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Bacterial cell cycle classification. Application to DNA synthesis and DNA content at any cell age
Journal of Theoretical Biology 419 (2017) 8–12
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Bacterial cell cycle classification. Application to DNA synthesis and DNA content at any cell age
MARK
A R T I C L E I N F O
A BS T RAC T
Keywords: rcd Cell cycle parameters Replication rate DNA per cell Rate of growth
A proposal is presented for classifying bacterial cell cycles into twelve discrete groups. This classification translates the three temporal parameters that define a cell cycle into numbers that facilitate an algorithmic approach to analyse the replication state of a single bacterium and of a bacterial population during steady-state of exponential growth. The classification and its implementation offer easy ways to obtain the rate of DNA synthesis and the amount of DNA per cell at any age in batch cultures.
1. Introduction The model of the bacterial cell cycle proposed by Cooper and Helmstetter (1968) has been used frequently in many studies where it was very helpful in understanding the relationship between the different processes of DNA replication (initiation and elongation) and cell division. The bacterial cell cycle was defined by three temporal parameters: τ, the time needed for a newborn cell to reach cell division or, in a population, the time required for the number of cells to double; C, the time required for a chromosome to be replicated from initiation to termination; D, the time between the moment replication ends and completion of the subsequent cell division. The relationship between cell cycle discrete events revealed by this model are easy to represent graphically. In this model, one and only one initiation and termination of replication must occur per cell cycle, and these events happen at ages ai and at respectively. The Cooper-Helmstetter model enables analysis of the cell cycle via equations such as 2(C+D)/τ for the average number of replication origins per cell, C/τ for the number of overlapping replication rounds, D/τ, for the time during which the cells have two chromosomes. (See ciclon (JiménezSánchez, 2016a) for a comprehensive list of these parameters and equations). Here, I suggest the use of the integer values of these simple equations to obtain a new view of the cell cycle and a new way to analyse it mathematically in a readily computable form. This method simplifies considerably calculations of the rate of DNA synthesis and the amount of DNA per cell at any cell age. 2. Results and conclusions 2.1. Proposal for a bacterial cell cycle classification A new approach for studying the replication parameters related to the cell cycle can be developed by defining new parameters to describe every type of cell cycle. These parameters are r, c and d, where r=⌊(C+D)/τ⌋, c=⌊C/τ⌋, and d=⌊D/τ⌋. (The symbols ⌊ and ⌋ are for the floor function which means that the expression between them is rounded down to the nearest integer, e.g., ⌊2.7855⌋=2). The rationale of these parameters is based on the biological meaning of the above equations. For instance, C is the time taken to complete a replication cycle whereas c is the integer of the ratio between C to the doubling time, τ. The latter gives information that can be more useful to describe the cell cycle; for example, the cell cycles of bacteria with C=60 min differ greatly depending on whether τ=28 or τ=62 min, which is captured by the fact that c equals 2 or 0 respectively. The rcd values can also be used to determine numerically, in a cell cycle, whether the initiation step of replication goes before the termination step, that I named an it-cycle, or in the reverse order, or ti-cycle, by the equation
h = r–c –d
(1)
which yields a value of h =0 for it-cycles, and h =1 for ti-cycles. Moreover, if a growing bacterium has an rcd number of 220, it can be deduced that D < τ and C > 2τ; for example, one possible combination is τ=28, C =60 and D =19 min so that D < τ (d=⌊D/τ⌋ =⌊19/28⌋ =0), and C > 2τ (c=⌊60/28⌋=2). It can also be deduced for this culture that, in its cell cycle, initiation of replication occurs sooner than termination (h=r–c–d=0, see Eq. (1)), and the replication that terminates in that cycle (permitting the forthcoming division), was actually initiated two cycles earlier (r =⌊ (C+D)/τ⌋ = 2) or, in other words, in the grandmother. Any treatment that changes the doubling time of this culture from 28 to 62 min will change its rcd from 220 to 100 since r=⌊ (C+D)/τ⌋=⌊ (60+19)/62⌋ =1, c=⌊C/τ⌋=⌊60/62⌋=0, and d =⌊D/τ⌋ =⌊19/62⌋=0; this will have vast consequences on the replication process since rounds of replication will no longer overlap and the initiation step will occur during the same cycle as the termination step. http://dx.doi.org/10.1016/j.jtbi.2017.01.045 Received 19 December 2016Received in revised form 26 January 2017Accepted 31 January 2017 Available online 02 February 2017 0022-5193/ © 2017 Elsevier Ltd. All rights reserved.
