∆ V ( v ) is independent of the initial volume V (see Fig 1for a depiction of this). This is in agreement with the experimental observations that thedistribution of the added volume does not depend on the size of the cell at birth [7, 12].Further, we can also use the expression in (13) to find moments of ∆ V . We will first discussthe expression of mean ∆ V , particularly emphasizing its dependence on the growth rate.The higher order moments are taken up next.6 ean of added volume The distribution of ∆ V given in (13) is an Erlang distribution conditioned on N , theminimum number of burst events required for the protein to cross the threshold. Theformula for (cid:104) ∆ V (cid:105) can be written as (cid:104) ∆ V (cid:105) = αk m (cid:104) N (cid:105) = αk m (cid:18) Xb + 1 (cid:19) , (14)where we have assumed that protein is produced in geometric bursts with mean burst size b (see SI, sections S1, S2). It can be seen that the if we consider the parameters k m , b , and X to be independent of the growth rate α , average volume added is a linearly increasingfunction of α . This is consistent with the analysis and experimental data on Pseudomonasaeruginosa [14]. In contrast, other studies have mentioned that there is an exponentialrelationship between (cid:104) ∆ V (cid:105) and α instead [9, 10, 12]. This nebulous aspect of the data hasbeen attributed to narrow range of achievable growth rates which makes it difficult to discerna linear dependency from an exponential one [14]. While suitable forms of k m , b , and X as functions of α can generate a desired growth rate dependency of the added volume, wediscuss a modification in the model to see an exponential relationship between them for thecase when k m , b , and X are constants.Let us consider that instead of accounting for the time between birth to division, theprotein accounts for time between other two events in the cell cycle. More specifically, weconsider initiation of DNA replication takes place when sufficient time-keeper protein hasbeen accumulated per origin of replication [6, 22, 24]. The corresponding division event isassumed to occur with a constant delay of T after an initiation. Here the delay T is whatis called C + D period whereby C represents the time to replicate the DNA and D denotesthe time between DNA replication and division [44, 45]. As shown in [24] (section 3.2),the volume added between two consecutive initiation events for each origin of replication issame as ∆ V in our previous model. Further, the average volume added between divisionsassociated associated with these initiations (denoted by ∆ V ∗ ) is related with ∆ V as (cid:104) ∆ V ∗ (cid:105) ≈ (cid:104) ∆ V (cid:105) e αT = αk m (cid:18) Xb + 1 (cid:19) e αT . (15)When k m , b , and X are constants, the expression in (15) suggests two different regimesof how ∆ V div depends upon α . For small values of α , we have α exp( αT ) ≈ α . Therefore7he mean added volume increases linearly with the growth rate. For large enough valuesof α , the exponential term starts dominating which leads to exponential dependence of theaverage volume added with change in the growth rate. This also means that if the growthrate α is small, it is not possible to distinguish whether the underlying mechanism accountsfor volume added between two division events or two initiation events as the data will showlinear dependence of the average added volume with changes in α [14]. Notice that a pureexponential relationship between ∆ V ∗ and α can also be obtained by having k m a linearlyincreasing function of α . For this particular case, the volume accounted for by each originof replication ∆ V will become invariant of the growth rate as suggested in [24, 46].To sum up, the case when the protein accounts for volume added between two divisionsgives a linear dependency of the mean added volume on growth rate. However, an exponen-tial dependence between these quantities can be achieved by having the protein account forvolume added between two initiation events. We now go back to discussing the higher ordermoments of the division to division model. Higher order moments of added volume and scale invariance of distributions
