VBA Macro Simulation of the Model

Supplemental document 1

VBA macro simulation of P1 plasmid segregation

General scheme The program follows the fates and positions of foci along the long cell axis with all measurements made as a proportion of cell length. An initial population of 1000 two focus cells is created with the foci randomly placed between a range of probable nucleoid end positions. The deduced rules for focus pairing and separation are applied, such that foci that lie close to each other pair and separate. One of the foci is replicated, and the products separate according to the rules. Several cycles of potential pairing and separation are applied to simulate foci that lie close to each other or which pass each other closely. The second original focus is replicated and further rounds of pairing and separation are considered. All separations occur over a random distance between pre- determined limits Lmin, Lmax. Should a focus encounter the nucleoid boundary at any point, it is arrested there. Small random movements occur to each focus at each calculation cycle to simulate the movements experienced by otherwise “resting” foci in vivo. The cells are assumed to double in size and then divide. A randomly chosen half of the divided cells are discarded and the remainder re-cycled to the start of the program for the next generation. A running record of plasmid loss (cells that receive no focus due to the inequitable distribution of foci at division) is recorded. The typical output of the program at critical stages in the simulation is shown in supplemental figure 1. At the end of the preset number of generations, a sub-routine reconstructs the experimentally observed population by randomly choosing the appropriate numbers of cells with 2, 3 or 4 foci. The rare cells with one focus are reconstructed by dividing the appropriate proportion of two and three focus cells. As the rare cells with five or six foci have central nucleoid divisions, they can be regarded as pairs of cells with two or three foci each. The program therefore reconstructs these by joining randomly chosen cells with two or three foci. Histograms of the focus distributions in cells with one-to-six foci are recorded after a preset number of generations of growth. The short axis The model does not assume any particular orientation for focus separation so that short axis position will vary widely. However, the program does not consider position along the short axis directly as this does not affect plasmid loss or the distributions of the foci, which are expressed in terms of long axis position only. However, the actual proximity of two foci that are passing each other on the long axis will vary according to their relative short axis paths. To allow for this, a user variable efficiency of pairing of foci on passing is incorporated into the program. Short axis path will also affect arrest at the nucleoid end as the ends are presumed to be hemispherical. Therefore a user defined variability of nucleoid end position is set for each cell to allow both for this effect and also for the experimentally determined variability in the positions of the extreme ends of the nucleoid.

Cell growth As all measurements in the program are relative to cell length, the program does not describe growth directly. However, it is assumed that the key measurements (separation etc.) are independent of cell growth, and therefore decrease progressively relative to cell length as the cell cycle progresses. To allow this, distance functions pertaining to foci movement are factored to reduce their relative averages as the cycle progresses. This is set to occur at three points in each cell cycle in the program. Addition of further steps in the time progression were not found to have a significant effect on the results. As the probability of a given relative distance of focus movement decreases with cell length, variation in distance traveled in one cell vs. another can represent either a real difference in distance traveled or a difference in time in the cell cycle when the two events occurred. Thus a sufficient range of probability of distance traveled by a focus in a particular cell can allow both for variation in the magnitude of focus separation and a variation in the time that the separation occurred. In conjunction with the stepwise decreases in average distance traveled, this allows the program to represent pairing and separation of foci to occur at a broad variety of different stages in the cell cycle. Copies of the spreadsheet and macro are available on request. Supplemental figure 1. The distribution of foci along the cell axis as the mathematical model progresses. A. The initial random population. B. The population after pairing and resolution is considered. C. After replication of one copy and the separation of the sister copies. D. After six potential rounds of pairing and separation are considered. E. After a second copy has replicated and separation, further rounds of pairing and separation are considered. F. The seed population for the next generation after cell division has occurred. a). b). Loss per generation Minimum travel

2 . 5 2

1 . 5 no variation 1 2X variation % plasmid1 loss

0 . 5 0 . 5 Loss per generation (%)

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 2 7 2 9 Minimum travel L (relative to cell length) generation min c). d). Maximum travel Close pair separation

0 . 7 3

0 . 5 0 . 1 Loss per generation (%) Loss per generation (%)

0 0 . 1 0 . 2 0 . 3 0 . 4 0 0 . 2 0 . 4 0 . 6 0 . 8 Separation of close pairs L (relative to cell length) Maximum travel L max(relative to cell length) sep

e). Random movement f). Efficiency of pairing when passing

0 . 8 0 . 7

0 . 7 0 . 6

0 . 2 0 . 2

0 . 1 0 . 1 Loss per generation (%) Loss per generation (%)

0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3 0 0 . 2 0 . 4 0 . 6 0 . 8 1

Maximum random movement Lran (relative to cell length) Efficiency of pairing (e.o.p.)

Supplemental Figure 2. Loss rates in the simulation on changing variable parameters. a). loss per generation over the first 30 generations of the simulation run under the following standard conditions where length unit = average length of newborn cell =1.9* : Lmin = 0.2, Lmax= 0.4, Lsep =0.1, Lran = 0.1, e.o.p.= 60% , N3 =6, N4 =6. b). through f). Average loss per generation for generations 4 through 10 for otherwise standard conditions when b). Lmin is varied; c). When Lmax is varied, d). when Lsep is varied; d). when Lran is varied; d). when e.o.p. is varied. User-determined variables

1. Lnr = Range of possible positions of the nucleoid ends centered on a point 0.2 newborn cell lengths (nbl) from the cell pole .

2. Lsep = Distance apart that will ensure that two foci form a pair.

3. N3 = Maximum number of cycles of focus separation and pairing in three focus cells considered in each cell cycle.

4. N4 = Maximum number of cycles of focus separation and pairing in four focus cells considered in each cell cycle.

5. Lran = Limit of random movement to left or right of its resting position achieved by each non-separating focus in each movement operation cycle.

6. Lmax =Maximum travel of focus after separation.

7. Lmin =Minimum travel of focus after separation. 8. e.o.p =Efficiency with which to foci that pass each other will pair at the closest point of interception. The effect on plasmid loss rates of varying several of these parameters is shown in supplemental figure 2.