Journal of Theoretical Biology 419 (2017) 8–12
Fig. 1. Bacterial cell cycle groups ordered by the value of their rcd identities. it/ti type of cycle, structure of a chromosome equivalent at different times: at birth, a0, at first replication event, a1, at second replication event, a2, and at cell division, d. Black dots represent new initiations of replication.
The cell cycle of Escherichia coli growing at different rates of mass doubling, replication and division processes (τ, C, D) can be fully described using the rcd parameters. Each cell cycle can be assigned to one of twelve different groups with the following values: 000, 100, 101, 110, 201, 210, 211, 220, 311, 320, 321, and 330 (Fig. 1). Thus 000 designates the cycle of all cells in which C+D is shorter than τ, and 330 designates a culture in which the chromosome that initiates its replication in that cycle will terminate it three cycles later. Other groups cannot be considered because the value of c+d must be equal or lower than r (e.g., 010 would mean that (C+D)/τ < C/τ which is impossible); this excludes cycles 010, 120, 121, 221, 222, 322 and 331; furthermore, r cannot be equal to or greater than c+d+2, therefore cycles 200, 300, and 310 are excluded; and as cycles with d =1, that is with a D period longer than τ, are very rarely reported in the literature (two strains out of the 104 mentioned by Helmstetter (1996)), no cycles with d =2 have been considered; finally, a cycle with an r equal to 4 could only be found in bacteria with very long C periods (corresponding, for instance, to a culture growing with a doubling of 70 min and a C+D period longer than 280 min), which is not easy to find in the literature. The following features emerge from this classification: 1. In seven of the cell cycle groups, initiation of replication (ai) takes place before termination occurs (at): 000, 101, 110, 211, 220, 321, and 330, 9
Journal of Theoretical Biology 419 (2017) 8–12
that I termed it-cycles. In the other five cycles groups, termination of replication occurs before initiation occurs and can be termed ti-cycles. As the order of the two replication events depends on the cell cycle group, I termed the age for the first replication event a1, and that of the second replication event a2. Therefore, period 1 goes from birth to a1, period 2 goes from a1 to a2, and period 3 goes from a2 to cell division (Fig. 1). 2. The rcd value can be used to determine numerically the it or ti order by the Eq. (1) which gives a value of h=0 for it-cycles, and h=1 for ti-cycles. 3. In the five cycles that have an rcd number ending with 1, the D period is longer than τ. Consequently, all bacteria will initiate the cycle with two chromosomes, and will end the cycle containing four chromosomes. 4. In all classes, initiation of replication occurs r cycles before the one where the two sister chromosomes segregate and the cell divides. 1. All bacteria begin their cell cycle with a number of replication origins equal to 2r; this value is also the number of origins that initiates replication in each cell. 2. All but one cell cycle class shows a total or partial overlapping of “C+D” periods; the number of these overlaps is equal to r from birth to ai, and r +1 from ai to cell division. In addition to the features mentioned, this way of classifying the cell cycles facilitates mathematical analysis; e.g., all cell cycles with the same rcd are described by the same equations. 3. Rate of DNA synthesis and DNA content Assuming that all forks in a given cell cycle replicate DNA at the same rate, each fork replicates half a genome in C minutes, consequently, the rate per fork is 1/2 C genomes per minute per fork. The rate of DNA synthesis per cell at each period of a cell cycle only depends on the number of active forks (except for those mutants affected in the replication process) (See Appendice for calculation of forks per period). As I want to obtain the rate of DNA synthesis per cell, and one fork replicates just half chromosome, the rate of synthesis per period is the number of forks times the rate per fork in that period:
GP1 = 2(2r – 2d )/2C = (2r – 2d )/ C GP2 = 2(2r (2–h )–2d (1 + h ))/2C = (2r (2 – h )–2d (1 + h ))/ C GP3 = 4(2r –2d )/2C = 2(2r –2d )/ C As the number of forks is the same over one cell period, it follows that the rate of DNA synthesis per period is constant. When considering a growing batch culture, the rate per unit volume also depends on the relative proportion of cells in each period (JiménezSánchez, 2015; Powell, 1956)