We can use the distribution of ∆ V to get its higher order statistics (see SI, section S2).In particular when the protein production is considered in geometric bursts, the coefficientof variation squared ( CV V ) and skewness ( skew ∆ V ) are given by CV V = b + 2 bX + X ( b + X ) , skew ∆ V = 2 ( b + 3 b X + 3 bX + X )( b + 2 bX + X ) / . (16)These formulas show that both CV and skewness do not depend on the growth rate α . Itturns out an even more general property is true: an appropriately scaled j th order momentof ∆ V , i.e., (cid:104) ∆ V j (cid:105) / (cid:104) ∆ V (cid:105) j is independent α . This arises from the fact that the distributionof ∆ V can be written in the following form f ∆ V ( v ) = 1 (cid:104) ∆ V (cid:105) g (cid:18) v (cid:104) ∆ V (cid:105) (cid:19) , (17)regardless of the distribution of the burst size B . An important implication of above formof distribution of ∆ V is the scale invariance property: the shape of the distribution acrossdifferent growth rates is essentially same, and a single parameter (cid:104) ∆ V (cid:105) is sufficient to char-acterize the distribution of ∆ V [47]. Recent experimental data have also exhibited the scaleinvariance property [12, 21, 48]. 8nterestingly, the above invariance property is not limited to the distribution of the addedvolume ∆ V . Ignoring the partitioning errors in the volume, it can be seen that in steady-state the cell-size distribution at birth is approximately same as the distribution of ∆ V [12].Also, the size at division is 2∆ V . Thus, the scale invariance of ∆ V immediately impliesscale invariance of the distributions of cell sizes at birth and division [12]. Moreover, thedistribution of the division time can also be determined by unconditioning (11) with respectto the distribution of the initial volume V . As shown in the SI (section S3), the distributionof the division time also has the scale invariance property which is in agreement with theresults in [7, 49]. Discussion
In this work, we studied a molecular mechanism that can lead to adder principle of cellsize control. This mechanism imagines a protein sensing the volume added between birth todivision [11, 14, 19–21] or two other events in the cell cycle [6, 22–24]. Our work shows thatthis mechanism can exhibit the stochastic traits observed in the data [12]. In particular, itis shown that the distribution of volume added between birth to division is independent ofthe initial cell volume [12]. Further, the distributions of key quantities such as the addedvolume, division time, volume at birth, volume at division, etc. show the scale invarianceproperty [12]. Our study also revealed that the noise in division time increases with increasein cell size at birth, which was validated from the available data from [12]. Here, we discussthe implications of these results.
Potential candidates for the time-keeper protein
Among many proteins involved in the process, prominent candidates for the time-keeperare FtsZ and DnaA. More specifically, if the constant volume addition is considered betweendivision to division, FtsZ is a potential candidate for the protein [15, 50–53]. It playsan important role in determination of the timing of cell division which is triggered uponassembly of FtsZ into a ring structure [54]. It has been proposed that the accumulationof FtsZ up to a critical level is required for cell division [50, 55]. Likewise, the proteinDnaA is known to regulate the timing of initiation of replication, thus presenting a strong9andidature for the protein if the constant volume is added between two initiation events[15, 56, 57]. In this case, initiation is thought to occur when a critical number of DnaA-ATPmolecules are available [18]. Upon initiation, these DnaA-ATP molecules get deactivated byconverting to DnaA-ADP [18, 58]. While it is not clear yet whether the production of DnaAor its conversion to DnaA-ATP is a rate limiting step in the initiation process, the modelpresented here can account for both cases as long as the conversion to DnaA-ATP happensat a volume dependent rate.We can employ the closed-form expressions for the moments of ∆ V developed in this workto investigate roles of these candidate proteins. Considering geometrically distributed burstof proteins, the expressions of mean and coefficient of variation squared ( CV ) are given by(14) and (16) respectively. Thus, increasing the threshold X or decreasing the mean burstsize of the time-keeper protein b should result in decrease in the CV of the added volume.Experimentally, the mean burst size can be altered by changing the translation rate of theproteins using techniques such as mutations in the Shine-Dalgarno sequence. Changingthe threshold can be achieved by changing the protein sequence which affects its functionand thus leads to a different number of protein molecules being required for division. It isimportant to point out cell-size control can possibly have mechanisms in place to overrulesuch tweaking. One possible way to overcome this could be to appropriately change thetranscription rate scaling factor k m by promoter mutations, along with changes in b or X such that the added volume is same in the mean-sense.We also acknowledge that a complex process like cell division may have a lot more goingon than a simple protein carrying out the size and time control. For instance, there is someevidence of DnaA not being solely responsible for the timing of initiation [59], a cell com-pensating for a larger or smaller initiation time by adjusting the genome replication time C [55, 60, 61], etc. Along the same lines, FtsZ ring formation is inhibited upon DNA damagewhich suggests that a viable copy of DNA is required for division to proceed [62, 63]. It ap-pears that several key proteins follow the dynamics of the hypothetical protein we consideredand at important stages, check points are established for proper coordination. Nonetheless,our model can shed light into how perturbations in expression of these proteins can lead toexperimentally observable changes. This provides exciting avenues for investigating differentcandidate proteins by examining the effect of alterations in gene expression parameters.10 ther sources of noise The source of noise accounted for in this work is the intrinstic noise arising becauseof random birth events of mRNA/protein molecules, and death of mRNA molecules. Inprinciple, there are other sources of noise such as cell-to-cell variation in cell specific factorssuch as enzyme levels which could affect the expression of the time-keeper protein and, inturn, influence the distributions of division time, cell size, etc. It is also possible that noisearising out of other factors dominates the noise from stochastic expression.One important parameter in our model is the event threshold X which we have assumedto be fixed. It is possible that instead of a strict requirement of exactly X molecules, thedivision event has an increased propensity as the protein count x ( t ) gets closer to X . Ouranalysis in SI, section 4 shows that in order to get (cid:104) ∆ V (cid:105) independent of the cell volume atbirth V , the propensity of division requires a strict attainment of X molecules. Thus, thealternate mechanism can be ruled out.Recall our discussion in prevision section that CV V decreases as the threshold X isincreased. Thus for a very large threshold, the contribution from expression of the proteinis negligible. To get a CV of ∼
20% without other factors being counted in, we need athreshold of about 20 molecules. Interestingly, the number of DnaA-ATP molecules requiredfor initiation are around 20 [18]. The threshold for FtsZ, however, is thought to be somewherebetween 4000 molecules [64] to 15000 molecules [65]. Therefore the stochastic expressionof protein suffices to account for noise in ∆ V if the initiation to initiation mechanism viaDnaA is the key regulator of cell cycle. However, it becomes inevitable that other sourcesof noise are also considered in the model if the regulation is from division to division viaFtsZ. One possibility is to consider the cell-to-cell variations in the growth rate. Likewise,in the case when the protein accounts for the volume added between two initiation events,partitioning errors can be introduced upon division of a cell. Accounting for these factorswould provide a better insight into the process. Deviations from the adder principle