GV = GP1(2–21− a1) + GP2(21− a1–21− a2) + GP3(21− a2–1) This value gives the rate of DNA synthesis per unit volume of a culture in G/min/cell; to show the rate per milliliter one has to multiply that rate by the number of cells/ml in that culture. Multiplying the number of forks per period by the length of that period in minutes, the amount of DNA synthesized during the first, second and third period respectively, are obtained:
ΔGP1 = τ a12(2r –2d )/2C = τ a1(2r –2d )/ C r
(2)
d
ΔGP2 = τ (a 2 –a1) 2 (2 (2–h )–2 (1 + h ))/2C = τ (a 2 –a1) (2r (2–h )–2d (1 + h ))/C
(3)
ΔGP3 = τ (1–a 2 ) 4 (2r –2d )/2C = τ (1–a 2 ) 2 (2r –2d )/C
(4)
With the same reasoning, one can calculate the rate of DNA synthesis per minute and per fork by dividing every ΔG divided by the time of every period and the number of forks. These results can be obtained in number of base pairs (bp) per minute and per fork by multiplying the previous results times the bacterial genome in bp. Consider as an example the bacterial culture with τ=28, C=60 and D=19, with an rcd number of 220; h=r–c–d =0 so that a1=ai and a2=at; an initiation time, relative to τ, of ai =1+ r–(C+D)/τ=0.1786, and a termination time at =1+d – D/τ=0.3214 (Jiménez-Sánchez, 2016a). The rate of DNA synthesis in G per minute and per cell will be:
GP1 = (2r –2d )/ C = (22–20)/60 = 0.0500 GP2 = (2r (2–h )–2d (1 + h ))/ C = (22(2–0)–20 (1 + 0))/60 = 0.1167 GP3 = 2(2r –2d )/ C = 2(22–20)/60 = 0.1000 The rate of chromosome replication in bp min−1 of such a culture, containing 106 cells ml−1, will depend on the rate of DNA synthesis per minute and per cell converted to bp min−1 by multiplying the value of GP1 times the genome size, which in E. coli is 4,639,221 bp. It will also depend on the fraction of cells per period:
4, 639, 221 × 106 × GV = 4, 639, 221 × 106 × [GP1(2–21− a1) + GP2(21− a1–21− a2) + GP3(21− a2–1)] = (5.4 + 9.02 + 2.79) × 1011 = 4.23 × 1011bp min−1ml−1. An application of these equations was performed by simulating the rate of DNA synthesis in cultures growing with seven different mass doubling 10
Journal of Theoretical Biology 419 (2017) 8–12
Fig. 2. Simulation of the rate of DNA synthesis per cell in bacteria growing with C periods ranging from 40 to 220 min, a D period of 20 min and mass doublings between 30 and 150 min.
times and with a wide range of C periods (Fig. 2). The simulation shows that (i) the rate of DNA synthesis per cell increases as the C period increases (i.e., as the rate per fork decreases), which is counter-intuitive, and (ii) the change in the rate of DNA synthesis per cell (i.e., the slopes of the curves) decreases as the mass doubling time increases. As C periods overlap less at longer mass doubling times, the predicted higher rate of replication per cell at these longer times (i.e., when the rate per fork is lower) suggests that the increase in DNA synthesis in the cell due to having overlapping replication cycles is greater than the decrease in synthesis due to the lower activity of the forks. 4. DNA content at any cell age The rcd number can also help calculate the DNA content at any cell age. The only attempt to determine these values is based on another algorithm (Jiménez-Sánchez, 2015) but its complexity renders it impossible to be used as part of another equation. In steady state conditions, the DNA content of a cell along the cell cycle must pass from an initial amount at birth to double that amount at division. DNA synthesis is not a continuous process during the cell cycle but is divided in three steps: the period P1 from birth to a1, P2 runs from a1 to a2, and P3 runs from a2 to division. Consequently, to determine the amount of DNA at any age, it is necessary to determine which periods a cell has been through since its birth and the amount of DNA synthesized in each period. The amount of DNA per cell that accumulates in each of these periods (ΔGp, Eqs. (2)–(4)) depends on the length of the period, on the number of forks, and on the rate of DNA synthesis per fork (see above). As the DNA content, G, doubles from birth to division, the amount of DNA synthesized in the whole cell cycle must be the same as the initial amount at birth,
G0 = ΔGP1 + ΔGP2 + ΔGP3
(5)
Ga1 = G0 + ΔGP1
(6)
Ga2 = G0 + ΔGP1 + ΔGP2
(7)
Gd = G0 + ΔGP1 + ΔGP2 + ΔGP3 = 2G0.