Recently it has been proposed that cells employ a generalized version of the adder prin-ciple wherein the volume added between divisions depends upon the cell volume at birth1113, 66]. An important implication of this is on the time a cell will take to converge tosteady-state value. For example if the added volume decreases with V then a large cell willconverge faster than it would have in a perfect adder strategy.There could be several ways to get deviations from the adder principle. For instance, if weconsider that the time-keeper protein does not degrade fully upon division and the remainingproteins are divided in the daughter cells in proportion to their respective volumes at birth,the added volume decreases as volume of daughter cell is increased. This is because if thereare already time-keeper proteins present in the cell at the time of its birth, the thresholdwill be achieved earlier than the case when there were no proteins at birth. As a result,the added volume will be smaller as compared to an adder principle. Another possibleway of getting such deviation could be if the mean burst size is an increasing function ofthe cell volume. It could also be explained by similar reasoning that it leads to a smallertime to reach the copy number threshold of the protein. In contrast, to get a higher addedvolume for a increase in V , we can curb the scaling of protein accumulation with the cellvolume. One example is to assume a transcription rate of the form r ( t ) = k m V ( t ) V ( t )+ V wherethe transcription rate saturates with increase in volume. Alternatively, this effect could alsobe achieved by considering that instead of cell division occurring upon achieving a constantvolume addition, its propensity increases as the added volume increases. Summary
This paper shows that the stochastic accumulation of a time-keeper protein can lead toadder principle of cell size control. We derived analytical formulas for the division timeand volume added between birth to division. These expressions were used to show thatthe volume added is independent of the cell size at birth, consistent with experimentaldata. Furthermore, the distributions of added volume and division time also show scaleinvariance property wherein the distribution can be uniquely determined by its mean in agiven growth condition. We also discussed the implications of these results in identifyinga possible molecular mechanism underlying the cell-size control. Finally, we discussed howthe proposed mechanism can be modified to get more general behaviors. Future work willinvolve accounting for other sources of noise such as growth rate fluctuations, partitioningerrors, etc. 12
UPPLEMENTARY INFORMATIONS1. REMARKS ON THE CONDITIONAL DISTRIBUTION OF
F P T
GIVENCELL VOLUME AT BIRTH
As discussed in (11) in the main text, the probability density function of the first-passagetime given cell volume at birth V is given by f F P T | V ( t ) = ∞ (cid:88) n =1 ( R ( t )) n − ( n − r ( t ) exp( − R ( t )) f N ( n ) . (S1.1)Here, f N ( n ) represents the probability mass function of N (the minimum number of tran-scription events required for protein level to cross a threshold X ). The relation in (7) canbe used to quantify the distribution N from the distribution of protein burst size B i . Wepresent the form of distribution of N for two relevant cases here: when protein burst size isone with probability one, and when the protein burst size is geometric [31, 33, 35–38].When the burst size B i is one with probability one, exactly X events are required for theprotein level x ( t ) to reach X for the first time. That is, we have f N ( n ) = δ ( n − X ) , (S1.2)where δ ( n − X ) is the Kronecker’s delta which is one when n = X and zero otherwise.For the case where the burst size B i follows a geometric distribution [31, 33, 35–38], thecalculation of the minimum number of transcription events N for this distribution has beenpreviously done in our works [40, 41]. The probability mass function of N is given by f N ( n ) = (cid:18) n + X − n − (cid:19) (cid:18) b + 1 (cid:19) n − (cid:18) bb + 1 (cid:19) X . (S1.3)Here b represents the mean protein burst size. Further, the first three statistical momentsof N given by the above probability mass function are (cid:104) N (cid:105) = Xb + 1 , (S1.4) (cid:10) N (cid:11) = b + 3 bX + X + X b , (S1.5) (cid:10) N (cid:11) = b + 7 b X + 6 bX ( X + 1) + X ( X + 3 X + 2) b . (S1.6)13 ean FPT given newborn size The expression of
F P T probability density function in (S1.1) can be used to determinethe mean numerically. As we mentioned in the main text that, consistent with experiments,the mean
F P T decreases with increase in the volume of the cell at birth ( V ). That is, alarge cell on average divides earlier than a small cell. This is depicted in Fig. S1.1. S2. DISTRIBUTION OF ∆ V Let us assume that the initial volume of a (newborn) cell is V . Our primary assumptionis that the cell divides at the first-passage time we computed in the previous section. Rep-resenting the volume added to the cell’s volume at birth until the event takes place by ∆ V ,we have ∆ V = V (cid:0) e αF P T − (cid:1) . (S2.1)We already have the distribution of F P T in (S1.1). We use it to determine the distributionof ∆ V as follows. P rob { ∆ V ≤ v } = P rob (cid:8) V (cid:0) e αF P T − (cid:1) ≤ v (cid:9) (S2.2)= P rob (cid:26)
F P T ≤ α ln (cid:18) vV + 1 (cid:19)(cid:27) (S2.3)= (cid:90) α ln (cid:16) vV +1 (cid:17) f F P T | V ( t ) dt. (S2.4)Therefore the probability density function of ∆ V is given by f ∆ V ( v ) = ddv ( P rob { ∆ V ≤ v } ) (S2.5)= ddv (cid:90) α ln (cid:16) vV +1 (cid:17) f F P T ( t ) dt (S2.6)= f F P T | V (cid:18) α ln (cid:18) vV + 1 (cid:19)(cid:19) ddv (cid:18) α ln (cid:18) vV + 1 (cid:19)(cid:19) . (S2.7)Note that ddv (cid:18) α ln (cid:18) vV + 1 (cid:19)(cid:19) = 1 α ( V + v ) . Also R (cid:18) α ln (cid:18) vV + 1 (cid:19)(cid:19) = k m vα , r (cid:18) α ln (cid:18) vV + 1 (cid:19)(cid:19) = k m ( v + V ) . V as f ∆ V ( v ) = 1 α ( V + v ) ∞ (cid:88) n =1 (cid:0) k m vα (cid:1) n − ( n − k m ( v + V ) exp (cid:18) − k m vα (cid:19) f N ( n ) (S2.8)= ∞ (cid:88) n =1 (cid:0) k m vα (cid:1) n − ( n − k m α exp (cid:18) − k m vα (cid:19) f N ( n ) . (S2.9)Notice that this distribution is an Erlang distribution conditioned to the random variable N . Moments of ∆ V a. Mean ∆ V Since the distribution of ∆ V is conditional Erlang, we have the followingexpression for mean ∆ V . (cid:104) ∆ V (cid:105) = ∞ (cid:88) n =1 nk m /α f N ( n ) = αk m (cid:104) N (cid:105) . (S2.10) b. Second order moment The second order moment of ∆ V is given by (cid:10) ∆ V (cid:11) = ∞ (cid:88) n =1 n + n ( k m /α ) f N ( n ) = α k m (cid:0)(cid:10) N (cid:11) + (cid:104) N (cid:105) (cid:1) . (S2.11) c. Calculation of third order moment The third order moment of ∆ V is given by (cid:10) ∆ V (cid:11) = ∞ (cid:88) n =1 n + 3 n + 2 n ( k m /α ) f N ( n ) = α k m (cid:0)(cid:10) N (cid:11) + 3 (cid:10) N (cid:11) + 2 (cid:104) N (cid:105) (cid:1) . (S2.12)When the burst size is one with probability one, we have N = X with probability one.The formulas of mean, CV and skewness of ∆ V simplify to (cid:104) ∆ V (cid:105) = αXk m , CV V = 1 X , skew ∆ V = 2 √ X . (S2.13)When the burst size is geometric, we get the following expressions: (cid:104) ∆ V (cid:105) = αk m (cid:18) Xb + 1 (cid:19) , (S2.14) CV V = var (∆ V ) (cid:104) ∆ V (cid:105) = b + 2 bX + X ( b + X ) , (S2.15) skew (∆ V ) = 2 ( b + 3 b X + 3 bX + X )( b + 2 bX + X ) / . (S2.16)We also note that that the skewness of ∆ V is positive in both cases considered abovewhich is consistent with previous results [17].15 cale invariance of the distribution of ∆ V It has been shown in [12] that the distributions of the added volume ∆ V in differentgrowth conditions collapse when rescaled by respective (cid:104) ∆ V (cid:105) . Mathematically, we want toshow that the probability density function f ∆ V ( v ) has the following form [12, SupplementaryInformation equation 36]: f ∆ V ( v ) = 1 (cid:104) ∆ V (cid:105) g (cid:18) v (cid:104) ∆ V (cid:105) (cid:19) , (S2.17)where g ( . ) is an arbitrary normalized function. For the distribution in (S2.9), let us considerthe following function for gg ( w ) = ∞ (cid:88) n =1 ( w (cid:104) N (cid:105) ) n − ( n − (cid:104) N (cid:105) exp ( − w (cid:104) N (cid:105) ) f N ( n ) , (S2.18)where (cid:104) N (cid:105) is the expected value of the minimum number of transcription events requiredfor the protein to cross the threshold X and is given by (cid:104) N (cid:105) = Xb + 1. Also, it is relatedwith (cid:104) ∆ V (cid:105) as described in (S2.10).For the function g given in (S2.18), we have1 (cid:104) ∆ V (cid:105) g (cid:18) v (cid:104) ∆ V (cid:105) (cid:19) = 1 (cid:104) ∆ V (cid:105) ∞ (cid:88) n =1 (cid:16) v (cid:104) N (cid:105)(cid:104) ∆ V (cid:105) (cid:17) n − ( n − (cid:104) N (cid:105) exp (cid:18) − v (cid:104) N (cid:105)(cid:104) ∆ V (cid:105) (cid:19) f N ( n ) (S2.19)= ∞ (cid:88) n =1 (cid:0) k m vα (cid:1) n − ( n − k m α exp (cid:18) − k m vα (cid:19) f N ( n ) (S2.20)= f ∆ V ( v ) . (S2.21)This establishes the scale invariance of the distribution f ∆ V ( v ).One consequence of the scale invariance property of f ∆ V ( v ) is that the normalized mo-ments (cid:104) ∆ V j (cid:105) / (cid:104) ∆ V (cid:105) j are independent of the growth conditions [12, Supplementary Infor-mation]. This