(8)
Using the same reasoning, the amount of DNA in a cell at any given age can be obtained by determining the time of that age relative to the time within its period. As every period has a different number of replication forks, they have a different rate of DNA synthesis, therefore the G content at a cell age requires a different equation depending on the period. There are three ages possible for a cell, ay1, ay2, and ay3, such that ay1 < a1, or a1 < ay2 < a2, or a2 < ay3, corresponding to whether the cell is in the first, second, or third period, respectively. The G content at each of these ages is:
Gay1 = G0 + τ ay1(2r –2d )/ C r
(9) d
Gay2 = Ga1 + τ (ay2 –a1) (2 (2–h ) –2 (1 + h ))/C
(10)
Gay3 = Ga2 + τ (ay3–a 2 ) 2 (2r –2d )/ C
(11)
The cells in the previous example in which τ=28, C =60, D =19, a1 =0.1786 and rcd =220, have a DNA content at birth from Eq. (5):
(where ΔGP1 = τ a1 (2r –2d )/ C = 28 × 0.1786 × (4–1)/60 = 0.25) G0 = 0.25 + 0.4667 + 1.9 = 2.6167 genomes per cell. 11
Journal of Theoretical Biology 419 (2017) 8–12
To determine the amount of DNA per cell at the age of 0.55, or minute 15.4, which is after a2 and therefore in the third period, Eq. (11) is used:
G(0.55) = G0 + ΔGP1 + ΔGP2 + τ (0.55–a 2 ) 2 (2r –2d )/ C = 2.6167 + 0.25 + 0.4667 + 28 (0.55–0.321) (23–21)/60 = 3.9733 genomes per cell. 5. Appendice 5.1. Number of forks per period Period 1. It is known that during the first period, the number of replication rounds is r, which gives the number of origins per cell of 2r that corresponds to a number of forks equal to 2 (2r – 1) for cycles with d =0 and 2 (2r – 2) for cycles with d =1 (Fig. 1). Therefore, the general equation is.
2(2r –2d ) Period 2. This period starts either with the initiation of replication if a1 = ai, or with termination of replication if a1 = at. In the first case, the number of replication origins is twice that number in previous period, 2r+1, and the number of termini is 2d; in the second case the number of termini is double, 2d+1, while that of origins does not change, 2r. The number of forks in each case will be 2r+1 – 2d, and 2r – 2d+1 respectively. One way of writing these alternatives in a single equation is by making use of the value of h, which differentiates between these two cases, since h=0 is for it-cycles and h =1 is for ti-cycles. Therefore, the number of forks will be:
(2r +1–2d ) – (2r +1–2d )h + (2r –2d +1)h = 2r (2–h ) – 2d (1 + h ) Period 3. Number of forks in this period is twice that number in period 1:
4(2r –2d ) 5.2. Proportion of cells per period Eq. (9) from Powell (1956) shows that the age distribution of bacterial cells in a culture is. ϕ(a)=2νe–νt the integral of that between two ages will give the number of cells between those two ages to be. [–2e–ν(a2–a1)] (between a1 and a2) =21–a1 – 21–a2. Consequently, the three periods will contain a proportion of cells of 2 – 21–a1, 2 1–a1 – 21-a2. and 21–a2 – 1, respectively. Note. All equations included in this work can be easily applied to any growing bacterium whose τ, C, and D are known, or any other simulated values, in the published spreadsheet (Jiménez-Sánchez, 2016b). Acknowledgements I thank Elena Guzmán and Arieh Zaritsky for critically reading the manuscript and for their effective comments, and Victor Norris for his useful comments and invaluable corrections. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References Cooper, S., Helmstetter, C.E., 1968. Chromosome replication and the division cycle of Escherichia coli B/r. J. Mol. Biol. 31, 619–644. Jiménez-Sánchez, A., 2015. Chromosome replication status and DNA content at any cell age in a bacterial cell cycle. J. Theor. Biol. 380, 585–589. http://dx.doi.org/10.1016/ j.jtbi.2015.06.008. Jiménez-Sánchez, A., 2016a. Ciclon. 〈http://goo.gl/IhQ2NF〉 Jiménez-Sánchez, A., 2016b. 〈http://goo.gl/fNGSsZ〉 Powell, E.O., 1956. Growth rate and generation time of bacteria, with special reference to continuous culture. J. Gen. Microbiol. 15, 492–511. Helmstetter, C.E., 1996. Timing of synthetic activities in the cell cycle. In: Neidhardt, F.C. (Ed.), Escherichia coli and Salmonella. Cellular and Molecular Biology, 2nd edition. ASN Press, 1627–1639.
Alfonso Jiménez-Sánchez Dpto. Genética, Universidad de Extremadura, E06071 Badajoz, Spain E-mail address: [email protected]
12
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Bacterial cell cycle classification. Application to DNA synthesis and DNA content at any cell age
MARK
A R T I C L E I N F O
A BS T RAC T
Keywords: rcd Cell cycle parameters Replication rate DNA per cell Rate of growth
A proposal is presented for classifying bacterial cell cycles into twelve discrete groups. This classification translates the three temporal parameters that define a cell cycle into numbers that facilitate an algorithmic approach to analyse the replication state of a single bacterium and of a bacterial population during steady-state of exponential growth. The classification and its implementation offer easy ways to obtain the rate of DNA synthesis and the amount of DNA per cell at any age in batch cultures.
1. Introduction The model of the bacterial cell cycle proposed by Cooper and Helmstetter (1968) has been used frequently in many studies where it was very helpful in understanding the relationship between the different processes of DNA replication (initiation and elongation) and cell division. The bacterial cell cycle was defined by three temporal parameters: τ, the time needed for a newborn cell to reach cell division or, in a population, the time required for the number of cells to double; C, the time required for a chromosome to be replicated from initiation to termination; D, the time between the moment replication ends and completion of the subsequent cell division. The relationship between cell cycle discrete events revealed by this model are easy to represent graphically. In this model, one and only one initiation and termination of replication must occur per cell cycle, and these events happen at ages ai and at respectively. The Cooper-Helmstetter model enables analysis of the cell cycle via equations such as 2(C+D)/τ for the average number of replication origins per cell, C/τ for the number of overlapping replication rounds, D/τ, for the time during which the cells have two chromosomes. (See ciclon (JiménezSánchez, 2016a) for a comprehensive list of these parameters and equations). Here, I suggest the use of the integer values of these simple equations to obtain a new view of the cell cycle and a new way to analyse it mathematically in a readily computable form. This method simplifies considerably calculations of the rate of DNA synthesis and the amount of DNA per cell at any cell age. 2. Results and conclusions 2.1. Proposal for a bacterial cell cycle classification A new approach for studying the replication parameters related to the cell cycle can be developed by defining new parameters to describe every type of cell cycle. These parameters are r, c and d, where r=⌊(C+D)/τ⌋, c=⌊C/τ⌋, and d=⌊D/τ⌋. (The symbols ⌊ and ⌋ are for the floor function which means that the expression between them is rounded down to the nearest integer, e.g., ⌊2.7855⌋=2). The rationale of these parameters is based on the biological meaning of the above equations. For instance, C is the time taken to complete a replication cycle whereas c is the integer of the ratio between C to the doubling time, τ. The latter gives information that can be more useful to describe the cell cycle; for example, the cell cycles of bacteria with C=60 min differ greatly depending on whether τ=28 or τ=62 min, which is captured by the fact that c equals 2 or 0 respectively. The rcd values can also be used to determine numerically, in a cell cycle, whether the initiation step of replication goes before the termination step, that I named an it-cycle, or in the reverse order, or ti-cycle, by the equation
h = r–c –d
(1)
which yields a value of h =0 for it-cycles, and h =1 for ti-cycles. Moreover, if a growing bacterium has an rcd number of 220, it can be deduced that D < τ and C > 2τ; for example, one possible combination is τ=28, C =60 and D =19 min so that D < τ (d=⌊D/τ⌋ =⌊19/28⌋ =0), and C > 2τ (c=⌊60/28⌋=2). It can also be deduced for this culture that, in its cell cycle, initiation of replication occurs sooner than termination (h=r–c–d=0, see Eq. (1)), and the replication that terminates in that cycle (permitting the forthcoming division), was actually initiated two cycles earlier (r =⌊ (C+D)/τ⌋ = 2) or, in other words, in the grandmother. Any treatment that changes the doubling time of this culture from 28 to 62 min will change its rcd from 220 to 100 since r=⌊ (C+D)/τ⌋=⌊ (60+19)/62⌋ =1, c=⌊C/τ⌋=⌊60/62⌋=0, and d =⌊D/τ⌋ =⌊19/62⌋=0; this will have vast consequences on the replication process since rounds of replication will no longer overlap and the initiation step will occur during the same cycle as the termination step. http://dx.doi.org/10.1016/j.jtbi.2017.01.045 Received 19 December 2016Received in revised form 26 January 2017Accepted 31 January 2017 Available online 02 February 2017 0022-5193/ © 2017 Elsevier Ltd. All rights reserved.