can be checked as follows.The j th order conditional moment of ∆ V (conditioned with respect to N ) is given by j th order moment of an Erlang distribution. Thus (cid:10) ∆ V j | N = n (cid:11) = (cid:18) αk m (cid:19) j ( n ( n + 1)( n + 2) · · · ( n + j − ⇒ (cid:10) ∆ V j (cid:11) = (cid:18) αk m (cid:19) j (cid:104) N ( N + 1)( N + 2) · · · ( N + j − (cid:105) . (S2.23)16herefore using (S2.10), we have (cid:104) ∆ V j (cid:105)(cid:104) ∆ V (cid:105) j = (cid:104) ( N ( N + 1)( N + 2) · · · ( N + j − (cid:105)(cid:104) N (cid:105) j (S2.24)which is independent of the growth rate.This fact can be used to show that statistical measures such as noise ( CV ) and skewnessare independent of the growth rate. Take CV for instance. It is defined as CV V = (cid:104) ∆ V (cid:105)(cid:104) ∆ V (cid:105) − . (S2.25)By the scale invariance property, (cid:104) ∆ V (cid:105) / (cid:104) ∆ V (cid:105) is independent of the growth rate α . Thus,the noise CV is also independent of α . Similarly, skewness is given by skew ∆ V = (cid:10) ∆ V (cid:11) − (cid:104) ∆ V (cid:105) var (∆ V ) − (cid:104) ∆ V (cid:105) ( var (∆ V )) / (S2.26)= (cid:104) ∆ V (cid:105) (cid:104) ∆ V (cid:105) − (cid:18) (cid:104) ∆ V (cid:105) (cid:104) ∆ V (cid:105) − (cid:19) − (cid:16) (cid:104) ∆ V (cid:105)(cid:104) ∆ V (cid:105) − (cid:17) / , (S2.27)which is again independent of α by the scale invariance property. S3. DISTRIBUTION OF FPT
FPT distribution can be obtained from (Equation S1.10) by solving f F P T ( t ) = (cid:90) ∞ f F P T | V ( t | v ) f V ( v ) dv, (S3.1)where f V ( v ) is the probability distribution of cell volumes at birth. Ignoring the errors inpartitioning of volume, it can be seen that f V ( v ) ≈ f ∆ V ( v ) [12]. Using this relation, theFPT distribution is given by f F P T ( t ) = k m e αt ∞ (cid:88) n =1 ∞ (cid:88) l =1 f N ( n ) f N ( l ) (cid:0) k m α (cid:1) n + l − ( e αt − n − ( n − l − (cid:90) ∞ v n + l − exp (cid:18) − k m vα e αt (cid:19) dv (S3.2)= ∞ (cid:88) n =1 ∞ (cid:88) l =1 f N ( n ) f N ( l )( n + l − n − l − α ( e αt − n − ( e αt ) n + l − (S3.3)= ∞ (cid:88) n =1 ∞ (cid:88) l =1 f N ( n ) f N ( l )( n + l − n − l − α n − (cid:88) i =0 n − i ( − i e − αt ( l + i ) . (S3.4)17PT moments can be written as (cid:10) F P T j (cid:11) = j ! α j ∞ (cid:88) n =1 ∞ (cid:88) l =1 f N ( n ) f N ( l )( n + l − n − l − n − (cid:88) i =0 n − i ( − i ( i + l ) j +1 (S3.5)= K j α j . (S3.6)The normalized moments are (cid:104) F P T j (cid:105)(cid:104) F P T (cid:105) j = K j K . (S3.7)This implies, as shown for ∆ V , that the noise, skewness and higher order moments areindendent of growth rate. Furthermore, if we use the function g ( w ) = ∞ (cid:88) n =1 ∞ (cid:88) l =1 f N ( n ) f N ( l )( n + l − n − l − K n − (cid:88) i =0 n − i ( − i e − K w ( l + i ) , (S3.8)then FPT distribution can be written as f F P T ( t ) = 1 (cid:104) F P T (cid:105) g (cid:18) t (cid:104) F P T (cid:105) (cid:19) . (S3.9)Thus, the FPT distribution is also scale invariant. In [12] constant K is equal to log 2. Wefound that K ≈ . ≈ log 2 for different distributions of burst size B and several values ofthreshold X . S4. THEORETICAL CONSTRAINTS ON THE MOLECULAR MECHANISMUNDERLYING ADDER PRINCIPLE
Let us consider a phenomenological description of the adder principle in terms of a hybridsystem as shown in Fig. S4.1. Assuming the volume of a cell at birth as V , the added volume∆ V follows a deterministic dynamics as ˙∆ V = α ( V + ∆ V ). The hazard rate for divisionevent is assumed to be some general function h (∆ V ). The adder principle states that thedivision occurs, on average, when a constant volume has been added, irrespective of theinitial cell volume V . 18sing the Dynkin’s formula [67], the expected value of the added volume (cid:104) ∆ V (cid:105) follows ddt (cid:104) ∆ V (cid:105) = (cid:68) α (∆ V + V ) − h (∆ V ) ∆ V (cid:69) (S4.1) ≈ α ( (cid:104) ∆ V (cid:105) + V ) − h (∆ V ) (cid:104) ∆ V (cid:105) . (S4.2)We have made the mean-field approximation in the second equation above. In steady state,a solution which has (cid:104) ∆ V (cid:105) independent of V is only possible if the hazard rate h (∆ V ) isa dirac delta function h (∆ V ) = δ (cid:0) ∆ V − ∆ V (cid:1) . Therefore any mechanism which activelysenses the added volume has to trigger the division the moment it reaches the prescribedthreshold. The time-keeper protein based mechanism proposed in this work satisfied thisnecessary condition, and since it produced the added model in distribution sense as well, itshows that this theoretical constraint is both necessary and sufficient. S5. ADDITIONAL INFORMATION ON FIG. 2 IN THE MAIN TEXT