Journal of Theoretical Biology 419 (2017) 8–12
Fig. 1. Bacterial cell cycle groups ordered by the value of their rcd identities. it/ti type of cycle, structure of a chromosome equivalent at different times: at birth, a0, at first replication event, a1, at second replication event, a2, and at cell division, d. Black dots represent new initiations of replication.
The cell cycle of Escherichia coli growing at different rates of mass doubling, replication and division processes (τ, C, D) can be fully described using the rcd parameters. Each cell cycle can be assigned to one of twelve different groups with the following values: 000, 100, 101, 110, 201, 210, 211, 220, 311, 320, 321, and 330 (Fig. 1). Thus 000 designates the cycle of all cells in which C+D is shorter than τ, and 330 designates a culture in which the chromosome that initiates its replication in that cycle will terminate it three cycles later. Other groups cannot be considered because the value of c+d must be equal or lower than r (e.g., 010 would mean that (C+D)/τ < C/τ which is impossible); this excludes cycles 010, 120, 121, 221, 222, 322 and 331; furthermore, r cannot be equal to or greater than c+d+2, therefore cycles 200, 300, and 310 are excluded; and as cycles with d =1, that is with a D period longer than τ, are very rarely reported in the literature (two strains out of the 104 mentioned by Helmstetter (1996)), no cycles with d =2 have been considered; finally, a cycle with an r equal to 4 could only be found in bacteria with very long C periods (corresponding, for instance, to a culture growing with a doubling of 70 min and a C+D period longer than 280 min), which is not easy to find in the literature. The following features emerge from this classification: 1. In seven of the cell cycle groups, initiation of replication (ai) takes place before termination occurs (at): 000, 101, 110, 211, 220, 321, and 330, 9
Journal of Theoretical Biology 419 (2017) 8–12
that I termed it-cycles. In the other five cycles groups, termination of replication occurs before initiation occurs and can be termed ti-cycles. As the order of the two replication events depends on the cell cycle group, I termed the age for the first replication event a1, and that of the second replication event a2. Therefore, period 1 goes from birth to a1, period 2 goes from a1 to a2, and period 3 goes from a2 to cell division (Fig. 1). 2. The rcd value can be used to determine numerically the it or ti order by the Eq. (1) which gives a value of h=0 for it-cycles, and h=1 for ti-cycles. 3. In the five cycles that have an rcd number ending with 1, the D period is longer than τ. Consequently, all bacteria will initiate the cycle with two chromosomes, and will end the cycle containing four chromosomes. 4. In all classes, initiation of replication occurs r cycles before the one where the two sister chromosomes segregate and the cell divides. 1. All bacteria begin their cell cycle with a number of replication origins equal to 2r; this value is also the number of origins that initiates replication in each cell. 2. All but one cell cycle class shows a total or partial overlapping of “C+D” periods; the number of these overlaps is equal to r from birth to ai, and r +1 from ai to cell division. In addition to the features mentioned, this way of classifying the cell cycles facilitates mathematical analysis; e.g., all cell cycles with the same rcd are described by the same equations. 3. Rate of DNA synthesis and DNA content Assuming that all forks in a given cell cycle replicate DNA at the same rate, each fork replicates half a genome in C minutes, consequently, the rate per fork is 1/2 C genomes per minute per fork. The rate of DNA synthesis per cell at each period of a cell cycle only depends on the number of active forks (except for those mutants affected in the replication process) (See Appendice for calculation of forks per period). As I want to obtain the rate of DNA synthesis per cell, and one fork replicates just half chromosome, the rate of synthesis per period is the number of forks times the rate per fork in that period:
GP1 = 2(2r – 2d )/2C = (2r – 2d )/ C GP2 = 2(2r (2–h )–2d (1 + h ))/2C = (2r (2 – h )–2d (1 + h ))/ C GP3 = 4(2r –2d )/2C = 2(2r –2d )/ C As the number of forks is the same over one cell period, it follows that the rate of DNA synthesis per period is constant. When considering a growing batch culture, the rate per unit volume also depends on the relative proportion of cells in each period (JiménezSánchez, 2015; Powell, 1956)
GV = GP1(2–21− a1) + GP2(21− a1–21− a2) + GP3(21− a2–1) This value gives the rate of DNA synthesis per unit volume of a culture in G/min/cell; to show the rate per milliliter one has to multiply that rate by the number of cells/ml in that culture. Multiplying the number of forks per period by the length of that period in minutes, the amount of DNA synthesized during the first, second and third period respectively, are obtained:
ΔGP1 = τ a12(2r –2d )/2C = τ a1(2r –2d )/ C r
(2)
d
ΔGP2 = τ (a 2 –a1) 2 (2 (2–h )–2 (1 + h ))/2C = τ (a 2 –a1) (2r (2–h )–2d (1 + h ))/C
(3)
ΔGP3 = τ (1–a 2 ) 4 (2r –2d )/2C = τ (1–a 2 ) 2 (2r –2d )/C
(4)
With the same reasoning, one can calculate the rate of DNA synthesis per minute and per fork by dividing every ΔG divided by the time of every period and the number of forks. These results can be obtained in number of base pairs (bp) per minute and per fork by multiplying the previous results times the bacterial genome in bp. Consider as an example the bacterial culture with τ=28, C=60 and D=19, with an rcd number of 220; h=r–c–d =0 so that a1=ai and a2=at; an initiation time, relative to τ, of ai =1+ r–(C+D)/τ=0.1786, and a termination time at =1+d – D/τ=0.3214 (Jiménez-Sánchez, 2016a). The rate of DNA synthesis in G per minute and per cell will be:
GP1 = (2r –2d )/ C = (22–20)/60 = 0.0500 GP2 = (2r (2–h )–2d (1 + h ))/ C = (22(2–0)–20 (1 + 0))/60 = 0.1167 GP3 = 2(2r –2d )/ C = 2(22–20)/60 = 0.1000 The rate of chromosome replication in bp min−1 of such a culture, containing 106 cells ml−1, will depend on the rate of DNA synthesis per minute and per cell converted to bp min−1 by multiplying the value of GP1 times the genome size, which in E. coli is 4,639,221 bp. It will also depend on the fraction of cells per period:
4, 639, 221 × 106 × GV = 4, 639, 221 × 106 × [GP1(2–21− a1) + GP2(21− a1–21− a2) + GP3(21− a2–1)] = (5.4 + 9.02 + 2.79) × 1011 = 4.23 × 1011bp min−1ml−1. An application of these equations was performed by simulating the rate of DNA synthesis in cultures growing with seven different mass doubling 10
Journal of Theoretical Biology 419 (2017) 8–12
Fig. 2. Simulation of the rate of DNA synthesis per cell in bacteria growing with C periods ranging from 40 to 220 min, a D period of 20 min and mass doublings between 30 and 150 min.
times and with a wide range of C periods (Fig. 2). The simulation shows that (i) the rate of DNA synthesis per cell increases as the C period increases (i.e., as the rate per fork decreases), which is counter-intuitive, and (ii) the change in the rate of DNA synthesis per cell (i.e., the slopes of the curves) decreases as the mass doubling time increases. As C periods overlap less at longer mass doubling times, the predicted higher rate of replication per cell at these longer times (i.e., when the rate per fork is lower) suggests that the increase in DNA synthesis in the cell due to having overlapping replication cycles is greater than the decrease in synthesis due to the lower activity of the forks. 4. DNA content at any cell age The rcd number can also help calculate the DNA content at any cell age. The only attempt to determine these values is based on another algorithm (Jiménez-Sánchez, 2015) but its complexity renders it impossible to be used as part of another equation. In steady state conditions, the DNA content of a cell along the cell cycle must pass from an initial amount at birth to double that amount at division. DNA synthesis is not a continuous process during the cell cycle but is divided in three steps: the period P1 from birth to a1, P2 runs from a1 to a2, and P3 runs from a2 to division. Consequently, to determine the amount of DNA at any age, it is necessary to determine which periods a cell has been through since its birth and the amount of DNA synthesized in each period. The amount of DNA per cell that accumulates in each of these periods (ΔGp, Eqs. (2)–(4)) depends on the length of the period, on the number of forks, and on the rate of DNA synthesis per fork (see above). As the DNA content, G, doubles from birth to division, the amount of DNA synthesized in the whole cell cycle must be the same as the initial amount at birth,
G0 = ΔGP1 + ΔGP2 + ΔGP3
(5)
Ga1 = G0 + ΔGP1
(6)
Ga2 = G0 + ΔGP1 + ΔGP2
(7)
Gd = G0 + ΔGP1 + ΔGP2 + ΔGP3 = 2G0.