To study the effect of volume at birth on the noise in generation time ( CV F P T ) , weselected the rescaled mean and standard deviation of the generation time given the smallestand the largest volumes at birth from Fig. 2D in [12]. We assumed that the generation timehas a normal distribution with mean and standard deviation given by the volume at birth(small and large). We drew 150 generation time samples for each volume at birth from theassumed distribution (150 is the number of samples per point used to generate plot in Fig.2D). Using bootstrapping we tested if the CV of the small cells is larger than the CV oflarge cells. The p value obtained is 0 . Lower bound Median Upper boundSmall volume 0.0424 0.0545 0.0686Large volume 0.0796 0.1030 0.1313
TABLE I: CV of timing increases with increase in cell size at birth . Using the data fromFig. 2D in [12], the bootstrapped values of CV with 95% CI are provided.19 CKNOWLEDGMENTS
AS is supported by the National Science Foundation Grant DMS-1312926. The authorsthank Prof. Suckjoon Jun for providing experimental data to compare with model predic-tions on noise in division time (Fig. 2).
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Monographs on Statistics andApplied Probability . Chapman & Hall, London, 1993. i z e P r o t e i n c o un t A dd e d s i z e Time (minutes)
Initial size
SmallMediumLarge mm Time (minutes)
Time (minutes) (a)(b)
Rate of protein productionProtein (c)(d)
FIG. 1:
A molecular mechanism that explains adder principle . (a) A rod-shaped growingcell starts synthesizing a protein right after its birth. Th production rate of the proteinscales with the size (volume) of the cell. When the protein’s copy number attains a certainlevel, the cell divides and the protein is degraded.26IG. 1: (b)
The stochastic evolution of the protein is shown for cells of three differentsizes at birth. Each cell divides when the protein’s level achieves a specific threshold. Thedistribution of the first-passage time (generated via 1000 realizations of the process) foreach newborn cell volume is shown above the three corresponding trajectories. Thefirst-passage time distribution is dependent upon the newborn cell size; on average theprotein in a smaller cell takes more time to reach the threshold as compared to the proteinin a larger cell. (c)
The time evolution of size is shown for cells of different initial volume.The size is assumed to grow exponentially until the protein in the top figure reaches acritical threshold. The division is assumed to take place then. (d)
The size added to theinitial size when the division event takes place is shown for cells of different volume. Thedistribution of the added size is independent of the initial size of the cell, thus reproducingan adder model.
Initial size ( μ m) Initial size Small Large C V c e ll d i v i s i o n t i m e C V c e ll d i v i s i o n t i m e FIG. 2:
The noise in the cell division time increases with initial volume . Left : The noisein division time ( CV of F P T ) increases as the cell volume at birth V is increased. Usingthe expression of the first-passage time in (11), the noise ( CV ) is numerically computedfor different values of the newborn cell volume. The parameters used for the model are k m = 1 (1/min), X = 100 (molecules), α = 0 .
03 (1/min), and geometric distribution of theburst size B with mean b = 3 (molecules). Right : The prediction from the model isvalidated using the experimental data from [12]. Using bootstrapping, a statisticallysignificant (p-value=0 . F P T ) given a newborn volume is computed numerically for each value of initial cellvolume V . The protein production is assumed in geometric bursts and the parametersassumed are: k m =1 per minute, X=100 molecules, b =3 molecules, α =0.03 per minute. ΔV̇ = 𝛼 Δ𝑉 + 𝑉 ℎ Δ𝑉 Δ𝑉 → 0𝑉 → 𝑉 + Δ𝑉 /2 FIG. S4.1: Description of the cell division process as a stochastic hybrid system. Theadded volume ∆ V evolves as per a deterministic dynamics until the division event takesplace. The hazard rate for division is h (∆ V, ∆ V ). Upon division, the added volume ∆ V and the cell volume at birth V reset to 0 and ( V + ∆ V ) //