(8)
Using the same reasoning, the amount of DNA in a cell at any given age can be obtained by determining the time of that age relative to the time within its period. As every period has a different number of replication forks, they have a different rate of DNA synthesis, therefore the G content at a cell age requires a different equation depending on the period. There are three ages possible for a cell, ay1, ay2, and ay3, such that ay1 < a1, or a1 < ay2 < a2, or a2 < ay3, corresponding to whether the cell is in the first, second, or third period, respectively. The G content at each of these ages is:
Gay1 = G0 + τ ay1(2r –2d )/ C r
(9) d
Gay2 = Ga1 + τ (ay2 –a1) (2 (2–h ) –2 (1 + h ))/C
(10)
Gay3 = Ga2 + τ (ay3–a 2 ) 2 (2r –2d )/ C
(11)
The cells in the previous example in which τ=28, C =60, D =19, a1 =0.1786 and rcd =220, have a DNA content at birth from Eq. (5):
(where ΔGP1 = τ a1 (2r –2d )/ C = 28 × 0.1786 × (4–1)/60 = 0.25) G0 = 0.25 + 0.4667 + 1.9 = 2.6167 genomes per cell. 11
Journal of Theoretical Biology 419 (2017) 8–12
To determine the amount of DNA per cell at the age of 0.55, or minute 15.4, which is after a2 and therefore in the third period, Eq. (11) is used:
G(0.55) = G0 + ΔGP1 + ΔGP2 + τ (0.55–a 2 ) 2 (2r –2d )/ C = 2.6167 + 0.25 + 0.4667 + 28 (0.55–0.321) (23–21)/60 = 3.9733 genomes per cell. 5. Appendice 5.1. Number of forks per period Period 1. It is known that during the first period, the number of replication rounds is r, which gives the number of origins per cell of 2r that corresponds to a number of forks equal to 2 (2r – 1) for cycles with d =0 and 2 (2r – 2) for cycles with d =1 (Fig. 1). Therefore, the general equation is.
2(2r –2d ) Period 2. This period starts either with the initiation of replication if a1 = ai, or with termination of replication if a1 = at. In the first case, the number of replication origins is twice that number in previous period, 2r+1, and the number of termini is 2d; in the second case the number of termini is double, 2d+1, while that of origins does not change, 2r. The number of forks in each case will be 2r+1 – 2d, and 2r – 2d+1 respectively. One way of writing these alternatives in a single equation is by making use of the value of h, which differentiates between these two cases, since h=0 is for it-cycles and h =1 is for ti-cycles. Therefore, the number of forks will be:
(2r +1–2d ) – (2r +1–2d )h + (2r –2d +1)h = 2r (2–h ) – 2d (1 + h ) Period 3. Number of forks in this period is twice that number in period 1:
4(2r –2d ) 5.2. Proportion of cells per period Eq. (9) from Powell (1956) shows that the age distribution of bacterial cells in a culture is. ϕ(a)=2νe–νt the integral of that between two ages will give the number of cells between those two ages to be. [–2e–ν(a2–a1)] (between a1 and a2) =21–a1 – 21–a2. Consequently, the three periods will contain a proportion of cells of 2 – 21–a1, 2 1–a1 – 21-a2. and 21–a2 – 1, respectively. Note. All equations included in this work can be easily applied to any growing bacterium whose τ, C, and D are known, or any other simulated values, in the published spreadsheet (Jiménez-Sánchez, 2016b). Acknowledgements I thank Elena Guzmán and Arieh Zaritsky for critically reading the manuscript and for their effective comments, and Victor Norris for his useful comments and invaluable corrections. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References Cooper, S., Helmstetter, C.E., 1968. Chromosome replication and the division cycle of Escherichia coli B/r. J. Mol. Biol. 31, 619–644. Jiménez-Sánchez, A., 2015. Chromosome replication status and DNA content at any cell age in a bacterial cell cycle. J. Theor. Biol. 380, 585–589. http://dx.doi.org/10.1016/ j.jtbi.2015.06.008. Jiménez-Sánchez, A., 2016a. Ciclon. 〈http://goo.gl/IhQ2NF〉 Jiménez-Sánchez, A., 2016b. 〈http://goo.gl/fNGSsZ〉 Powell, E.O., 1956. Growth rate and generation time of bacteria, with special reference to continuous culture. J. Gen. Microbiol. 15, 492–511. Helmstetter, C.E., 1996. Timing of synthetic activities in the cell cycle. In: Neidhardt, F.C. (Ed.), Escherichia coli and Salmonella. Cellular and Molecular Biology, 2nd edition. ASN Press, 1627–1639.
Alfonso Jiménez-Sánchez Dpto. Genética, Universidad de Extremadura, E06071 Badajoz, Spain E-mail address: [email protected]
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